Karnataka 2nd PUC Question Bank for Maths

Government of Karnataka
Department of School Education
(Pre-University)

QUESTION BANK

II PUC CHAPTERWISE WEIGHTAGE FRAMEWORK
CONTENTS
TIME: 3 HOURS 35 – MATHEMATICS Max Marks: 80
UNIT Number of Marks Page
UNIT NAME
NUMBER Teaching Hours Allotted Number
1 RELATIONS AND FUNCTIONS 9 9 4 – 11
INVERSE
2 6 6 12 – 19
TRIGONOMETRIC FUNCTIONS

3 MATRICES 9 9 20 – 30

4 DETERMINANTS 12 12 31 – 36
CONTINUITY AND
5 20 17 37 – 50
DIFFERENTIABILITY

6 APPLICATION OF DERIVATIVES 10 8 51 – 62

7 INTEGRALS 22 18 63 – 81

8 APPLICATION OF INTEGRALS 5 5 82

9 DIFFERENTIAL EQUATIONS 10 8 83 – 90

10 VECTOR ALGEBRA 11 8 91 – 97

11 THREE-DIMENSIONAL GEOMETRY 8 6 98 – 102

12 LINEAR PROGRAMMING 7 6 103 – 105

13 PROBABILITY 11 8 106 – 114

TOTAL 140 120 ---

14
2025 Board Exam Question papers --- --- 115 – 123
(Three Papers)

15
Previous Year Model Question papers and --- --- 124 – 158
Practice Papers (Six Papers)

SUBJECT: 35 – MATHEMATICS Page 3 of 164

CHAPTER -01
RELATIONS AND FUNCTIONS
MCQ /FB questions.

1. A relation R in a set A, If each element of A is related to every element of A , then R is called
(A) empty relation (B) universal relation (Easy)
(C) Trivial relation (D) function
2. Both the empty relation and the universal relation are (Easy)
(A) empty relations (B) universal relations
(C) Trivial relations. (D) equivalence relations.
3. Let A be the set of all students of a boys school. Then the relation R in A given by
R = {(a, b) : a is sister of b} is (Easy)
(A) empty relation (B) transitive relation
(C) symmetric relation (D) reflexive relation
4. A relation R in the set A is called a reflexive relation, if (Easy)
(A) (a,a) ∈ R, for every a ∈ A
(B) (a,a) ∈ R, at least one a ∈A
(C) (a,b) ∈ R implies that (b, a) ∈ R, for all a, b ∈ A
(D) (a,b) and (b, c) ∈ R implies that (a, c) ∈ R, for all a, b, c ∈ A
5. A relation R in the set {1, 2, 3} given by
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1,3)}. Then R is (Average)
(A) reflexive and symmetric (B) reflexive and transitive
(C) reflexive , symmetric and transitive (D) reflexive but neither symmetric nor transitive
6. A relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is (Easy)
(A) reflexive and symmetric (B) symmetric but not transitive
(C) symmetric and transitive (D)neither symmetric nor transitive.
7. A relation R in the set {1,2,3} given that 𝑅 = {(1,2), (2,1), (1,1)} is (Average)
(A) transitive but not symmetric (B) symmetric but not transitive
(C) symmetric and transitive (D) neither symmetric nor transitive.
8. Let R be the relation in the set {1, 2, 3, 4} given by
R = {(1, 2), (2, 2), (1, 1), (4,4),(1, 3), (3, 3), (3, 2)}. Choose the correct answer. (Average)
(A) R is reflexive and symmetric but not transitive
(B) R is reflexive and transitive but not symmetric
(C) R is symmetric and transitive but not reflexive
(D) R is an equivalence relation
9. Let R be the relation in the set N given by R = {(a, b) :a = b – 2, b > 6}.
Choose the correct answer. (Average)
(A) (2, 4) ∈R (B) (3, 8) ∈R (C) (6, 8) ∈R (D) (8, 6) ∈R.
10. Consider the non-empty set consisting of children in a family and a relation R defined as
aRb if a is brother of b. Then R is (Average)
(A) symmetric but not transitive (B) transitive but not symmetric
(C) neither symmetric nor transitive (D) both symmetric and transitive.
11. If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is (Easy)
(A) reflexive (B) transitive (C) symmetric (D) none of these
12. Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and
only if l is perpendicular to m ∀l, m ∈L. Then R is (Easy)
(A) reflexive (B) symmetric (C) transitive (D) none of these.
13. Let R be the relation in the set {1, 2, 3, 4} given by R = {(2, 2), (1, 1), (4, 4), (3, 3)}. Choose the
correct answer. (Easy)
(A) R is reflexive and symmetric but not transitive
(B) R is reflexive and transitive but not symmetric
(C) R is symmetric and transitive but not reflexive
(D) R is an equivalence relation

SUBJECT: 35 – MATHEMATICS Page 4 of 164

14. Let W denote the words in the English dictionary. Define the relation R by
R = {(x, y) ∈ W × W ∶ the words x and y have at least one letter in common}. Then R is (Average)
(A) not reflexive, symmetric and transitive (B) reflexive, symmetric and not transitive
(C) reflexive, symmetric and transitive (D) reflexive, not symmetric and transitive
15. Let S = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is (Average)
(A) 1 (B) 2 (C) 3 (D) 4
16. The number of equivalence relation in the set {1, 2, 3} containing (1, 2) and (2, 1) is (Average)
(A) 5 (B) 2 (C) 4 (D) 3
17. Let S = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and
symmetric but not transitive is (Average)
(A) 1 (B) 2 (C) 3 (D) 4
18. If a relation R on the set {1, 2, 3} be defined by R = {(1, 1)}, then R is (Easy)
(A) symmetric but not transitive (B) transitive but not symmetric
(C) symmetric and transitive (D) neither symmetric nor transitive
19. Let R be a relation on the set N of natural numbers defined by nRm if n divides m.
Then R is (Average)
(A) Reflexive and symmetric (B) Transitive and symmetric
(C) Equivalence (D) Reflexive, transitive but not symmetric
20. Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as
aRb if a is congruent to b  a, b  T . Then R is (Easy)
(A) reflexive but not transitive (B) transitive but not symmetric
(C) equivalence (D) symmetric but not reflexive
21. Let A = 2,3, 4,5 & B = 36, 45, 49,60,77,90 and let R be the relation ‘is factor of’ from A to B
Then the range of R is the set (Average)
(A) 60 (B) { 36,45,60,90 } (C) 49,77 (D) 49,60,77
22. The maximum number of equivalence relation on the set A = 1, 2,3 is (Average)
(A) 1 (B) 2 (C) 3 (D) 5
23. Let us define a relation R in R as aRb if a  b . Then R is (Average)
(A) an equivalence relation (B) reflexive, transitive but not symmetric
(C) symmetric, transitive but not reflexive (D) neither transitive nor reflexive but symmetric
24. A relation R in set A = {1,2,3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the
following ordered pair in R shall be removed to make it an equivalence relation in A? (Easy)
(A) (1, 1) (B) (1, 2) (C) (2, 2) (D) (3, 3)
25. Let A = 1, 2,3 and consider the relation R = {(1, 1), (2, 2), (1, 2), (2, 3), (3, 3)}.
Then R is (Easy)
(A) reflexive but not symmetric (B) reflexive but not transitive
(C) symmetric and transitive (D) neither symmetric, nor transitive
26. If a relation R on the set {1, 2, 3} be defined by R = {(1,1),(2,2)}, then R is (Easy)
(A) symmetric but not transitive (B) transitive but not symmetric
(C) symmetric and transitive (D) neither symmetric nor transitive
27. Let f :R→ R be defined by f(x) = x 4 , x ∈R. Then (Average)
(A) f is one-one but not onto (B) f is one-one and onto
(C) f is many-one onto (D) f is neither one-one nor onto
28. Let f :R→ R be defined by f(x) = 3x, x ∈R. Then (Average)
(A) f is one-one but not onto (B) f is one-one and onto
(C) f is many-one onto (D) f is neither one-one nor onto
29. Let f : R→ R be defined by f(x) = x 3 , x ∈R. Then (Average)
(A) f is one-one but not onto (B) f is one-one and onto
(C) f is many-one onto (D) f is neither one-one nor onto
1
30. Let f :R→ R be defined by f(x) = x , x ∈R. Then f is (Easy)
(A) one-one (B) onto (C) bijective (D) f is not defined.

SUBJECT: 35 – MATHEMATICS Page 5 of 164

31. Let f: R → R defined by f(x) = 2x + 6 which is a bijective mapping then f −1 (x) is given by
(Average)
x
(A) 2 − 3 (B) 2x + 6 (C) x − 3 (D) 6x + 2
32. If the set A contains 5 elements and the set B contains 6 elements, then the number of
one-one and onto mappings from A to B is (Average)
(A) 720 (B) 120 (C) 0 (D) 30
33. Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is (Average)
(A) nP2 (B) 2n − 2 (C) 2n − 1 (D) 2n
34. If the set A contains 5 elements and the set B contains 6 elements, then the number of one-
one mappings from A to B is (Easy)
(A) 720 (B) 120 (C) 0 (D) 30
35. A contains 5 elements and the set B contains 6 elements, then the number of onto mappings
from A to B is (Easy)
(A) 720 (B) 120 (C) 0 (D) 30
36. Let N be the set of natural numbers and the function f : N → N be defined by
f ( n ) = 2n + 3  n  N . Then f is (Easy)
(A) surjective (B) Injective (C) bijective (D) Many to one
37. Which of the following functions from Z into Z are bijections? (Average)
(A) f ( x ) = x3 (B) f ( x ) = x + 2 (C) f ( x ) = 2 x + 1 (D) f ( x ) = x + 1
2

−1
38. Let f :R→ R be defined by f(x) = 3x − 4. Is invertible. Then f (x) is given by (Average)
x+4 x 4
(A) 3 (B) 3 − 4 (C) 3x + 4 (D) x +
3
39. Let S = {a, b, c} and T = {1, 2, 3} then which of the following functions f from S to T, f −1 exists.
(Easy)
(A) f = {(a, 3), (b, 2), (c, 1)} (B) f = {(a, 1), (b, 1), (c, 1)}
(C) f = {(a, 2), (b, 1), (c, 1)} (D) f = {(a, 1), (b, 2), (c, 1)}
40. Find the number of all one-one functions from set A = {1, 2, 3,4} to itself. (Easy)
(A) 8 (B) 24 (C)16 (D) 256
41. Statement 1 : A relation R = { (1,1),(1,2),(2,1) } defined on the set A = { 1,2,3 } is transitive.
Statement 2 : A relation R on the set A is transitive if ( a, b ) and ( b, c ) ϵ R, then( a, c ) ∈ R
a,b,c  A (Average)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is true
42. Let f :{1, 2, 3} → {1,2,3} is a function,
Statement 1: If f is one-one, then f must be onto.
Statement 2: If f is onto, then f must be one-one. Choose the correct answer. (Easy)
A) Statement 1 is true, and Statement 2 is false
B) Statement 1 is false, and Statement 2 is true
C) Statement 1 is true, and Statement 2 is true
D) Statement 1 is false, and Statement 2 is false
43. Statement 1: Let f :R→ R be defined by f(x) = 3x is bijective. (Average)
Statement 2: A function f :A→ B is a bijective function if f is one-one and onto
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false

SUBJECT: 35 – MATHEMATICS Page 6 of 164

44. Statement 1: Consider the set A = {1,2,3} and R be the smallest equivalence relation on A,
then R is an identity relation. (Average)
Statement 2: R is an equivalence relation, then R is reflexive, symmetric and transitive.
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false
45. Assertion (A): In set A = {1, 2, 3} a relation R defined as R = {(1, 1), (2, 2)} is reflexive.
Reason (R): A relation R is reflexive in set A if (a, a)  R for all a  A (Easy)
A) A is false and R is true B) A is true and R is true
C) A is true and R is false D) A is false and R is false
46. Assertion (A): In set A = {1, 2, 3} relation R in set A, given as R = {(1, 2)} is transitive.
Reason (R): A singleton relation is transitive. (Easy)
A) A is false and R is true B) A is false and R is false
C) A is true and R is false D) A is true and R is true
47. Assertion (A): If n (A) =3, then the number of reflexive relations on A is 3 (Easy)
Reason(R) : A relation R on the set A is reflexive if (a, a) ∈ R, ∀a∈ A.
A) A is false and R is true B) A is true and R is false
C) A is true and R is true D) A is false and R is false
48. Assertion (A): A relation R = { (a,a), (b,b), (b,c), (c,c) } defined on the set A = {a,b,c} is symmetric
Reason(R): A relation R on the set A is symmetric if (a, b) ∈ R ⟹ (b, a) ∈ R (Easy)
A) A is true and R is true B) A is false and R is true
C) A is true and R is false D) A is false and R is false
0 , if x is rational
49. Statement 1 : The function f : R→R defined as f ( x ) =  is bijective
1 ,if x is irrational
Statement 2: A function is said to be bijective if it is both one-one and onto (Difficult)
A) Statement 1 is true, and Statement 2 is false
B) Statement 1 is false, and Statement 2 is true
C) Statement 1 is true, and Statement 2 is true
D) Statement 1 is false, and Statement 2 is false
50. Statement 1 : A function 𝑓: 𝐴 → 𝐵, can not be an onto function if 𝑛 (𝐴) <𝑛 (𝐵).
Statement 2: A function 𝑓 is onto if every element of co-domain has at least one pre-image in
the domain (Easy)
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false
51. Consider the set A containing 3 elements. Then, the total number of injective functions from A
onto itself is _________ (Easy)
52. Set A has 3 elements, and set B has 4 elements. Then the number of injective mappings that
can be defined from A to B is_________ (Average)
53. Let A={1,2,3} and B={a, b}. Then the number of surjections from A into B is_______ (Average)
54. The number of equivalence relations containing (2,1) on the set A = 1, 2,3 is_______(Easy)
55. A contains 4 elements and the set B contains 5 elements, then the number of onto mappings
from A to B is______ (Easy)

SUBJECT: 35 – MATHEMATICS Page 7 of 164

56. If f: {2,8} →{-2,2,4} ,for the following figure f is (Easy)

(A) f is one-one but not onto (B) f is one-one and onto
(C) f is neither one-one nor onto (D) f is not a function

57. If f : R → R, then graph of the function is
(Average)
(A) f is one-one but not onto
(B) f is one-one and onto
(C) f is neither one-one nor onto
(D) f is onto but not one-one

58. If f : R → R, then graph of the function is
(Average)
(A) f is one-one but not onto
(B) f is one-one and onto
(C) f is neither one-one nor onto
(D) f is onto but not one-one

59. The maximum number of equivalence relations on the set A = {1, 2 } is ------ (Easy)
60. Given set A ={1, 2, 3} and a relation R = {(3, 1),(1, 3),(3,3)}, the relation R will be (Average)
(A) reflexive if (1, 1) is added (B) symmetric if (2, 3) is added
(C) transitive if (1, 1) is added (D) symmetric if (3, 2) is added.
61. Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defined by y = 2x4, is (Average)
(A) one-one and onto (B) one-one into (C) many-one onto (D) many-one into.
62. Let A be the set of all 100 students of Class XII in a college. Let f : A → N be function
defined by f (x)= roll number of the student Class XII. (Easy)
(A) f is neither one-one nor onto. (B) f is one-one but not onto
(C) f is not one-one but onto (D) f is bijective .
63. Statement 1 : If R and S are two equivalence relations on a set A, then R ∩ S is
also an Equivalence relation on A. (Difficult)
Statement 2 : The union of two equivalence relations on a set is not necessarily an equivalence
relation on the set.
Statement 3 : The inverse of an equivalence relation is an equivalence relation.
(A) All 3 Statements are true (B) 1 and 2 Statements are true but 3 false
(C) All 3 Statements are false (D) 1 and 3 Statements are true but 2 false
64. The number of bijective functions from set A to itself is 120, then A contains __________
elements. (Average)
65. Let R be an equivalence relation on a finite set A having n elements. Then, the number of
ordered pair in R is (Average)
(A) <n (B) >or= n (C) < or =n (D) >n

SUBJECT: 35 – MATHEMATICS Page 8 of 164

Two Mark Questions.
1. Define a reflexive relation and give an example of it. (Easy)
2. Define a symmetric relation and give an example of it. (Easy)
3. Define a transitive relation and give an example of it. (Easy)
4. Define an equivalence relation and give an example of it. (Easy)
5. Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}
is reflexive but neither symmetric nor transitive. (Average)
6. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)}
is symmetric but neither reflexive nor transitive. (Average)
7. Show that the function f : N→N, given by f (1) = f (2) = 1 and f (x) = x – 1,
for every x > 2, is onto but not one- one. (Average)
8. Show that an onto function f : {1, 2, 3} →{1, 2, 3} is always one-one. (Average)
9. Show that one-one function f : {1, 2, 3} →{1, 2, 3} is always onto. (Average)
10. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1, 4), (2, 5), (3, 6)} be a function from A to B.
Show that f is one-one. (Average)
11. Show that the Signum Function f : R→R, is neither one-one nor onto. (Average)
12. Let f : {2, 3, 4, 5}→{3, 4, 5, 9}and g : {3, 4, 5, 9}→{7, 11, 15} be functions defined as
f (2) = 3, f(3) = 4, f (4) = f (5) = 5 and g (3) = g (4) = 7 and g(5) = g (9) = 11. Find gof. (Average)
13. If f : R → R is defined by f(x) = 3x – 2. Show that f is one-one. (Average)
14. If f : N → N given by f ( x ) = x check whether f is one-one. Justify your answer.
2
(Average)
15. If f : Z → Z given by f ( x ) = x 2 check whether f is one-one and onto. (Average)
16. If f : N → N given by f ( x ) = x check whether f is one-one and onto.
3
(Average)
17. If f : Z → Z given by f ( x ) = x check whether f is one-one and onto.
3
(Average)
18. Show that the function f : N → N , given by f(x) = 2x is one-one but not onto. (Average)
19. Show that the function given by f(1) = f(2) = 1 and f(x) = x – 1, for every x > 2,
is onto but not one-one. (Average)
20. If f : R → R given by f ( x ) = 3x check whether f is one-one and onto (Average)
21. Prove that f : R → R given by f(x) = x3 is onto. (Average)
22. Let f :2, 3, 4, 5 → 3, 4, 5, 9 and g : 3, 4, 5, 9 → 7, 11, 15 be functions defined
f(2) = 3, f(3) = 4, f(4) = f(5) = 5 and g(3) = g(4) = 7 and g(5) = g(9) = 11. Find gof. (Average)
23. Let f :1, 3, 4 → 1, 2, 5 and g:1, 2, 5 → 1, 3 given by f = {1, 2), (3, 5), (4, 1)} and
g = {(1, 3), (2, 3), (5, 1)} write down gof. (Average)
24. Determine, with justification whether the function
f : 1, 2, 3, 4 → 10 with f = (1, 10 ) , ( 2, 10 ) , ( 3, 10 ) ,(4,10) has an inverse function? (Easy)
25. Determine, with justification whether the function g = 5, 6, 7, 8 → 1, 2, 3, 4 with
g = ( 5, 4 ) , ( 6, 3) , ( 7, 4 ) , (8, 2 ) has an inverse function? (Easy)
26. Determine, with justification whether the function h :2, 3, 4, 5 → 7, 9, 11, 13 with
h = ( 2, 7 ) , ( 3, 9 ) , ( 4, 11) , ( 5, 13) has an inverse function? (Easy)
Three Mark Questions.
1. A relation R on the set A = {1, 2, 3......14} is defined as R = {(x, y) : 3x – y =0}.
Determine whether R is reflexive, symmetric and transitive. (Difficult)
2. A relation R in the set N of natural number defined as R = {(x, y) : y = x + 5 and x < 4}.
Determine whether R is reflexive, symmetric and transitive. (Difficult)
3. A relation ‘R’ is defined on the set A = {1, 2, 3, 4, 5} as R = {(x, y) : y is divisible by x}.
Determine whether R is reflexive, symmetric, transitive. (Difficult)
4. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1}is reflexive, symmetric or transitive. (Difficult)
SUBJECT: 35 – MATHEMATICS Page 9 of 164

5. Let f : X →Y be a function. Define a relation R in X given by
R = {(a, b): f(a) = f(b)}. Examine whether R is an equivalence relation or not. (Difficult)
6. Relation R in the set Z of all integers is defined as R = {(x, y) : x – y is an integer}.
Determine whether R is reflexive, symmetric and transitive. (Difficult)
7. Determine whether R, in the set A of human beings in a town at a particular time is given by
R = {(x, y) : x and y work at the same place} (Difficult)

8. Show that the relation R in R, the set of reals defined as R = ( a,b ) : a  b is 
reflexive and transitive but not symmetric. (Average)

9. Show that the relation R on the set of real numbers R is defined by R = ( a,b ) : a  b is
2

neither reflexive nor symmetric nor transitive. (Average)
10. Check whether the relation R in R the set of real numbers defined as
R = ( a,b ) : a  b3  is reflexive, symmetric and transitive. (Average)
11. Show the relation R in the set Z of integers given by R = {(a, b) : 2 divides (a – b)} is
an equivalence relation. (Average)
12. Show the relation R in the set Z of integers given by R = {(a, b) : (a – b) is divisible by 2} is
an equivalence relation. (Average)
13. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) :|a-b| is even} is
an equivalence relation. (Average)
14. Show that the relation R on the set A of point on coordinate plane given by
R = {(P, Q) distance OP = OQ, where O is origin is an equivalence relation. (Average)
15. Show that the relation R on the set A = x  Z : 0  x  12 given by
R = {(a,b): a − b is a multiple of 4 } is an equivalence relation. (Average)
16. Show that the relation R on the set A = x  Z : 0  x  12 given by
R = {(a,b) : a=b } is an equivalence relation. (Average)
17. Let T be the set of triangles with R – a relation in T given by
R = {(T1, T2) : T1 is congruent to T2} Show that R is an equivalence relation. (Average)
18. Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is
perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive. (Average)
19. Let L be the set of all lines in the XY plane and R is the relation on L by
R = {(l1, l2) :l1 is parallel to l2}. Show that R is an equivalence relation.
Find the set of all lines related to the line y = 2x + 4. (Average)
20. Show that the relation R defined in the set A of polygons as
R = {(P1, P2) : P1 and P2 have same number of side} is an equivalence relation. (Average)
21. If R1 and R2 are two equivalence relations on a set, is R1  R 2 also an equivalence relation?
Justify your answer. (Difficult)
22. If R1 and R2 are two equivalence relations on a set, then prove that R1  R 2 is
also an equivalence relation. (Difficult)
23. Find gof and fog if f : R → R and g : R → R are given by f(x) = cos x and g(x) = 3x .
2
Show that gof  fog. (Average)
24. If f & g are functions from f : R → R defined by f ( x ) = sin x and g ( x ) = x 2 Show that
gof  fog. (Average)
25. Show that the modulus function f : R → R given by f(x) = | x | is neither one-one nor onto.
(Average)
26. Prove that the greatest integer function f : R → R given by f(x) = [x] is neither one-one
nor onto (Average)

SUBJECT: 35 – MATHEMATICS Page 10 of 164

1
27. Show that the function f :𝐑 𝟎→𝐑 𝟎, given by f (x) = x is one-one and onto, where 𝐑 𝟎 is the set of all
non-zero real numbers. Is the result true, if the domain𝐑 𝟎 is replaced by N with co-domain
being same as 𝐑 𝟎 ∗? (Average)
28. Let A and B be sets. Show that f : A × B →B × A such that f (a, b) = (b, a) is
bijective function. (Average)
29. If f : R → R given by f ( x ) = x check whether f is one-one and onto. Justify our answer.
2

(Average)
30. If f : R → R is defined by f(x) = 1 + x2, then show that f is neither 1-1 nor onto. (Average)

x
 ,x0
31. Show that the Signum function f : R → R defined by f ( x ) =  x is
0 , x = 0

neither one-one nor onto. Justify your answer. (Average)

Five Mark Questions
1. Let f : N → Y be a function defined as f (x) = 4x + 3, where, Y = {y ∈ N: y = 4x + 3 for
some x ∈ N}. Show that f is invertible. Find the inverse. (Average)
OR
Consider f : R → R given by 𝑓(𝑥) = 4𝑥 + 3. Show that 𝑓 is invertible. Find the
inverse of 𝑓.
2. Consider f : R → R given by f ( x ) = 3 − 4 x . Show that f is invertible.
Find the inverse of f. (Average)
3. Consider f : R → R given by f(x) = 10x + 7. Show that f is invertible.
Find the inverse of f. (Average)
𝑥−2
4. If 𝐴 = 𝑅 − (3) and 𝐵 = 𝑅 − {1} and 𝑓: 𝐴 → 𝐵 is a function defined by 𝑓(𝑥) = ( ) is 𝑓 one-one and
𝑥−3
onto? Justify your answer (Average)

ADDITIONAL QUESTIONS:
5. Consider 𝑓 ∶ 𝑅+ → [4, ∞) given by 𝑓(𝑥) = 𝑥 2 + 4. Show that 𝑓 is invertible with the
inverse 𝑓 −1 of 𝑓 given by 𝑓 −1 (𝑦) = √𝑦 − 4, where 𝑅+ is the set of all non-negative real
numbers. (Average)
 n +1
 2 if n is odd
6. Let f : N → N defined by f ( n ) =  State whether f is bijective.
 n if n is even
 2
Justify your answer. (Average)
 4 4x
7. Let f : R − −  → R be a function defined by define f ( x ) = .
 3 3x + 4
 4
Find the inverse of the function f : R − −  → Range of f. (Average)
 3

SUBJECT: 35 – MATHEMATICS Page 11 of 164

CHAPTER -02
INVERSE TRIGONOMETRIC FUNCTIONS
MCQ /FB questions.
1. The principal value branch of sin-1x. (Easy)
π π π π
(A) (− 2 , 2 ) (B)[0, π] (C) [− 2 , 2 ] (D)[−1,1]
2. The domain of f(x)=sin x.-1 (Easy)
(A) (−1,1 ) (B)[0, π] (C) (−∞, ∞) (D)[−1,1]
3. The principal value branch of cos-1x. (Easy)
π π
(A) [−1,1] (B)(0, π) (C) [− 2 , ] (D)[0, π]
2
4. The domain of cos-1x (Easy)
(A) (−1,1 ) (B)[0, π] (C) (0, π) (D)[−1,1]
5. The principal value branch of tan-1x (Easy)
π π
(A) (− , ) (B)(0, π) (C) [−∞, ∞] (D)(−∞, ∞)
2 2
6. The domain of tan x. -1 (Easy)
π π
(A) (− , )
2 2
(B)(0, π) (C) (−∞, ∞) (D)[−1,1]
7. The domain of cot-1x. (Easy)
π π
(A) (− 2 , 2 ) (B)(0, π) (C) (−∞, ∞) (D)[−1,1]
8. The principal value branch of cot-1x. (Easy)
π π
(A) (− 2 , )
2
(B)(0, π) (C) (−∞, ∞) (D)[0, π]
9. The range of sec −1 x is (Easy)
π π π π
(A) [− 2 , 2 ] − {0} (B)(0, π) − 2 (C) [0, π] − { 2 } (D)R − (−1,1 )
10. The principal value branch of sec-1x. (Easy)
π π π π π π
(A) (− , ) − {0} (B)(0, π) − { 2 } (C) [− 2 , 2 ] − {0} (D)[0, π] − { 2 }.
2 2
11. The principal value branch of cosec-1x. (Easy)
π π π π π π
(A) (− , ) − {0} (B)(0, π) − { 2 } (C) [− 2 , 2 ] − {0} (D)[0, π] − { 2 }.
2 2
12. The domain of sec −1 x is (Easy)
(A) (−1,1 ) (B)R − (−1,1 ) (C) R − [−1,1]. (D)R.
1
13. Principal value of sin−1 (− ) is (Easy)
2
π
(A) −
6
(B) π/3 (C) π/6 (D)-π/3 .
1
14. Principal value of cos −1 (− 2) is (Easy)
π
(A) −
6
(B) -π/3 (C) 5π/6 (D) 2π/3 .
−1
15. Principal value of cosec (−√2) is (Easy)
  3 7
(A) − (B) (C) (D)
4 4 4 4
16. Principal value of tan−1 (−1) is (Easy)
  3 5
(A) (B) − (C) (D)
4 4 4 4
−1 (1) −1 1 −1 1
17. The value of tan + cos (− 2) +sin (− ) is equal to (Average)
2
3 3  
(A) (B) (C) (D)
2 4 2 3
18. The value of tan−1 (√3) + sec −1 (−2) is equal to (Average)
2  
(A)  (B) (C) − (D)
3 3 3

SUBJECT: 35 – MATHEMATICS Page 12 of 164

1 1
19. The value of cos−1 ( ) +2sin−1 ( ) is equal to (Average)
2 2
2 3  
(A) (B) (C) (D)
3 2 2 6
−1 −1
20. The value of tan (√3) + cot (−√3) is equal to (Average)
 7
(A)  (B) (C) 0 (D)
6 6
π 1
21. sin ( − sin (− )) is equal to (Easy)
3 2
1 1 1
(A) (B) (C) (D) 1
2 3 4
7π
22. The principal value of cos−1 (cos ( 6 )) is (Average)
7π 5π π π
(A) 6 (B) 6 (C) 6 (D) 3
2π
23. The principal value of sin−1 (sin ( 3 )) is (Easy)
2π π π √3
(A) 3 (B) -3 (C) 3 (D) .
2
3π
24. The principal value of sin−1 (sin ( 5 )) is (Average)
3π π 2π 4π
(A) 5 (B) -5 (C) 5 (D) .
5
−1
25. sin ( tan x), | x| < 1 is equal to (Average)
√1−x2 x 1 x
(A) x (B) (C) 1+x2
(D) .
√1−x2 √1+x2
π
26. The value of x, if sin−1(1 − x) − 2 sin−1 x = 2 is (Average)
1 1 1
(A) 0, (B) 1, (C) 0 (D) .
2 2 2
27. If sin−1 x = y then (Easy)
π π π π
(A) 0 ≤ y ≤ π ( B) − ≤ y ≤ (C) − 1 ≤ y ≤ 1 (D) − < y <
2 2 2 2
−1 1
28. The value of cot ( ) , |x| > 1 (Average)
√x2 −1
(A)cot −1 x (B) tan x −1 (C) sec −1
x (D)cosec −1 𝑥
−1 cosx
29. tan (1−sinx) is equal to (Difficult)
π x π x π x π x
(A) 4 + 2 (B) 4 − 2 (C) 4 + 4 (D) 4 − 4..
−1 −1
30. The set of value of x, if sin [2x√1 − x 2 ] = 2 sin x, holds is (Average)
1 1 1 1 1
(A) ≤ x ≤ 1. (B) − ≤ x ≤ . (C) − 1 ≤ x ≤ 1 (D) − ≤ x ≤
√2 √2 √2 2 2
31. The set of value of x, if sin−1 [2x√1 − x 2 ] = 2 cos−1 x, holds is (Average)
1 1 1 1 1
(A) ≤ x ≤ 1. (B) − ≤ x ≤ . (C) − 1 ≤ x ≤ 1 (D) − ≤ x ≤
√2 √2 √2 2 2
32. The set of value of x, if sin−1 [3x − 4x 3 ] = 3 sin−1 x, holds is (Average)
1 1 1 1 1 1
(A) − < x < (B) ≤ x ≤ 1. (C) < x < 1 (D) − ≤ x ≤
2 2 2 2 2 2
33. The set of value of x, if cos −1 [4x3 − 3x] = 3 cos−1 x, holds is (Average)
1 1 1 1 1 1
(A) − < x < (B) ≤ x ≤ 1 . (C) < x < 1 (D) − ≤ x ≤
2 2 2 2 2 2
34. The set of value of x, if sin−1 [sinx] = x, holds is (Easy)
π π
(A) 0 ≤ x ≤ 1 (B) − ≤ x ≤ (C) − 1 < x < 1 (D) − 1 ≤ x ≤ 1
2 2
 π π
35. Write the range of f ( x ) = sin-1x in 0, 2π other than - ,
 2 2 
(Easy)

  3    3    3    3 
(A)  , (B)  ,  (C)  −, (D)  − ,− 
 2 2  2 2   2 2   2 2 

SUBJECT: 35 – MATHEMATICS Page 13 of 164

36. What is the reflection of the graph of the function y = sinx along the line y = x (Easy)
−1 −1 −1 −1
(A) sin x (B) − sin x (C) cos x (D) − cos x
37. In which of the following the inverse of the function y=sinx does not exist. (Average)
𝜋 𝜋 𝜋 3𝜋 3𝜋 𝜋
(A)[0,π ] (B)[− 2 , ] (C)[ 2 , 2 ] (D)[− 2 , − 2 ]
2
38. The graph of the function y = cos −1 x is the mirror image of the graph of the
function y= cosx along the line (Easy)
(A)x=0 (B)y=x (C)y=1 (D)y=0
  2   2
39. Statement 1: sin −1  sin   = (Easy)
  3  3

Statement 2: sin
−1
( sin ( ) ) =  , if   − 2 , 2 
 
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is false and Statement 2 is true
D) Statement 1 is false and Statement 2 is false
−1 −1
40. Assertion (A): Domain of f(x) = sin x + cos x is [-1, 1] (Average)
Reason (R): Domain of a function is the set of all possible values for which function will be
defined.
A) A is false and R is true B) A is false and R is true
C) A is true and R is true D) A is false and R is false.
  3 
41. Assertion (A): One branch of sin−1 ( x ) other than the principal value branch is  ,
 2 2 
  3 
Reason (R): sin(x) is invertible in the interval  , (Easy)
 2 2 
A) A is false and R is true B) A is false and R is false
C) A is true and R is false D) A is true and R is true.
 7   5
−1 
42. Statement 1: Principal value of cos  cos    is (Average)
  6  6
Statement 2: Principal value branch of cos x is [ 0, π] and cos −1 ( cos x ) = x if x  [ 0, π]
−1

A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
D) Statement 1 is false and Statement 2 is false.
1
43. Statement 1: If ≤ 𝑥 ≤ 1, 𝑡ℎ𝑒𝑛 𝑠𝑖𝑛−1 [2𝑥√1 − 𝑥 2 ] = 2 𝑐𝑜𝑠 −1 𝑥 (Easy)
√2
Statement 2: If 0 ≤ 𝑥 ≤ 𝜋, 𝑡ℎ𝑒𝑛 𝑠𝑖𝑛−1 (𝑠𝑖𝑛 𝑥) = 𝑥.
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Statement 1 is true and Statement 2 is true
D) Statement 1 is false and Statement 2 is false

SUBJECT: 35 – MATHEMATICS Page 14 of 164

44. Match List I with List II (Average)
List I List II
a) Domain of 𝑠𝑖𝑛−1 𝑥 i) (−∞ , ∞)
b) Domain of 𝑡𝑎𝑛−1 𝑥 ii) [0 , 𝜋]
c) Range of 𝑐𝑜𝑠 −1 𝑥 iii) [-1, 1]

Choose the correct answer from the options given below
A) a-i , b-ii, c-iii B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-iii, b-i, c-ii
45. Match List I with List II (Average)

List I List II
a) Range of 𝑐𝑜𝑡 −1 𝑥 𝜋 𝜋
i)(− 2 , 2 )
b) Range of 𝑡𝑎𝑛−1 𝑥 ii)(0, 𝜋)
c) Range of 𝑠𝑖𝑛−1 𝑥 𝜋 𝜋
iii) [− , ]
2 2

Choose the correct answer from the options given below:
A) a-i , b-ii, c-iii B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-iii, b-i, c-ii
46. Match Column I with Column II (Average)
Column I Column II
a) Domain of 𝑠𝑒𝑐 −1 𝑥 i) 𝑅 − (−1,1 )
−1 ii) (0, 𝜋)
b) Range of 𝑐𝑜𝑠𝑒𝑐 𝑥
c) Range of 𝑐𝑜𝑡 −1 𝑥 𝜋 𝜋
iii) [− 2 , 2 ] − {0}

Choose the correct answer from the options given below:
A) a-i , b-ii, c-iii B) a-iii, b-ii, c-i C) a-i, b-iii, c-ii D) a-iii, b-i, c-ii
𝜋 −1 1
47. 𝑐𝑜𝑠 ( − 𝑠𝑖𝑛 (− 2)) 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜_________
3
(Average)
1 𝜋
48. 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑐𝑜𝑠 −1 (− 2) = 𝑘 , 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘 𝑖𝑠_________ (Average)
𝜋
49. 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑐𝑜𝑠𝑒𝑐 −1 (−2) = − 𝑘 , 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘 𝑖𝑠_________ (Average)
50. Principal value sin −1
( −1) = 𝑘𝜋, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘 𝑖𝑠_________ (Average)
1
51. The values of 2 𝑐𝑜𝑠 (2 𝑠𝑖𝑛−1 (2)) 𝑖𝑠_________ (Average)
52. The graph shown represents one of the inverse trigonometric functions.
Identify the correct function from the following options:(Easy)

−1 −1
(A) sin x (B) 𝑐𝑜𝑠𝑒𝑐 −1 x (C) cos x (D) 𝑠𝑒𝑐 −1 x

53. The graph shown below represents one of the inverse
trigonometric functions.Identify the correct function from the
following options: (Average)

−1 −1
(A) sin x (B) 𝑐𝑜𝑠𝑒𝑐 −1 x (C) cos x (D) 𝑠𝑒𝑐 −1 x

SUBJECT: 35 – MATHEMATICS Page 15 of 164

54. The given graph is for which equation? (Average)

−1
(A) y = 𝑠𝑒𝑐 −1 x (B)𝑦 = 𝑐𝑜𝑡 −1 x (C) y= cos x (D) 𝑦 = 𝑡𝑎𝑛−1 x.
𝜋
55. If sin-1 x – cos-1 x = 6 , then x =------ (Average)
56. The domain of the function defined by f(x) =sin−1 √𝑥 − 1 is (Difficult)
(A) [1, 2] (B) [-1, 1] (C) [0, 1] (D) [-1,0]
57. Which value is similar to sin-1sin(6 π/7)? (Average)
A) sin(sin-1(π/7)) B) sin(cos-1(π/7)) C) sin(sin-1(2π/7)) D) sin(coses-1(π/7))
58. What is the value of cos-1(-x) for all x belongs to [-1, 1]? (Easy)
A) cos-1(-x) B) π – cos-1(x) C) π – cos-1(-x) D) π + cos-1(x).
3
59. The value of cos[𝑡𝑎𝑛−1 4] is ……. (Difficult)
−1 −1 −1 𝜋
60. 𝑡𝑎𝑛 √3 + 𝑠𝑒𝑐 2 – cos 1 = 𝑘 , 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘 𝑖𝑠_________ (Difficult)
61. The domain of sin-1 (3x) is equal to (Average)
A) [-1, 1] B) [−1/3,1/3] C) [-3, 3] D) [-3π, 3π]
62. What is the value of cos-1(cos (2π/3)) + sin-1 (sin (π/3)) is ? (Average)
A) π B) π/2 C) 3π/4 D) 4π/3
63. The value of 𝑡𝑎𝑛2 (sec−1 2) + 𝑐𝑜𝑡 2 (cosec−1 3) is (Average)
(A) 5 (B) 11 (C) 13 (D) 15.
64. The value of the expression sin [cot-1(cos (tan-11))] is (Average)
A) 0 B) 1 C) 1/√3 D) √(2/3).
65. The given graph is for which equation? (Easy)

A) y = cos-1x B) y = cot-1x C) y = cosec-1x D) 𝑦 = 𝑡𝑎𝑛−1 x
66. The domain of sine function is 𝑅 and function sine : 𝑅 → 𝑅 is neither one-one nor onto.
The following graph shows the sine function. (Average)

Let sine function be defined from set 𝐴 to [−1,1] such that inverse of sine function exists, i.e.,
sin−1 x is defined from [−1,1] to A. On the basis of the above information, The interval A other
than principal value branch is
  3       3 
(A)  − ,  (B)  − ,   (C)  ,  (D)  − ,0
 2 2  2  2 2 

SUBJECT: 35 – MATHEMATICS Page 16 of 164

Two Mark Questions
−1 −1  −1  −1  −1 
1. Find the value of tan 1 + cos   + sin   (Average)
 2   2 
−1 1 1
2. Find the value of cos + 2sin −1   (Average)
2 2
3. Find the values of tan
−1
3 − sec ( −2 )
−1
(Average)
4. Find the value of tan −1 3 − cot −1 − 3 ( ) (Average)

  1 
5. Find the value of sin  − sin −1  −   (Average)
3  2 
−1
 3
6. Find sin  −  (Easy)
 2 

−1
7. If y = 𝑐𝑜𝑡 −1 ( 3 ), then find value of y. (Easy)
√
8. Find the principal value sin −1 ( −1) . (Easy)
9. Find the principal value of tan −1 − 3 ( ) (Easy)

−1  1
10. Find the principal value of cos  −  (Easy)
 2
11. Find the principal value of cosec−1 − 2 ( ) (Easy)

−1  1 
12. Find the principal value of tan  −  (Easy)
 3
13. Find the principal value of cos ( −1)
−1
(Easy)
−1
14. Find the principal value of sec (−2) (Easy)
  3 
15. Find the value of sin  − sin1  −   (Average)
 2  2  
−1  2 
16. Evaluate sin sin (Average)
 3 
−1  13 
17. Evaluate cos cos (Average)
 6 
−1  7 
18. Evaluate tan  tan
6
(Average)
 
 3 
−1
19. Evaluate sin sin (Average)
 5 
−1  3 
20. Evaluate tan  tan (Average)
 4 
−1  7 
21. Evaluate cos cos (Average)
 6 
 x 
22. Express tan −1   , x  a in simplest form. (Average)
 a −x 
2 2

SUBJECT: 35 – MATHEMATICS Page 17 of 164

1  1− x 
 , x   0,1
−1
23. Prove that tan x = cos −1  (Average)
2  1+ x 
 3a 2 x − x3  −a a
24. Express tan−1  3 2
,a  0, x in simplest form (Average)
 a − 3ax  3 3
−1
 1 − cos x 
25. Write tan 
  , 0<x   in simplest form (Average)
 1 + cos x 
  1
26. Express cot −1   , x>1 in the simplest form (Average)
 x 2
− 1 

(
27. Prove that sin −1 2x 1 − x 2 = 2sin −1 x,
−1
2
)x
1
2
(Average)

(
28. Prove that sin −1 2x 1 − x 2 = 2 cos −1 x,
1
2
) x 1 (Average)

 −1 1 
−1 −1
29. Prove that 3sin x = sin 3x − 4x , x   , 
3
( 2 2
) (Average)

1 
−1 −1
30. Prove that 3cos x = cos 4x − 3x , x   , 1
3
(
2 
) (Average)

(
31. Prove that sin tan x =
−1
) x
, x 1 (Average)
1+ x 2

1
32. Find the values of 𝑡𝑎𝑛−1 (2𝑐𝑜𝑠 (2 𝑠𝑖𝑛−1 2 )) (Average)
3 24
33. Prove that 2𝑠𝑖𝑛−1 = 𝑡𝑎𝑛−1 . (Average)
5 7
Additional Questions
  −1  
34. Find sin  − sin −1    (Average)
2  2 
  3 
35. Find sin  − sin −1  −  (Average)
 2  
 3 
1
36. Find sin  sin
−1
( −1) (Average)
2 
 1 
37. Express tan −1   , x  1, in the simplest form. (Average)
 x −1 
2

 1 + cos x 
38. Write tan −1 
 1 − cos x  ,x  2n in simplest form (Average)
 
 3cos x − 4 sin x  3
39. Simplify tan −1   , if tan x  −1. (Average)
 4cos x + 3sin x  4
 a cos x − b sin x  a
40. Simplify tan −1   , if tan x  −1. (Average)
 b cos x + a sin x  b
9 9 −1 1 9 −1 2 2
41. Prove that − sin = sin (Average)
8 4 3 4 3

SUBJECT: 35 – MATHEMATICS Page 18 of 164

Three Mark Questions
 cos x − sin x  
−1 3
1. Write tan   ,−  x  in simplest form (Difficult)
 cos x + sin x  4 4
−1
 1+ x − 1− x   1 −1 −1
2. Prove that tan   = − cos x,  x 1 (Difficult)
 1 + x + 1 − x  4 2 2
 1 + sin x + 1 − sin x  x  
Prove that cot −1  = , x  0
 1 + sin x − 1 − sin x  2 
3. (Difficult)
   4
−1  cos x  3 
4. Express tan   ,− x in the simplest form (Difficult)
 1 − sin x  2 2
−1 8 3 77
5. Prove that sin + sin −1 = tan −1 (Difficult)
17 5 36
−1 63 5 3
6. Prove that tan = sin −1 + cos −1 (Difficult)
16 13 5
−1 4 12 33
7. Prove that cos + cos −1 = cos −1 (Difficult)
5 13 65
−1 12 3 56
8. Prove that cos + sin −1 = sin −1 (Difficult)
13 5 65
9. Solve : 2 tan ( cos x ) = tan ( 2 cosec x )
−1 −1
(Difficult)
 1 − x  1 −1
 = tan x ( x>0 )
−1
10. Solve : tan  (Average)
 1+ x  2

11. Solve : sin −1 (1 − x ) − 2sin −1 x = (Average)
2
 1 + x2 − 1 
−1
12. Write the function tan   ,x  0 in the simplest form (Difficult)
 x 
 
1 2x −1 1 − x 
2
13. Find the values of tan−1  sin−1 + cos , x  1, y  0 and xy  1
1 + x 2 
(Difficult)
2 1 + x2

Additional Questions
3 8 84
14. Prove that sin −1 − sin −1 = cos −1 (Difficult)
5 17 85
12 4 63
15. Prove that sin −1 + cos −1 + tan −1 = (Difficult)
13 5 16

SUBJECT: 35 – MATHEMATICS Page 19 of 164

CHAPTER -03
MATRICES
MCQ /FB questions.
1. If A is a matrix of order 3 × 4, then each row of A has__ (Easy)
(A) 3 elements (B) 4 elements (C) 12 elements (D) 7 elements
2. If every row of a matrix A contains m elements and its column contains n elements,
then the order of A is__ (Average)
(A) m × m (B) m × n (C) n × m (D) n × n
𝑇
3. If the order of A is 4 × 3 and the order of B is 4 ×5, then the order of (𝐴 𝐵)𝑇 is (Average)
(A) 3 × 5 (B) 3 × 4 (C) 4 × 3 (D) 5 ×3
4. If a matrix has 8 elements, then total number the possible different orders matrices (Easy)
(A) 8 (B) 6 (C) 4 (D) 2
5. If a matrix has 13 elements, then total number the possible different orders matrices (Easy)
(A) 1 (B) 2 (C) 3 (D) 4.
6. For any square matrix A = [aij], aij = 0, when i ≠ j, then A is- (Easy)
(A) unit matrix (B) scalar matrix (C) diagonal matrix (D) row matrix
(𝑖+𝑗)2
7. For 2 × 2 matrix, A = [aij], whose elements are given by 𝑎𝑖𝑗 = then A is equal to (Average)
2
9 9 9 9
1 2 2 2
2 2 2 2
(A) [ 9 ] (B) [ 9 ] (C) [ 9 ] (D) [ 9 ]
8 4 8 2
2 2 2 2

( i + j ) then A is equal to
2

8. For 2x3 matrix A =  aij  whose elements are given by aij = (Average)
2
 9   9   9   25 
2 8 2  2 2 8  2 2 8  2 2 
8
(A)   (B)   (C)   (D)  
8 9 25   9 8 25  8 9 25  9 9
8 
 2 2   2 2   2 2   2 2  .
9. A row matrix has only- (Easy)
(A) one element (B) one row with one or more columns
(C) one column with one or more rows (D) one row and one column.
10. A matrix A = (aij) m x nis said to be a square matrix if- (Easy)
(A) m = n (B) m ≥n (C) m ≤ n (D) m < n.
11. If A and B are matrices of order m × n and n × n respectively, then which of the following are
defined- (Easy)
(A) Both AB, BA (B) AB, A2 (C) A2, B2 (D) AB, B2
12. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is: (Average)
(A) 27 (B) 18 (C) 81 (D) 512.
13. The values of x and y make the following pair of matrices equal
3𝑥 + 7 5 5 𝑦−2
[ ]= [ ] (Average)
𝑦 + 1 2 − 3𝑥 8 4
2 2 7 −2 1
(A) x =− 3 , 𝑦 = 7 (B) x= 3 , 𝑦 = 3 (C) x= 3 , 𝑦 = −7 (D) x =− 3 , 𝑦 = 7.
14. In the following, scalar matrix is- (Easy)
 −1 3 0 3 4 0 4 0
(A)   (B)   (C)   (D) [ ].
 2 4 2 0 0 4 0 0

SUBJECT: 35 – MATHEMATICS Page 20 of 164

15. In the following, diagonal matrix is- (Easy)
0 3 4 3 1 0 0  3 0 
(A)   (B) [ ]. (C)   (D)  
4 0 0 0 0 0 1  0 4
5 2 3 6 
16. If X+ Y =   and X – Y =   then the matrix X is- (Easy)
0 9 0 −1
8 8  2 − 4 1 − 2  4 4
(A)   (B)   (C)   (D)  
0 8 0 10  0 5  0 4
7 0 3 0
17. If A, B are two matrices such that A + B =   ,A–B=   then A equals- (Easy)
 2 5 0 3
 4 0 10 0 2 0 5 0
(A)   (B)   (C)   (D)  
2 8  2 8 1 4 1 4
18. If X is a matrix of order 2 × n and Z is a matrix of order 2 × p.
If n = p, then the order of the matrix 7X – 5Z is: (Easy)
(A) p× 2 (B) 2 × n (C) n × 3 (D) p × n
19. For matrices A and B, AB = 0, then- (Easy)
(A) A = 0 or B = 0 (B) It is not necessary that A = 0 or B = 0
(C) A = 0 and B = 0 (D) All above statements are wrong.
 −1 5 
 −1 0 2   
20. If A =   and B =  2 7  , then- (Easy)
 3 1 2   3 10 
(A) AB and BA both exist (B) AB exists but not BA
(C) BA exists but not AB (D) both AB and BA do not exist.
21. Which one of the following is not true (Easy)
(A) Matrix addition is commutative (B) Matrix addition is associative
(C) Matrix multiplication is commutative (D) Matrix multiplication is associative
22. If A and B are two matrices such that A+B and AB are both defined, then (Easy)
(A) A and B are two matrices not necessarily of same order
(B) A and B are square matrices of same order
(C) Number of columns of A= number of rows of B
(D) A and B are symmetric matrices
𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼
23. If 𝐴(𝛼) = ( ) then A(𝛼) ∙A(𝛽) is equal to (Average)
−𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼
(A) A(𝛼) – A(𝛽) (B) A(𝛼) + A(𝛽) (C) A(𝛼 –𝛽) (D) A(𝛼 + 𝛽)
24. For suitable matrices A, B; the false statement is- (Easy)
(A) (BA)T =ATBT (B) (AT)T= A (C) (A – B)T = BT - AT (D) (A + B)T = AT + BT
3 x
25. If A =  and A = AT, then - (Easy)
y 0 
(A) x = 0, y = 3 (B) x + y = 3 (C) x = y (D) x = – y
26. Which one of the following is not true (Easy)
(A)A is a symmetric matrix if AT= A. (B)A is a skew symmetric matrix if AT= – A.
(C)For any square matrix A with real number entries,
A + A′ is a skew symmetric matrix and A – A′ is a symmetric matrix.
(D)Every square matrix can be expressed as the sum of a symmetric and a skew symmetric
matrix.
SUBJECT: 35 – MATHEMATICS Page 21 of 164

0 5 −7 

27. Matrix −5 0 11  is a- (Easy)

 7 −11 0 
(A) diagonal matrix (B) scalar matrix (C) skew-symmetric matrix (D) symmetric matrix.
28. If A is symmetric as well as skew symmetric matrix, then - (Easy)
(A) A is a diagonal matrix (B) A is a null matrix
(C) A is a unit matrix (D) A is a triangular matrix
29. If A, B are symmetric matrices of the same order then (AB – BA) is : (Average)
(A) symmetric matrix (B) skew symmetric matrix (C) null matrix (D) unit matrix
2 x −3x − 2
30. If A=  3 −2 −1  is a symmetric matrix then x = (Easy)

 4 −1 −5 
(A) 0 (B) 3 (C) 6 (D) 8
31. If A is a square matrix then A – A is
T (Easy)
(A) Unit matrix (B) null matrix (C) skew-symmetric (D) Zero matrix
1 2
32. If A =[ ], then A−1 = (Average)
−4 −1
1 1 2 − 1 − 2 1 1 2  1 − 1 − 2
(A)   (B)  (C) (D)
7 − 4 − 1 4 1  9 4 1  7  4 1 
33. Matrices A and B will be inverse of each other only if (Easy)
(A) AB = BA (B) AB = BA = 0 (C) AB = 0, BA = I (D) AB = BA = I
a b
34. If  is skew symmetric matrix then a + b + c + d =
d 
(Average)
c
(A)-b (B)-c (C)0 (D)1
35. Statement1: If A is a symmetric as well as a skew symmetric matrix, then A is a null matrix
Statement 2: A is a symmetric matrix if AT=A and A is a skew symmetric matrix if AT =– A.
A) Statement 1 is true and Statement 2 is false. (Average)
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
0 1 1 1 0 0
36. Statement 1:[ ][ ]=[ ] (Average)
0 1 0 0 0 0
Statement 2: For matrices A and B, AB = 0, then it is not necessary that A = 0 or B = 0
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
D) Statement 1 is false and Statement 2 is false.
1 2 3
37. Assertion (A): The matrix A = [−2 1 4] is a skew symmetric matrix (Average)
−3 −4 1
Reason (R): If matrix A is a skew symmetric matrix , then AT = - A .
A) A is false and R is true B) A is true and R is true
C)A is true and R is false D) A is false and R is false

SUBJECT: 35 – MATHEMATICS Page 22 of 164

0 1 
38. Statement 1 : A =   is an identity matrix. (Average)
1 0 
1 if i = j
Statement 2 : A square matrix A =  aij  is an identity matrix, if aij = 
0 if i  j
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Statement 1 is true and Statement 2 is true
D) Statement 1 is false and Statement 2 is false
 2 0 0
39. Statement1 : Matrix 0 3 0 is a scalar matrix (Average)
 
0 0 5
Statement 2 : Every scalar matrix is a diagonal matrix
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Statement 1 is true and Statement 2 is true
D) Statement 1 is false and Statement 2 is false
0 2 3
40. Assertion (A): The matrix A = [−2 0 4] is a skew symmetric matrix (Average)
−3 −4 1
Reason (R): If matrix A is a symmetric matrix , then AT = - A .
A) A is false and R is true B) A is true and R is true
C)A is true and R is false D) A is false and R is false
41. Statement 1 : Two matrices A23 and B32 can be multiplied and their product will be a
matrix of order 2×2
Statement 2 : Two matrices can be multiplied if number of columns in the first matrix must
be equal to the number of rows in the second matrix. (Easy)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false
42. Total number of possible matrices of order 2 × 2 with each entry 1 or 0 is____ (Average)
( )
43. A matrix A = aij
3n
is said to be a square matrix, then the value of n is _____ (Easy)
44. If A is a matrix of order 3 × 4, then each column of A has ______ elements. (Easy)
2 0
45. If A=[ ] is a symmetric matrix, then x = _____ (Easy)
𝑥−2 3
46. If the order of matrix A is 5 × 3 and matrix B is 4 ×3 and order of (𝐴𝐵𝑇 )𝑇
is 4 ×k , then k = _____ (Easy)
47. If a matrix A = (aij) m x n, Match Column I with Column II (Average)

Column I Column II
a) Square matrix i)𝑚 = 1,
b) Column matrix ii) 𝑛 = 1
c) Row matrix iii) 𝑚 = 𝑛
Choose the correct answer from the options given below:
A) a-i , b-ii, c-iii B) a-iii, b-ii, c-i C) a-ii, b-iii, c-i D) a-iii, b-i, c-ii
SUBJECT: 35 – MATHEMATICS Page 23 of 164

48. If a matrix A = (aij) n x n, Match List I with List II (Average)

List I List II
a) Scalar matrix i)𝑎𝑖𝑗 = 1, 𝑖 = 𝑗𝑎𝑛𝑑𝑎𝑖𝑗 = 0, 𝑖 ≠ 𝑗
b) Diagonal matrix ii) aij = k, i = jandaij = 0, i ≠ j
c) Identity matrix iii) aij = 0, i ≠ j
Choose the correct answer from the options given below:
A) a-i , b-ii, c-iii B) a-iii, b-ii, c-i
C) a-ii, b-iii, c-i D) a-iii, b-i, c-ii
49. If A is a matrix of order m × n and B is a matrix such that A BT and BT A are both defined, then
the order of matrix B is (Difficult)
(A) m × m (B) n × n (C) n × m (D) m × n.
50. If A is a square matrix such that A2 = I, then (A – I)3 + (A + I)3 – 7A is equal to (Difficult)
(A) A (B) I – A (C) I + A (D) 3A
51. If A is an m × n matrix such that AB and BA are both defined, then B is a (Easy)
(A) m × n matrix (B) n × m matrix (C) n × n matrix (D) m × n matrix.
52. Statement 1 : Every identity matrix is a scalar matrix (Difficult)
Statement 2 : Every scalar matrix is a diagonal matrix
Statement 3 : Every identity matrix is a diagonal matrix
Choose the correct answer
(A) All 3 Statements are true (B) 1 and 2 Statements are true and 3 false
(C) 1 and 3 Statements are true and 2 false (D) 2 and 3 Statements are true and 1 false
53. If A is a square matrix such that A2 = A, then (I – A)3 +A is equal to (Difficult)
(A) I (B) 0 (C) I – A (D) I + A
1 0 0
54. The matrix A = [0 2 0] is (Easy)
0 0 3
(A) identity matrix (B) symmetric matrix
(C) skew symmetric matrix (D) scalar matrix
55. If A is a skew-symmetric matrix, then A is a
2 (Easy)
(A) Skew symmetric matrix (B) Symmetric matrix
(C) Null matrix (D) Cannot be determined
56. A square matrix in which all elements except at least one element in diagonal (Easy)
are zeros is said to be a
(A) A is a diagonal matrix (B) A is a null matrix
(C) A is a unit matrix (D) A is a triangular matrix.
57. Which of the following is not a possible ordered pair for a matrix with 6 elements. (Easy)
A) (2,3) B) (3,2) C) (1,6) D) (3,1)
58. If A and B are symmetric matrices of the same order, then (Average)
Statement 1 : A + B is a symmetric matrix
Statement 2 : A – B is a symmetric matrix
Statement 3 : AB + BA is a symmetric matrix
Choose the correct answer
(A) All 3 Statements are true (B) 1 and 2 Statements are true and 3 false
(C) 1 and 3 Statements are true and 2 false (D) 2 and 3 Statements are true and 1 false

SUBJECT: 35 – MATHEMATICS Page 24 of 164

0 1 
59. The matrix A =   is a (Easy)
1 0 
(A) unit matrix (B) symmetric matrix (C) diagonal matrix (D) skew-symmetric matrix.
0 1
60. If A=[ ] then (A +I)2 - 2A is equal to (Average)
1 0
(A) 2I (B) 3I (C) -2I (D) null matrix

Two marks questions:
 1 3   y 0  5 6 
1. Find x and y, if 2  + =  (Average)
0 x   1 2 1 8 
 2  −1 10
2. If x   + y   =   , find the values of x and y. (Average)
 3 1 5
3 2   1 0
3. Find X, if Y =   and 2 X + Y =   (Average)
1 4   −3 2
4. Find the values of x and y from the following equation (Average)
x 5  3 − 4  7 6 
2 + =
7 y − 3 1 2  15 16
 a − b 2a + c   −1 5 
5. Find the value of a, b, c and d from the equation:  =  (Average)
 2a − b 3c + d   0 13
 1 −1 5
6. Show that the matrix A =  −1 2 1  is a symmetric matrix. (Easy)

 5 1 3
0 1 −1
7. Show that the matrix A =  −1 0 1  is a skew symmetric matrix. (Easy)

 1 −1 0 
2 4  1 3  −2 5 
8. Let A =   , B =  −2 5  , C=  3 4  . Find each of the following
3 2    
(i) A + B (ii) A – B (iii) 3A – C (iv) AB (v) BA (Average)
9. Consider the following information regarding the number of men and women workers in three
factories I, II and III (Difficult)
Men Workers Women workers
I 30 25
II 25 31
III 27 26
Represent the above information in the form of a 3 × 2 matrix. What does the entry in the third
row and second column represent?
2 5 1
 3 1 1  
10. Given A =   and B =  1  , find A + B (Easy)
 2 3 0 −2 3
 2 
1 2 3  3 − 1 3
11. If A =   and B =   , then find 2A – B (Easy)
2 3 1 − 1 0 2
6 9 2 6 0
12. Find AB, if A =   and B =   (Average)
 2 3 7 9 8 

SUBJECT: 35 – MATHEMATICS Page 25 of 164

1 0  0 1  0 1 0 − 1
13. If A =   and B =   , then prove that i) AB =   and ii) BA =   .(Average)
0 −1 1 0  − 1 0 1 0 
0 − 1 3 5
14. Find AB, if A =   and B =   (Average)
0 2  0 0
 cos  sin    sin  − cos  
15. Simplify cos    + sin    (Average)
− sin  cos   cos  sin  
16. Find the transpose of each of the following matrices: A = 3
 3 2 and B = 2 − 1 2 .(Easy)
 1 2 4 
4 2 0  
3 3 2  2 − 1 2
17. If A =   and B =   , verify that (Average)
4 2 0 1 2 4 
(i) ( A ) = A (ii) ( A + B ) = A + B
19. Compute the following (Average)
 a b  a b  a + b
2 2
b + c   2ab
2 2
2bc 
(i)  +  (ii)  2 +
 −b a   b a  a + c
2
a 2 + b 2  − 2ac − 2ab
− 1 4 − 6 12 7 6
cos 2 x sin 2 x   sin 2 x cos 2 x
(iii)  8 5 16  +  8 0 5 (iv)  2 + 
    cos 2 x cos 2 x sin 2 x 
 2 8 5   3 2 4  sin x
20. Compute the indicated products: (Average)
1 
 a b  a − b  1 − 2 1 2 3
(i)    (ii) 22 3 4 (iii)   
 − b 1  b a    2 3  2 3 1
3
2 3 4 1 − 3 5  2 1 2 − 3
 1 0 1  3 − 1 3 
(iv) 3 4 5 0 2 4 (v)  3 2  (vi)  1 0 
     − 1 2 1  
4 5 6 3 0 5 − 1 1     − 1 0 2  3 1 
 
 x + y 2  6 2 
21. Find the values of x, y and z from the following equations:  = . (Average)
 5 + z xy 5 8
x z  1 − 1  3 5
22. Solve the equation for x, y, z and t, if 2  + 3  = 3  (Average)
y t  0 2   4 6
x y   x 6  4 x + y
23. Given 3  =  +
3 
, find the values of x, y, z and w. (Average)
 z w − 1 2w  z + w
24. Find the values of a, b, c and d from the following equation
2a + b a − 2b   4 − 3
5c − d 4c + 3d  = 11 24  (Average)
   
 x + 3 z + 4 2 y − 7  0 6 3 y − 2

25. If − 6 a − 1  
0  =  −6 − 3 2c + 2  , Find the values of a, b, c, x, y and z. (Average)

b − 3 − 21 0  2b + 4 − 21 0 

SUBJECT: 35 – MATHEMATICS Page 26 of 164

Three marks questions:
7 0 3 0
1. Find X and Y, if X + Y =   and X – Y =   (Average)
 2 5 0 3
2 3  2 − 2
2. Find X and Y, if 2X + 3Y =   and 3X + 2Y =   (Average)
 4 0 − 1 5 
8 0   2 − 2
3. If A = 4 
− 2 and B =  4 2  , then find the matrix X, such that 2A + 3X = 5B. (Average)
 
3 6  − 5 1 
5 2 3 6 
4. Find X and Y, if X + Y =   and X – Y = 0 −1 (Average)
0 9  
2 5 2 3 
3 1   1
3 5 5
1 2 4 1 2 4
5. If A =   and B =   , then compute 3A – 5B (Average)
3 3 3 5 5 5
7 2
2 7 6 2
 3 3  5 5 5 
cos x − sin x 0
6. If F(x) =  sin x cos x 0 , show that F ( x )  F ( y ) = F ( x + y ) (Difficult)

 0 0 1
 cos  sin  
7. If (i) A =   , then verify that AA = I (Difficult)
− sin  cos  
 sin  cos  
(ii) If A =   , then verify that AA = I
− cos  sin  
5 − 1 2 1 2 1 5 − 1
8. Show that      (Average)
6 7  3 4 3 4 6 7 
 3 − 2 1 0
9. If A = 
−  and I =   , find k so that A2 = kA – 2I (Difficult)
 4 2   0 1 
1 5
10. For the matrix A =   , verify that (Average)
6 7
(i) (A + A ) is a symmetric matrix (ii) (A - A ) is a skew symmetric matrix
 0 a b
11. Find ( A + A') and ( A − A') , when A = − a 0 c 

1 1
(Average)
2 2  
 − b − c 0
12. Express the following matrices as the sum of a symmetric and skew symmetric matrix:
(Difficult)
 6 −2 2   2 − 2 − 4 3 3 −1
3 5   1 5      
(i)   (ii)   (iii) − 2 3 − 1 (iv) − 1 3
   4  (v)  −2 −2 1 
1 −1  −1 2  2 − 1 3   1 − 2 − 3  −4 −5 2 
13. If A and B are invertible matrices of the same order, then prove that ( AB ) = B A (Average)
−1 −1−1

14. Prove that for any square matrix A with real number entries, A + A is a symmetric matrix
and A − A is a skew symmetric matrix. (Average)

SUBJECT: 35 – MATHEMATICS Page 27 of 164

15. Prove that any square matrix can be expressed as the sum of symmetric and skew
symmetric matrix. (Difficult)
16. Prove that inverse of a square matrix, if it exist, is unique. (Average)

ADDITIONAL QUESTIONS:
1 2 3  −1 1 0   −1 1 0  1 2 3
17. Show that 0 1 0  0 −1 1    0 −1 1  0 1 0  (Difficult)
     
1 1 0  2 3 4   2 3 4  1 1 0 
 −1 2 3   −4 1 −5 
18. If A = 5 7 9 and B =  1 2 0  , then verify that
 
   
 −2 1 1   1 3 1 
(i) (A + B) = A + B (ii) (A – B) = A - B (Average)
 3 4
− 1 2 1
19. If A =  − 1 2  and B = 
1 2 3
, then verify that (Average)
  
 0 1 
(i) (A + B) = A + B (ii) (A – B) = A – B
 − 2 3 − 1 0
20. If A =   and B =  1 2 , then find (A + 2B) (Average)
 1 2  
  2 3
 1 − 2 3 4 5 then find AB, BA. Show that AB  BA.
21. If A =   and B = (Difficult)
− 4 2 5  
 2 1  
22. In a legislative assembly election, a political group hired a public relations firm to promote
its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise)
is given in matrix A as Cost per contact (Difficult)
 40 Telephone
A = 100 Hou sec all
 
 50  Letter
The number of contacts of each type made in two cities X and Y is given by
Telephone Hous sec all Letter
1000 500 5000  → X
B= 3000 10000 → Y
.
 1000
Find the total amount spent by the group in the two cities X and Y.
23. A trust fund has RS. 30,000 that must be invested in two different types of bonds. The first
bond pays5 % interest per year, and the second bond pays 7 % interest per year. Using
matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds.
If the trust fund must obtain an annual total interest of : (a) Rs. 1800 (b) Rs. 2000 (Difficult)
24. A book shop of a particular school has 10 dozen chemistry books, 8 dozen physics books,
10 dozen economics books. Their selling prices are Rs. 80, Rs. 60 and Rs. 40 each respectively.
Find the total amount the bookshop will receive from selling all the books using matrix
algebra. (Difficult)
25. If A and B are symmetric matrices of the same order, then show that AB is symmetric if and
only if A and B commute, that is AB = BA. (Average)

SUBJECT: 35 – MATHEMATICS Page 28 of 164

Five marks questions:
1. For the matrices A and B , verify that ( AB ) = B A , where

 −2  0  1
(i) A =  4  , B = 1 3 −6
 
(ii) A = 1 , B = 1 5 7 (iii) A =  −4  , B =  −1 2 1 (Difficult)
     
 5   2   3 
1 2 − 3 3 − 1 2  4 1 2

2. If A = 5 0
 2  , B = 4 2 5 and C = 0 3 2
   (Difficult)
1 − 1 1  2 0 3 1 − 2 3
Then compute (A + B) and (B – C). Also, verify that A + (B – C) = (A + B) – C
1 1 − 1  1 3
  1 2 3 − 4 
, B =  0 2  and C = 
2 0 − 2 1 
3. If A = 2 0 3 ,
    
3 − 1 2   − 1 4 
find A(BC), (AB)C and show that (AB)C = A(BC). (Difficult)
2 0 1
4. Find A – 5 A + 6 I , if A =  2 1 3 
2
(Difficult)
 
 1 −1 0 
0 6 7 0 1 1  2
5. If A = −6 0 8 , B = 1 0 2 , C =  −2 
   
     
 7 −8 0  1 2 0   3 
Calculate AC, BC and (A + B) C. Also, verify that (A + B)C = AC + BC. (Difficult)
1 1 1
6. If A = 1 2 −3 , verify that A − 6 A + 5 A + 11 I = O ,
 3 2
 
 2 −1 3 
where O is zero matrix of order 3 x 3. (Difficult)
 1 2 3
7. If A = A =  3 −2 1  , then show that A – 23 A – 40 I = O .
3
(Difficult)
 
 4 2 1
ADDITIONAL QUESTIONS:
1 0 2
8. If A =  0 2 1  , prove that A – 6 A + 7 A + 2 I = 0 .
3 2
(Difficult)
 
 2 0 3 
 2 −1 5 2  2 5
9. Let A = A =   ,B =   ,C =   . Find a matrix D such that CD – AB = O (Average)
3 4  7 4   3 8
 
 0 − tan
2
10. If A =   and I is the identity matrix of order 2, show that
 tan  0 
 2 
cos  − sin  
I + A = ( I – A)  . (Difficult)
 sin  cos  
3 3 2  2 −1 2 
11. If A =  ;B =  then verify ( A + B ) = A + B
1 1 1
 (Difficult)
4 2 0  1 2 4 
SUBJECT: 35 – MATHEMATICS Page 29 of 164

1 2  2 0  1 1
12. If A =   ,B =   & C=  calculate AC, BC and ( A + B ) C . Also verify that
2 1 1 3  2 3
( A + B ) C = AC + BC (Difficult)

****************************************

SUBJECT: 35 – MATHEMATICS Page 30 of 164

CHAPTER -4
DETERMINANTS
MCQ /FB questions:
1. If A = kB, where A and B are square matrices of order n, then | A| = (Easy)
(A) 𝑘|B| (B) k n |B| (C) 𝑘 𝑛+1 |B| (D) 𝑛𝑘|B|.
0 sin 𝛼 − cos 𝛼
2. The value of determinant | − sin 𝛼 0 sin 𝛽 |= (Easy)
cos 𝛼 − sin 𝛽 0
(A) 0 (B) 1 (C) sin 𝛼 (D) cos 𝛼.
3 𝑥 3 2
3. If | |=| | then x is equal to (Average)
𝑥 1 4 1
(A) 2 (B) 4 (C) 8 (D) ±2√2.
2𝑥 4 2 4
4. If | | =| |, then value of x is (Average)
6 𝑥 5 1
(A) √3 (B) ±√3 (C) ±√6 (D) √6.
5. If A is square matrix of order 3×3, then |𝑘A|is equal to (Easy)
(A) 𝑘|A| (B) k 2 |A| (C) 𝑘 3 |A| (D) 3𝑘|A|
1 2
6. If A= [ ], then |2A|is equal to (Easy)
4 2
(A) 2|A| (B) 3|A| (C) 4|A| (D) |A|.
1 0 1
7. If A=[0 1 2], then |3A|is equal to (Average)
0 0 4
(A) 27 (B) 4 (C) 54 (D) 108.
𝑥 2 6 2
8. If | | =| |, then value of x is (Average)
18 𝑥 18 6
(A) 3 (B) ±3 (C) ±6 (D) 6.
𝑥 3 2 3
9. If | | =| |, then value of x is (Easy)
2𝑥 5 4 5
(A) 2 (B) ±2 (C) -2 (D) 3.
10. Which of the following is correct (Easy)
(A) Determinant is a square matrix
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) Determinant is just an arrangement of numbers
2 3
11. Adjoint of a matrix A =[ ] (Easy)
1 4
2 3 2 −3 4 3 4 −3
(A) [ ] (B)[ ] (C) [ ] (D) [ ].
1 4 −1 4 1 2 −1 2
12. If A be a non singular matrix of order 3, then |adj A| is equal to (Easy)
(A) |A| (B)|A|2 (C)|A|3 (D)3|A|.
13. If A is an invertible matrix of order 2, then det (𝐴−1) is equal to (Easy)
1
(A) det (A) (B) det (A) (C) 0 (D) 1.
14. If A is a square matrix of order 2 and |A| = 3, then | 𝐴−1| = (Easy)
2 1
(A) 3 (B) 3 (C) 3 (D) 12.
15. If A is a square matrix of order n, then |adj(A) | = (Easy)
(A) |A| (B)|A|n (C)|A|𝑛−1 (D) 𝑛|A|.
16. If A and B are invertible matrices, then which of the following is not correct? (Easy)
(A) A (adj A) = (adj A) A = A I (B) A (adj A) = (adj A) A = |A|I
−1 −1 −1
(C) (AB) =B A (D) |A| ≠ 0 and |B| ≠ 0 .
SUBJECT: 35 – MATHEMATICS Page 31 of 164

17. For a square matrix A in matrix equation AX = B, Which of the following is not correct
(A) |A| ≠0, there exists unique solution (Easy)
(B) |A| = 0 and (adj A) B ≠ 0, then there exists no solution
(C) |A| = 0 and (adj A) B = 0, then system may or may not be consistent
(D) |A| ≠0, then system is inconsistent.
1 2 −1
18. If A = [1 𝑥 − 2 1 ] 𝑖𝑠 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑖𝑠 (Average)
𝑥 1 1
(A) 2 (B) 3 (C) 1 (D) 0.
19. If A is a square matrix of order 3 and |adjA|= 25, then |𝐴| is (Average)
1 1
(A) 25 (B) 25 (C) 5 (D) 5 .
2 𝜆 −4
20. If A =[0 2 5 ] , then A−1 exists if (Average)
1 1 3
(A) 𝜆 =- 2 (B)λ ≠2 (C) 𝜆≠ -2 (D) 𝜆≠2 and 𝜆≠ -2.
2 3
21. The inverse of the matrix [ ] is (Easy)
4 6
2 3 2 −3 6 −3
(A) [ ] (B)[ ] (C) [ ] (D) Does not exists.
4 6 −4 6 −4 2
2 2
22. The inverse of the matrix [ ] is (Easy)
4 3
1 3 −2 1 −3 2 1 3 2 1 3 2
(A) [
2 −4
] (B) 2 [ ] (C) 2 [ ] (D) − 2 [ ].
2 4 −2 4 2 4 2
23. Consider the system of linear equations: (Average)
3𝑥 − 2𝑦 + 3𝑧=8, 2𝑥 + 𝑦 − 𝑧=1 and 4𝑥 − 3𝑦 + 2𝑧=4.The system has
(A) exactly 3 solutions (B) a unique solution
(C) no solution (D) infinite number of solutions.
2
24. If 𝐴 − 4𝐴 + 𝐼 = 𝑂, then the inverse of A is (Average)
(A) A + I (B) A- 4I (C) A – I (D) 4I – A.
25. If area of triangle is 35 sq units with vertices (2, – 6), (5, 4) and (k, 4) then k is (Average)
(A) 12 (B) –2 (C) –12, –2 (D) 12, –2
26. If area of triangle is 4 sq units with vertices (k, 0), (4, 0) and (0, 2) then k is (Average)
(A) 2,6 (B) –2,6 (C) 0, 8 (D) 0, 4
27. In matrix equation AX = B, |𝐴|=0 and (𝑎𝑑𝑗𝐴)𝐵=0, then system of equations has (Easy)
A) unique solution B) finite solution
C) either infinity many solutions or no solution D) infinitely many solution
28.If 𝐴and 𝐵are square matrix of order 3 and |𝐴|=5, |𝐵|=3 then |3𝐴𝐵|= (Average)
A) 405 B) 45 C) 135 D) 675
29.If 𝐴is a matrix of order 3, such that 𝐴(𝑎𝑑𝑗𝐴)=10𝐼then |𝑎𝑑𝑗𝐴|= (Average)
A) 10 B) 30 C) 1 D) 100
30. Which of the following is not correct? (Easy)
(A) A square matrix A is said to be singular if |𝐴| = 0
(B) If elements of a row (or column) are multiplied with cofactors of any other row
(or column), then their sum is zero.
(C) A square matrix A is invertible if and only if A is singular matrix
(D) A square matrix A is said to be non-singular if |𝐴| ≠ 0

SUBJECT: 35 – MATHEMATICS Page 32 of 164

1 2 1
31. Statement 1: Matrix A = [1 2 1] 𝑖𝑠 a 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 matrix. (Easy)
0 1 1
Statement 2: A square matrix A is said to be singular if |𝐴| = 0.
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
D) Statement 1 is false and Statement 2 is false.
32.Assertion (A): If A is a square matrix of order 2 and |A| = 3, then|adj(A)|= 9 (Average)
Reason (R): If A is a square matrix of order n, then |adj(A) | =|A|𝑛−1
A) A is false and R is true B) A is true and R is false
C) A is true and R is true D) A is false and R is false.
0 1 2
33.Assertion (A): The matrix A = [−1 0 4],then |𝐴| = 0. (Average)
−2 −4 0
Reason (R): If Determinant of a skew-symmetric matrix of odd order is zero
A) A is true and R is true B) A is false and R is true
C) A is true and R is false D) A is false and R is false.
34.Statement 1: If A is a square matrix of order 2 and|𝑨| = 7, then |𝟒𝑨| = 112 (Average)
Statement 2: |𝝀𝑨| = λ |𝑨|, where n is order of square matrix
n

A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is true.
C) Statement 1 is true and Statement 2 is true.
D) Statement 1 is false and Statement 2 is false.
2 3
35.Statement I : The inverse of the matrix A=[ ]does not exist (Average)
4 6
Statement II : The inverse of singular matrix does not exist
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false.
36. If 𝐴 is matrix of order 3×3, then number of minors in determinant of 𝐴 is ___ (Easy)
37. If 𝐴=[𝑎𝑖𝑗] is a square matrix of order 3, |𝐴| = 3 and Aij is cofactor of aij then
𝑎11𝐴21+𝑎12𝐴22+𝑎13𝐴23 is equal to _____ (Easy)
38. If 𝐴is an invertible matrix of order 2 then |𝐴𝐴 |=______
−1 (Easy)
39. If 𝐴and 𝐵are square matrices of same order and|𝐴𝐵|=16, |𝐴|=8 then |𝐵|=______ (Easy)
40. A is a square matrix of order 2 and |𝑎𝑑𝑗𝐴|=9, then |𝐴| =______ (Easy)
1 1 T
41. If A= [ ], then |2A |=______ (Average)
3 2
42. If A is a square matrix of order 3 with |A| = 3, find the values of |AAT |= ------------- (Easy)
43. If A is a square matrix such that 𝐴2 = 𝐼 𝑎𝑛𝑑 |𝐴| ≠ 0, then A−1 is equal to (Easy)
(A) 2A (B) O (C) A (D) 𝐴2
°
44. The value of |cos 15° sin 15° | is (Average)
sin 15 cos 15°
√3−1 √3 1 √3+1
(A) (B) (C) 2 (D) 2 2 .
2√2 2 √

SUBJECT: 35 – MATHEMATICS Page 33 of 164

45. If A is a square matrix of order 3 and |A| = 5, then the value of |2A′| is (Average)
(A) -10 (B) 10 (C) -40 (D) 40.
3
46. Given that A = [aij] is a square matrix of order 3×3 and |A| = -7, then the value of a A ,
i =1
i2 i2

where Aij denotes the cofactor of element aij is (Average)
(A) 7 (B) -7 (C) 0 (D) 49.
47. If A and B are square matrices of order 3 such that A = −1, B = 3 , then 3AB −1 = (Average)
A)-9 B)-81 C)-27 D)81.
48. If A and B are invertible square matrices of order n, then which of the following is not
true? (Average)
(A) det (AB) = det(A)det(B) (B) det (kA) = kn det(A)
(C) det (A+B) = det(A) + det(B) (D) det (AT) = 1/ det(A−1 )
49. For a non singular matrix A, |A| is (Easy)
A) A  0 B) A  0 C) |A| = 0 D) A  0
50. For any singular matrix A, A−1 = (Easy)
𝐴𝑑𝑗𝐴 1
A) |𝐴| B) |A|𝐴𝑑𝑗𝐴 C) |A| adj A D) does not exist.
51. If A and B are square matrices of same order n × n and |A| = 3, |B| = 2. (Average)
n
i) det (AB) = 6 ii) det (A B ) = 1/6
T T iii) det (kAB) = k 6 .
A) only i) is true B) only i) and ii) statements are true.
C) only i) and iii) statements are true. D) all i), ii) and iii) statements are true.
52. The system of equations 4x + 6y = 5, 8x + 12y = 10 has (Easy)
A) No solution. B) Infinitely many solutions.
C) A unique solution. D) trivial solution.
53. A and B are invertible matrices of the same order such that ( AB )
−1
= 8 . If A = 2 , then |B| is
equal to (Average)
A)16 B) 4 C)6 D) 1/ 16.
10 0
54. For any 2×2 matrix, if A(adjA) = [ ], then |𝐴| is equal to (Easy)
0 10
A) 20 B) 100 C)10 D) 0
k −2
55. The Values of k for which the matrix [ ] has no inverse is (Average)
3 k−5
A) k = 3, 2 B) k = -2, 3 C) 𝑘 ≠ 3 ,2 D) 𝑘 ≠ 2, -3
56. If the value of a third- order determinant is 6, then the value of the determinant formed by
replacing each of its elements by its cofactor (Average)
(A) 12 (B)36 (C) 216 (D) 18
57. If A is square matrix of order 3 and |A|=4, then |A adjA| is (Average)
(A) 64 (B)16 (C) 4 (D) 12.
1 3 2 −1 ′
58. If A =[ ], B=[ ] then |ABB | = (Average)
4 2 1 2
(A) 50 (B) -250 (C) 100 (D) 250.
2 𝜆 3
59. If A =[2 0 5] is non singular matrix if (Average)
0 3 3
(A) 𝜆 = 3 (B)𝜆≠-3 (C)𝜆≠ - 2 (D) 𝜆≠ −2
60. For a square matrix A in matrix equation AX = B (Average)
i)|A| ≠ 0, system of equations is consistent
ii)|A| = 0 and (adj A) B ≠ 0, then system of equations is inconsistent.
iii)|A| = 0 and (adj A) B = 0, then system may or be either consistent or inconsistent
A) only i) is true B) i) and ii) statements are true.
C) i) and iii) statements are true. D) all i), ii) and iii) statements are true.
SUBJECT: 35 – MATHEMATICS Page 34 of 164

Two/Three marks questions:
1. Find the equation of the line joining the points (3, 1) and (9, 3) using determinants. (Average)
2. Find the equation of the line joining the points (1, 2) and (3, 6) using determinants. (Average)
3. Find the area of the triangle whose vertices are (3, 8), (−4, 2)and (5, 1) using determinants.
(Average)
4. Find the area of the triangle whose vertices are (2, 7), (1, 1) and (10,8) using determinants.
(Average)
5. If the area of the triangle with vertices (−2, 0), (0, 4) and (0, 𝑘) is 4 square units.
Find the values of 𝑘 using determinants. (Average)
6. If the area of the triangle with vertices (2, – 6), (5, 4) and (k, 4) is 35 square units.
Find the values of 𝑘 using determinants. (Average)
7. If the area of the triangle with vertices (k, 0), (4, 0) and (0, 2) is 4 square units.
Find the values of 𝑘 using determinants. (Average)
8. Find the area of the triangle whose vertices are (1, 0)(6, 0)and (4, 3) using determinants.
(Average)
9. Find𝑘, if the area of the triangle is 3 square units and whose vertices are (𝑘, 0)(1, 3) and (0, 0)
using determinants. (Average)
10. Prove that |𝑎𝑑𝑗𝐴| = |𝐴|2 , where 𝐴 is the matrix of order 3 × 3. (Average)
11. Solve the system of linear equations using matrix method: (Average)
(i) 2𝑥 + 5𝑦 = 1, 3𝑥 + 2𝑦 = 7 (ii) 5𝑥 + 2𝑦 = 4, 7𝑥 + 3𝑦 = 5
(iii) 5𝑥 + 2𝑦 = 3, 3𝑥 + 2𝑦 = 5 (iv) 4𝑥 − 3𝑦 = 3, 3𝑥 − 5𝑦 = 7
2 3
12. Find adjA for A =[ ]. (Average)
1 4
2 −2
13. Find the inverse of the matrices [ ]. (Average)
4 3
−1 5
14. Find the inverse of the matrices [ ]. (Average)
−3 2
15. Examine the consistency of the system of equations x + 2y = 2, 2x + 3y = 3. (Average)
16. Examine the consistency of the system of equations x + 3y = 5, 2x + 6y = 8. (Average)
17. Examine the consistency of the system of equations 3x–y – 2z = 2, 2y – z = –1 and
3x – 5y = 3. (Average)
18. Examine the consistency of the system of equations 5x – y + 4z = 5, 2x + 3y + 5z = 2 and
5x – 2y + 6z = –1. (Average)
19. If A be any given square matrix of order n, then prove that A(adj A) = (adj A) A = |A| I ,
where I is the identity matrix of order n (Difficult)
20. If A is a square matrix of order 3, then prove that |adj(A) | = |𝐴|2 . (Average)
10 −2 
21. Find P −1 , if it exits, given P =   (Average)
 −5 1 
Four marks questions:

3 1
1. If 𝐴 = [ ], show that 𝐴2 − 5𝐴 + 7𝐼 = 0 and hence find 𝐴−1 . (Difficult)
−1 2
2 3
2. Show that the matrix A = [ ] satisfies the equation A2 -4A+I =O, where I is 2 x 2 identity
1 2
matrix and 2 x 2 zero matrix. Using this equation, find A−1 . (Difficult)
1 3 3
3. If [1 4 3] then verify that A adj A = |A| I. Also find 𝐴−1 . (Difficult)
1 3 4
2 3 1 −2
4. If A =[ ] and B =[ ]then verify that (𝐴𝐵)−1 = 𝐵−1 𝐴−1 (Difficult)
1 −4 −1 3
1 3
5. If A =[ ] Verify A (adj A) = (adj A) A = |A| I . (Difficult)
2 4
3 7 6 8
6. If A =[ ] and B =[ ]then verify that (𝐴𝐵)−1 = 𝐵−1 𝐴−1 (Difficult)
2 5 7 9
3 −1 1 1 2 −2
7. If 𝐴−1 = [−15 6 −5] and B = [−1 3 0 ] ,find (𝐴𝐵)−1 . (Difficult)
5 −2 2 0 −2 1
SUBJECT: 35 – MATHEMATICS Page 35 of 164

Five marks questions:
1. Solve the system of equations 𝑥 + 𝑦 + 𝑧 = 6, 𝑦 + 3𝑧 = 11 and 𝑥 − 2𝑦 + 𝑧 = 0 by matrix method.
(Difficult)
2. Solve the system of equations 3𝑥 − 2𝑦 + 3𝑧 = 8, 2𝑥 + 𝑦 − 𝑧 = 1 and 4 𝑥 − 3𝑦 + 2𝑧 = 4
by matrix method. (Difficult)
2 −3 5
3. If 𝐴 = [3 2 −4], find 𝐴−1 . Using 𝐴−1 , solve the system of equations (Difficult)
1 1 −2
2𝑥 − 3𝑦 + 5𝑧 = 11, 3𝑥 + 2𝑦 − 4𝑧 = −5and 𝑥 + 𝑦 − 2𝑧 = −3.

4. Solve the system of equations 4 x + 3 y + 2 z = 60 , 2 x + 4 y + 6 z = 90 , 6 x + 2 y + 3 z = 70 by matrix
method. (Difficult)
3
5. Solve the equations 2𝑥 + 𝑦 + 𝑧 = 1, 𝑥 − 2𝑦 − 𝑧 = and 3𝑦 − 5𝑧 = 9 by matrix method. (Difficult)
2
1 −1 2   −2 0 1 
6. Use the product 0 2 −3  9 2 −3 to solve the system of equations
  
 3 −2 4   6 1 −2 
𝑥 − 𝑦 + 2𝑧 = 1, 2𝑦 − 3𝑧 = 1, 3𝑥 − 2𝑦 + 4𝑧 = 2. (Difficult)
7. Solve the system of equations 𝑥 − 𝑦 + 2𝑧 = 7, 3𝑥 + 4𝑦 − 5𝑧 = −5 and 2𝑥 − 𝑦 + 3𝑧 = 12
by matrix method. (Difficult)
8. Solve the system of equations 𝑥 − 𝑦 + 𝑧 = 4, 2𝑥 + 𝑦 − 3𝑧 = 0 and 𝑥 + 𝑦 + 𝑧 = 2
by matrix method. (Difficult)
9. Solve the system of equations 2𝑥 + 2𝑦 + 3𝑧 = 4, 𝑥 − 2𝑦 + 𝑧 = −3 and 3𝑥 − 4𝑦 − 2𝑧 = 5
by matrix method. (Difficult)
10. Solve the system of equations 2𝑥 + 3𝑦 + 3𝑧 = 5, 𝑥 − 2𝑦 + 𝑧 = −4 and 3𝑥 − 𝑦 − 2𝑧 = 3
by matrix method. (Difficult)

ADDITIONAL QUESTIONS :

11. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg
wheat and 6 kg rice is Rs 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs 70.
Find the cost of each item per kg by matrix method. (Difficult)
12. The sum of three numbers is 6. If we multiply the third number by 3 and add the second
number to it we get 11. By adding the first and third numbers, we get double the second
number. Represent it algebraically and find the numbers using matrix method. (Difficult)
2 3 10 4 6 5 6 9 20
13. Solve the system of equations + + = 4, − + = 1,& + − = 2 by matrix method
z y z x y z x y z
(Difficult)

SUBJECT: 35 – MATHEMATICS Page 36 of 164

CHAPTER-5
CONTINUITY AND DIFFERENTIABILITY
MCQ /FB questions:
1. Which of the following is true for the function f given by 𝑓(𝑥) = |𝑥 − 1| (Easy)
A) Discontinuous and differentiable at 𝑥 = 1
B) Continuous but not differentiable at 𝑥 = 1
C) Continuous and differentiable at 𝑥 = 1
D) Discontinuous and not differentiable at 𝑥 = 1
2. The left hand derivative of f ( x ) = x at x = 0 is (Easy)
A) 1 B) -1 C) 0 D) does not exist.
3. The right hand derivative of f ( x ) = x − 2 at x = 2 is (Average)
A) 1 B) -1 C) 0 D) does not exist.
4. The greatest integer function defined by𝑓(𝑥) = [𝑥] is (Average)
A) Continuous but not differentiable at 𝑥 = 1 B) Continuous and differentiable at 𝑥 = 1
C) Discontinuous but differentiable at 𝑥 = 1 D) Discontinuous and not differentiable at 𝑥 = 1
5. Number of points in the interval (-3, 3) in which f(x) = [x], where [ ] denotes the greatest integer
function, is not differentiable is (Average)
A) 0 B) 3 C) 5 D) 7
6. The greatest integer function f(x) = [x] is (Easy)
A) continuous at x = 0 B) differentiable at x = 0
C) both continuous and differentiable at x = 0 D) discontinuous at x = 0
7. The derivative of f ( x ) = x − 3 at x = 3 is (Average)
A) 0 B) 1 C) –1 D) does not exist.
8. The statement which is not true in the options given below is (Average)
A) Every polynomial function is continuous.
B) Every rational function is continuous.
C) Every differentiable function is continuous.
D) Every continuous function is differentiable.
9. The function 𝑓(𝑥) = |𝑥 − 𝑎| is (Average)
A) continuous and differentiable at 𝑥 = 𝑎
B) continuous but not differentiable at 𝑥 = 𝑎
C) not continuous but differentiable at 𝑥 = 𝑎
D) not continuous and not differentiable at𝑥 = 𝑎
10. The function 𝑓(𝑥)=|𝑥+1|+|𝑥−1| is (Difficult)
A) continuous at 𝑥=−1 as well as 𝑥=1 B) continuous at 𝑥=1 but not 𝑥=−1
C) continuous at 𝑥=−1 but not 𝑥=1 D) discontinuous at 𝑥 = −1 as well as 𝑥 = 1
𝑑𝑦
11. If 𝑦 = tan(2𝑥 + 3), then 𝑑𝑥 = (Easy)
A) 2𝑠𝑒𝑐(2𝑥 + 3) B) 2𝑠𝑒𝑐 2 (2𝑥 + 3) C) 𝑠𝑒𝑐 2 (2𝑥 + 3) D) 𝑠𝑒𝑐(2𝑥 + 3).
2 ), 𝑑𝑦
12. If 𝑦 = sin(𝑐𝑜𝑠 𝑥 then 𝑑𝑥
= (Average)
A) cos(𝑐𝑜𝑠 𝑥 2 ) B) cos(𝑐𝑜𝑠 𝑥 2 ) sin 𝑥 2 C) 2𝑥 cos(𝑐𝑜𝑠 𝑥 2 ) sin 𝑥 2 D) -2𝑥 cos(𝑐𝑜𝑠 𝑥 2 ) sin 𝑥 2 .
𝒅𝒚
13. If 𝒙 − 𝒚 = 𝝅 , then 𝒅𝒙 = (Easy)
A) 𝜋 B) 1 C) 1 + 𝜋 D) -1.
−1 −1 𝑑𝑦
14. If 𝑦 = sin 𝑥 + sin √1 − 𝑥 2 , 0 < 𝑥 < 1 then = (Difficult)
𝑑𝑥
1 2
A) B) −1 C) 0 D) .
√1−𝑥 2 √1−𝑥 2

SUBJECT: 35 – MATHEMATICS Page 37 of 164

𝜋 dy
15. If 𝑦 = cos−1(sin 𝑥) where 𝑥 ∈ (0, ), then = (Average)
2 dx
1 −1
A) B) C) 1 D) −1.
√1−𝑥 2 √1−𝑥 2
𝑥 𝑎 𝑎 𝑑𝑦
16. If 𝑦 = 𝑎 + 𝑥 + 𝑎 , then = (Average)
𝑑𝑥
𝑥 𝑎−1 𝑎−1 𝑥 𝑎−1
A) 𝑎 log 𝑒 𝑎 + 𝑎𝑥 + 𝑎𝑎 B) 𝑎 log 𝑒 𝑎 + 𝑥
𝑥 𝑎−1
C) 𝑎 log 𝑒 𝑎 + 𝑎𝑥 D) 𝑎 𝑥 log 𝑒 𝑎 + 𝑥 𝑎 𝑙𝑜𝑔𝑥 + 𝑎𝑎 𝑙𝑜𝑔𝑎.
𝑑𝑦
17. If 𝑦 = 𝑠𝑒𝑐(𝑡𝑎𝑛√𝑥), then = (Average)
𝑑𝑥
𝑠𝑒𝑐(𝑡𝑎𝑛√𝑥)𝑡𝑎𝑛(𝑡𝑎𝑛√𝑥)
A) 𝑠𝑒𝑐(𝑡𝑎𝑛√𝑥)𝑡𝑎𝑛(𝑡𝑎𝑛√𝑥)𝑠𝑒𝑐 2 (√𝑥) B) 2√𝑥
𝑠𝑒𝑐(𝑡𝑎𝑛√𝑥)𝑡𝑎𝑛(𝑡𝑎𝑛√𝑥)𝑠𝑒𝑐 2 (√𝑥) 𝑠𝑒𝑐(𝑡𝑎𝑛√𝑥)𝑡𝑎𝑛(𝑡𝑎𝑛√𝑥)𝑠𝑒𝑐 2 (√𝑥)
C) 2√𝑥
D) .
√𝑥
𝑑𝑦
18. If 𝑦 = cos(√𝑥), then 𝑑𝑥 = (Easy)
𝑠𝑖𝑛(√𝑥) −𝑠𝑖𝑛(√𝑥)
A) 𝑠𝑖𝑛(√𝑥) B) −𝑠𝑖𝑛(√𝑥) C) D) .
2√𝑥 2√𝑥
19. The derivative of 𝑒 2 log𝑒 𝑥 with respect to x is (Average)
2
2 𝑒 log𝑒 𝑥
A) 𝑒 2 log𝑒 𝑥 B) 𝑥 C) 2𝑥 D) 𝑥 .
−𝑥
20. The derivative of 𝑒 with respect to x is (Easy)
A) −𝑒 −𝑥 B) 𝑒 𝑥 C) −𝑒 𝑥
D) 𝑒 −𝑥
.
𝑑𝑦
21. If 𝑦 = cos −1(𝑒 𝑥 ), then 𝑑𝑥 = (Average)
𝑒𝑥 𝑒𝑥 1 1
A) B) − C) D) − .
√1−𝑒 2𝑥 √1−𝑒 2𝑥 √1−𝑒 2𝑥 √1−𝑒 2𝑥
−1 𝑑𝑦
22. If 𝑦 = sin (𝑥 √𝑥), then 𝑑𝑥 = (Difficult)
1 2√𝑥 3√𝑥 −3√𝑥
A) B) C) D) .
√1−𝑥 3 3√1−𝑥 3 2√1−𝑥 3 2√1−𝑥 3
𝑑𝑦
23. If 𝑦 = √𝑒 √𝑥 , then 𝑑𝑥 = (Average)
1 𝑒 √𝑥 𝑒 √𝑥 𝑒 √𝑥
A) B) C) D) .
2√𝑒 √𝑥 2√𝑒 √𝑥 2√𝑥 √𝑒 √𝑥 4√𝑥 √𝑒 √𝑥
−1 𝑥
24. The derivative of 𝑒 sin with respect to x is (Easy)
−1 −1 −1
𝑒 sin 𝑥 𝑒 sin 𝑥 𝑒 sin 𝑥 −1 𝑥
A) B) sin−1 𝑥 C) − D) 𝑒 sin .
√1−𝑥 2 √1−𝑥 2
𝑑𝑦
25. If y = 𝑠𝑖𝑛 (𝑙𝑜𝑔 𝑥), then 𝑑𝑥 = (Average)
sin (log 𝑥) √𝟏−𝒚𝟐 √𝟏−𝒚𝟐 √𝟏−𝒙𝟐
A) B) C) D) .
𝒙 𝒚 𝒙 𝒙
𝑑𝑦
26. If y = log(log x) , then = (Average)
𝑑𝑥
1 1 1 logx
A) x B) xlogx C) logx D) x .
𝑑𝑦
27. If y = log 7(𝑙𝑜𝑔𝑥), then 𝑑𝑥 = (Difficult)
1 1 𝑙𝑜𝑔7 log7
A) B) xlogx C) xlogx D) x .
𝑥𝑙𝑜𝑔7𝑙𝑜𝑔𝑥
𝑑2 𝑦
28. If y = 𝑙𝑜𝑔𝑥 , then 𝑑𝑥 2 = (Average)
1 1 1 1
A) 𝒙 B) xlogx C) x2 D)− x2.
𝑑2 𝑦
29. If y = 𝑥 20 , then 𝑑𝑥 2 = (Average)
A) 20x19 B) 20x18 C) 380 x18 D) 360x18.
𝑑𝑦
30. If 𝑦 = 𝑠𝑖𝑛 𝑥 2 , then 𝑑𝑥 = (Easy)
A) 2sin2x B) 2sinxcosx C) 2 cos 𝑥 2 D) 2𝑥 cos 𝑥 2
SUBJECT: 35 – MATHEMATICS Page 38 of 164

31. The derivative of f given by f (x) = sin−1 x exists, if x ∈ (Difficult)
A) (−1, 1) B) [−1, 1] C) (−∞, ∞) D) 𝑅 − (−1, 1)
−1
32. The derivative of f given by f (x) = tan x exists, if x ∈ (Average)
A) (−1,1) B) [−1,1] C) (−∞, ∞) D) 𝑅 − (−1,1)
33. Suppose f and g be two real functions continuous at 𝑥 = 𝑐,then (Easy)
A) 𝑓 + 𝑔 is discontinuous at x = c. B) 𝑓 − 𝑔 is discontinuous at x = c.
C) 𝑓 ∙ 𝑔 is discontinuous at x = c. D) f/g is continuous at x = c, (provided g(c) ≠ 0).
34. Derivative of 𝑠𝑖𝑛(𝑎𝑥 + 𝑏) with respect to 𝑥 is (Easy)
1 1
A) 𝑎 𝑐𝑜𝑠(𝑎𝑥 + 𝑏) B) 𝑎 𝑐𝑜𝑠(𝑎𝑥 + 𝑏) C) − 𝑎 𝑐𝑜𝑠(𝑎𝑥 + 𝑏) D) −a cos(ax + b)
𝑑𝑦
35. If 𝑦 = sin(𝑐𝑜𝑠𝑥), then 𝑑𝑥 = (Easy)
A) cos(𝑐𝑜𝑠𝑥) B) cos(𝑐𝑜𝑠𝑥) sin 𝑥 C) − cos(𝑐𝑜𝑠𝑥) sin 𝑥 D) − cos(𝑐𝑜𝑠𝑥) 𝑐𝑜𝑠𝑥.
2
36. Derivative of sin (𝑥 + 5) with respect to x is (Easy)
2 2 2 2
A) cos(𝑥 + 5) B) − cos(𝑥 + 5) C) −2𝑥 cos(𝑥 + 5) D) 2𝑥 cos(𝑥 + 5).
37. Derivative of 2√cot 𝑥 2 with respect to x is (Average)
−𝑐𝑜𝑠𝑒𝑐𝑥 2 2𝑥𝑐𝑜𝑠𝑒𝑐𝑥 2 2𝑥𝑐𝑜𝑠𝑒𝑐 2 𝑥 2 2𝑥𝑐𝑜𝑠𝑒𝑐𝑥
A) B) C) − D) − .
√cot 𝑥 2 √cot 𝑥 2 √cot 𝑥 2 √cot 𝑥 2
𝑑𝑦
38. If 𝑦 = cos (𝑙𝑜𝑔𝑥 + 𝑒 𝑥 ),then𝑑𝑥 = (Difficult)
1 sin(𝑙𝑜𝑔𝑥+𝑒 𝑥 ) −sin(𝑙𝑜𝑔𝑥+𝑒 𝑥 ) 1
A) sin(𝑙𝑜𝑔𝑥 + 𝑒 𝑥 ) B) C) D)−sin(𝑙𝑜𝑔𝑥 + 𝑒 𝑥 ) (𝑥 + 𝑒 𝑥 ).
𝑥+𝑒 𝑥 𝑥+𝑒 𝑥 𝑥+𝑒 𝑥
𝑑𝑦
39. If 𝑦 = log (cos 𝑒 𝑥 ), then 𝑑𝑥 = (Average)
A) −𝑒 𝑥 𝑡𝑎𝑛𝑒 𝑥 B) 𝑒 𝑥 𝑡𝑎𝑛𝑒 𝑥 C)−𝑒 𝑥 𝑐𝑜𝑡𝑒 𝑥 D)−𝑡𝑎𝑛𝑒 𝑥 .
𝑥2 𝑥3 𝑥4 𝑥5 𝑑𝑦
40. If 𝑦 = 𝑒 𝑥 + 𝑒 +𝑒 +𝑒 +𝑒 ,then 𝑑𝑥 at x =1 is (Average)
A) 5𝑒 B) 15e C) 15 D)5.
2 𝑑𝑦
41. If 𝑥 = 𝑎𝑡 and 𝑦 = 2𝑎𝑡, then 𝑑𝑥 = (Average)
1 1
A) 2𝑎𝑡 B) C) 𝑡 D) 𝑡
2𝑎𝑡
42. The greatest integer function defined by f (x) = [x], 0 < 𝑥 < 3 is not differentiable at (Average)
A) all non itegral points B) 𝑥 = 0, 1, 2, 3 C) 𝑥 = 1, 2 D) 𝑥 = 1, 2, 3
43. The number of points in the set of real number R in which the function f (x)= |x|+ |x + 1|
is not differentiable, is (Average)
A) 0 B) 1 C) 2 D) 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒.
𝑑2 𝑦
44. If 𝑦 = 𝐴 𝑠𝑖𝑛 𝑥 + 𝐵 𝑐𝑜𝑠 𝑥, 𝑡ℎ𝑒𝑛 𝑑𝑥 2 = (Easy)
A) 𝑦 B) −y C)𝑥 D)𝑦 2 .
𝑑2 𝑦
45. If y = 5 cosx – 3 sin x, 𝑡ℎ𝑒𝑛 +𝑦 = (Average)
𝑑𝑥 2
dy
A) 0 B) 1 C)dx D)2y.
46. Statement 1: The function f(x) = |𝒙| is discontinuous at x =0 (Difficult)
Statement 2: The function f(x) = |𝒙| is not differentiable at x =0
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Both Statement 1and 2 are true
D) Both Statement 1 and 2 are false

SUBJECT: 35 – MATHEMATICS Page 39 of 164

47. Assertion (A) : The function f (x) = [x], where [x] denotes the greatest integer less than or equal
to x , is discontinuous at x = 1.
Reason ( R ) : The greatest integer function is discontinuous at all integral points (Difficult)
A) A is false and R is true B) A is false and R is false
C) A is true and R is false D) A is true and R is true.
48. Assertion (A) : The function f (x) = |𝒙-1| is continuous and differentiable at x = 0.
Reason ( R ): Every differentiable function is continuous. (Difficult)
A) A is false and R is true B) A is false and R is false
C) A is true and R is false D) A is true and R is true
49. Assertion (A): f(x)=|x – 3| is continuous at x = 3.
Reason (R): f(x) =|x – 3| is differentiable at x = 3. (Average)
A) A is false and R is true B) A is false and R is false
C) A is true and R is false D) A is true and R is true
50. Statement I: If 𝑓(𝑥) is continuous at 𝑥 = 𝑎 , then lim f ( x ) = f ( a )
x →a

Statement II: Every differentiable function is continuous. (Average)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false.
51. Statement 1: Left hand derivative of f(x) = | x | at x = 0 is -1.
Statement 2: Left hand derivative of f(x) at x = a is lim f (a − h) (Average)
h→0

A) Statement 1 is true, and Statement 2 is false.
B) Statement 1 is true, and Statement 2 is true, Statement 2 is correct Explanation for
Statement 1
C) Statement 1 is true, and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false, and Statement 2 is false.
52. Right hand derivative of f(x) = |x| at x = 1 is _______ (Average)
𝑑2 𝑦
53. If y =2cosx +3 sinx , then 𝑑𝑥 2 + y = __________ (Average)
54. Left hand derivative of f(x) = |𝑥 − 7| at x = 7 is _________ (Average)
55. The number of points in R at which the function f(x) =|x|+|x + 1|is not differentiable, is-...
(Average)
56. For the figure given below the slope of tangent to the curve at x = a is (Average)

A) 𝑓(𝑎) − 𝑎
B) lim (𝑓(𝑥))
𝑥→𝑎
C) lim (𝑓(𝑥) − 𝑓(𝑎))
𝑥→𝑎
D) 𝑓 / (𝑎).

SUBJECT: 35 – MATHEMATICS Page 40 of 164

57. For the figure given below, consider the following statements 1 and 2 (Difficult)
Statement 1: The given function is
differentiable at x=0
Statement 2: The given function is
continuous at x=1
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Both Statement 1and 2 are true
D) Both Statement 1 and 2 are false

58. Consider the following statements about the function where f(x) = | x - 2|
1. f(x) is not continuous at x = 2 2. f(x) is differentiable at x = 0 (Average)
Choose the correct statement?
A) Statement 2 is correct
B) Both statement 1 and statement 2 are correct
C) Both statements 1 and statement 2 are wrong D) Statement 1 is correct
59. The derivative of 2𝑥 + 3𝑦 = 𝑠𝑖𝑛𝑥 is (Average)
cos 𝑥+2 cos 𝑥−2 sin 𝑥−2 sin 𝑥+2
A) B) C) D) .
3 3 3 3
60. If the function f(x) = ax (Difficult)
1. Its domain is (∞, ∞) 2. It is a continuous function 3. It is differentiable at x = 0
Which of the above statements are correct?
A) 1 and 2 only B) 2 and 3 only C) 1 and 3 only D) 1, 2 and 3.
61. The conditions for a function to be continuous on (a, b)? (Difficult)
1) The function is continuous at each point of (a, b).
2) The function is right continuous at each point of (a, b).
3) The function is left continuous at each point of (a, b).
Which of the above statements are correct?
A) 1 and 2 only B) 2 and 3 only C) 1 and 3 only D) 1, 2 and 3.
𝑥 2 −2𝑥
62. The value of f(x),if the function f(x)= 𝑥−2 is continuous at the point x = 6 is (Average)
A) 12 B) 36 C) 6 D) 0
𝑥 2 −2𝑥
63. Consider the function f:ℛ − {2} → ℛ defined by f(x)= 𝑥−2 . What should be the value of f(2) if
f(x) is a continuous function from ℛ to ℛ ? Where ℛ is the set of all real numbers. (Average)
A) 2 B) -2 C) 0 D) 1
64. For the figure given below, consider the following statements 1,2 and 3 (Difficult)
Statement 1: The given function is
continuous and differentiable at x=2
Statement 2: The given function is not
continuous but differentiable at x=6 and x=8
Statement 3:The given function is
continuous but not differentiable at x=9
Then which of the following is true

A) Statement 1 is true and Statements 2 and 3 are false
B) Statements 1 and 3 are true but Statement 2 is false
C) Statements 2 and 3 are true but Statement 1 is false
D) All the Statements 1 ,2 and 3 are true.

SUBJECT: 35 – MATHEMATICS Page 41 of 164

65. For the figure given below, consider the following statements 1 and 2 (Difficult)

Statement 1: The given function is continuous and differentiable at x=0
Statement 2: The given function is not continuous and not differentiable at x=1
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Both Statement 1and 2 are true
D) Both Statement 1 and 2 are false
 1
 x sin , x  0
66. If f (x ) =  x is continuous at x = 0 , then the value of k is = ________ (Difficult)
 =
 k , x 0

67. If f (x ) =| x − 3 |, then f is continuous but not differentiable at x=___ (Easy)
𝑑
68. (3𝑥 𝑥 ) at x =1 is= ____________ (Average)
𝑑𝑥
d
69. The value of (|𝑥| − |𝑥 − 2|)at x = 1is __________ (Difficult)
dx
𝑑𝑦
70. If x = acos2θ , y = asin2θ , then = __________ (Difficult)
𝑑𝑥
TWO MARK QUESTIONS
1. Check the continuity of the function f given by f ( x) = 2 x + 3 at x = 1 . (Easy)
2. Examine whether the function f given by f ( x) = x is continuous at x = 0 .
2
(Average)
3. Discuss the continuity of the function f given by f ( x) = x at x = 0 . (Average)

 x3 + 3, for x  0
4. Show that the function f given by f ( x) =  is not continuous at x = 0 .(Average)
1, for x = 0
5. Check the points where the constant function f ( x) = k is continuous. (Easy)
6. Prove that the identity function on real numbers given by f ( x) = x is continuous
at every real number. (Easy)
7. Is the function defined by f ( x) = x , a continuous function? Justify your answer. (Average)

8. Discuss the continuity of the function f given by f ( x) = x + x − 1 .
3 2
(Average)
1
9. Discuss the continuity of the function f defined by f ( x) = , x 0. (Average)
x
10. Prove that the function f ( x) = 5 x − 3 is, continuous at (i) x = 0 (ii) x = −3 (iii) x = 5 (Easy)
11. Examine the continuity of the function f ( x) = 2 x − 1 at x = 3 .
2
(Average)
12. Examine the following functions for continuity: (Each question of 2 Marks)
a) f ( x ) = x − 5 b) f ( x ) = x − 5
x 2 − 25 1
c) f ( x ) = , x  −5 d) f ( x ) = , x5. (Average)
x+5 x −5
SUBJECT: 35 – MATHEMATICS Page 42 of 164

13. Prove that the function f ( x) = x is continuous at x = n , where n is a positive integer.
n

(Average)
14. Discuss the continuity of the following functions: (Each question is of 2 Marks)
a) f ( x) = sin x + cos x b) f ( x) = sin x − cos x c) f ( x) = sin x  cos x . (Average)
dy
15. If y = (2 x + 1) , find
3
. (Easy)
dx
16. Find the derivative of the function given by f ( x) = sin( x ) .
2
(Easy)
dy
17. Find , if y + sin y = cos x . (Average)
dx
dy
18. Find , if 2 x + 3 y = sin x . (Easy)
dx
dy
19. Find , if 2 x + 3 y = sin y . (Average)
dx
dy
, if ax + by = cos y .
2
20. Find (Average)
dx
dy
, if x + xy + y = 100 .
2 2
21. Find (Average)
dx
dy
, if sin x + cos y = 1 .
2 2
22. Find (Average)
dx
dy y
23. If x + y = 10 , show that + = 0. (Easy)
dx x
, if y = sin −1 
dy 2x 
24. Find 2 
, -1 < x < 1. (Average)
dx  1+ x 
dy  3x − x3  1 1
25. Find , if y = tan −1  2 
,− x . (Average)
dx  1 − 3x  3 3

−1  1 − x 
2
dy
26. If y = cos  2 
, 0  x  1 , find . (Average)
 1+ x  dx

−1  1 − x 
2
dy
27. Find , if y = sin  2 
, 0  x 1. (Average)
dx  1+ x 
, if y = cos −1 
dy 2x 
28. Find 2 
, −1  x  1 . (Average)
dx  1+ x 

29. Find
dy
dx (
, if y = sin −1 2 x 1 − x 2 , )
−1
2
x
1
2
. (Average)

, if y = sec −1  2
dy 1  1
30. Find , 0  x  . (Average)
dx  2x −1  2
dy
31. Find , if y = log a x . (Difficult)
dx
dy ex
32. Find , if y = . (Easy)
dx sin x

SUBJECT: 35 – MATHEMATICS Page 43 of 164

33. Find
dy
dx
, if y = sin tan e(
−1 − x
. ) (Difficult)

34. Find
dy
dx
(
, if y = log cos e .
x
) (Difficult)

dy
, if y = e + e + e ..... + e .
x x2 x3 x5
35. Find (Average)
dx
dy
, if y = e , x  0 .
x
36. Find (Average)
dx
dy cos x
37. Find , if y = , x  0. (Easy)
dx log x
dy
, if y = cos(log x + e ), x  0 .
x
38. Find (Average)
dx
39. Differentiate ( log x )
cos x
with respect to x . (Average)
dy
40. If y = x , find
x
. (Easy)
dx
x
 1
41. Differentiate  x +  w. r. to x . (Average)
 x
 1
dy  x+ 
42. Find , if y = x  x  . (Average)
dx

, if (i ) y = ( log x ) (ii ) y = x
dy x ( log x )
43. Find . (Average)
dx

, if (i ) y = ( sin x ) (ii ) y = sin
dy x −1
44. Find x. (Average)
dx
dy ( cos x )
, if (i) y = x , x  0 (ii) y = (sin x )
sin x
45. Find (Average)
dx
dy
46. Find , if y = log7 (log x ) . (Average)
dx

, if y = cos ( sin x ) .
dy −1
47. Find (Average)
dx
dy
, if y = (3x − 9 x + 5) .
2 9
48. Find (Easy)
dx

, if y = ( 5 x )
dy 3cos 2 x
49. Find . (Average)
dx
x
cos −1
dy 2 , −2 x  2
50. Find , if y = . (Average)
dx 2x + 7
, if y = ( log x ) , x  1 .
dy log x
51. Find (Average)
dx
dy
52. Find , if y = cos(a cos x + b sin x), for some constant ‘a’ and ‘b’ (Average)
dx

SUBJECT: 35 – MATHEMATICS Page 44 of 164

dy
, if y = x log x .
3
53. Find (Easy)
dx
dy
, if y = e sin 3x .
x
54. Find (Easy)
dx
dy
, if y = e cos3x .
6x
55. Find (Average)
dx
dy 2
56. Find , if y = sin(cos( x )) (Average)
dx
dy
, if y = sin x + cos x .
3 6
57. Find (Average)
dx

( )
58. If (i) y = sec tan( x ) (ii) y = cos x  sin ( x ) find
3 2 5 dy
dx
. (Average)

THREE MARK QUESTIONS
 x + 2, if x  1
1. Discuss the continuity of the function f defined by f ( x) =  . (Easy)
 x − 2, if x  1
 x + 2, if x  1

2. Find all the points of discontinuity of the function f defined by f ( x ) =  0, if x = 1 . (Average)
 x − 2, if x  1

𝑥 + 2, 𝑖𝑓 𝑥 ≤ 0
3. Discuss the continuity of the function f defined by 𝑓(𝑥) = { (Easy)
−𝑥 + 2 , 𝑖𝑓 𝑥 > 0
 x, if x  0
4. Discuss the continuity of the function f defined by f ( x) =  . (Average)
 x , if x  0
2

𝑥, 𝑖𝑓 𝑥 ≤ 1
5. Is the function f defined by 𝑓(𝑥) = { continuous at x = 0 ? At x = 1 ?At x = 2 (Average)
5, 𝑖𝑓 𝑥 > 1
6. Prove that the function f given by f ( x ) = x −1 , x  R is not differentiable at x = 1 . (Easy)
dy
7. If y = 2 cot( x2 ) , find . (Average)
dx
dy
8. Find , if x + sin xy − y = 0 . (Easy)
dx
dy
, if xy + y = tan x + y .
2
9. Find (Easy)
dx
dy
, if x + x y + xy + y = 81 .
3 2 2 3
10. Find (Easy)
dx
dy
, if sin x + cos xy = k .
2
11. Find (Easy)
dx
( x − 3)( x 2 + 4)
12. Differentiate with respect to x . (Easy)
3x 2 + 4 x + 5
dy
13. Find , if y x + x y + x x = a b . (Easy)
dx
dy
14. Find , if y = cos x  cos 2 x  cos 3 x . (Easy)
dx

SUBJECT: 35 – MATHEMATICS Page 45 of 164

( x − 1)( x − 2)
15. Differentiate with respect to x . (Average)
( x − 3)( x − 4)( x − 5)
dy
16. Find , if 𝑦 = x x − 2sin x . (Difficult)
dx

, if y = ( x + 3)  ( x + 4 )  ( x + 5 ) .
dy 2 3 4
17. Find (Easy)
dx
dy
, if x = y .
y x
18. Find (Average)
dx
dy x− y
19. Find , if xy = e . (Average)
dx
20. Find the derivative of the function given by f ( x) = (1 + x)(1 + x )(1 + x )(1 + x )
2 4 8

and hence find f (1). (Easy)
21. Differentiate ( x − 5 x + 8)( x + 7 x + 9) with respect to x, by using product rule.
2 3
(Easy)
dy x2 + 1
22. Find , if (i) y = x x cos x (ii) y = 2 . (Any One) (Average)
dx x −1
dy 1
, if (i) y = ( x cos x ) (ii) y = ( x sin x ) x (Any One)
x
23. Find (Average)
dx
dy
24. Find , if x = a cos  , y = a sin  . (Easy)
dx
dy
, if x = at , y = 2at .
2
25. Find (Easy)
dx

, if x = a ( + sin  ) , y = a (1 − cos ) .
dy
26. Find (Easy)
dx
2 2 2
dy
27. Find , if x 3 + y 3 = a 3 . (Average)
dx
dy
, if x = 2at , y = at .
2 4
28. Find (Average)
dx
dy
29. Find , if x = a cos  , y = b cos  . (Easy)
dx
dy
30. Find , if x = sin t, y = cos 2t . (Average)
dx
dy 4
31. Find , if x = 4t , y = . (Average)
dx t
dy
32. Find , if x = cos  − cos 2 , y = sin  − sin 2 . (Average)
dx
33. If x = a ( − sin  ) and y = a (1 + cos  ) then prove that
dy  
= − cot   . (Average)
dx 2
dy sin 3 t cos3 t
34. Find , if x = , y= . (Difficult)
dx cos 2t cos 2t
dy
35. Find , if x = a sec , y = b tan  . (Easy)
dx
SUBJECT: 35 – MATHEMATICS Page 46 of 164

, if x = a ( cos +  sin  ) , y = a ( sin  −  cos ) .
dy
36. Find (Average)
dx
−1 −1 dy y
37. If x = asin t and y = acos t , then prove that =− . (Average)
dx x
38. If x = a ( + sin  ) and y = a (1 − cos ) . Prove that
dy  
= tan   . (Average)
dx 2
dy 1
39. Find , if y = 3x + 2 + . (Difficult)
dx 2x + 4
2

dy 2
40. Find ,if y = esec x + 3cos−1 x . (Average)
dx
−1  2 x +1 
41. Find f ( x ) ,if f ( x) = sin  x . (Difficult)
 1+ 4 
42. Find f ( x ) ,if f ( x) = tan −1 
sin x 
. (Difficult)
 1 + cos x 
43. Find f ( x ) if f ( x) = ( sin x ) for all 0  x   .
sin x
(Average)
a
dy t+
1
 1
44. For a positive constant ‘a’ find , where y = a t and y =  t +  . (Difficult)
dx  t
45. Differentiate sin 2 x with respect to e cos x . (Average)
dy −1
 1 + sin x + 1 − sin x  
46. Find , if y = cot  ,0  x  .
dx  1 + sin x − 1 − sin x  2
(Difficult)
dy ( sin x −cos x )  3
47. Find , if y = ( sin x − cos x ) , x . (Average)
dx 4 4
dy
, if x + x + a + a , for some fixed a  0 and x  0 .
x a x a
48. Find (Easy)
dx
dy  
49. Find , if y = 12(1 − cos t ), x = 10(t − sin t ), −  t  . (Average)
dx 2 2
dy
50. Find , if y = sin −1 x + sin −1 1 − x 2 , 0  x  1 . (Easy)
dx
3
51. If f ( x) = x , show that f ( x) exists for all real x and find it. (Average)

dy
52. If y = cos x  sin ( x ) , find
3 2 5
. (Average)
dx
53. Prove that the greatest integer function defined by f ( x ) = [ x], 0  x  3 is not differentiable at

x = 1 and x = 2 . (Average)

SUBJECT: 35 – MATHEMATICS Page 47 of 164

FOUR MARK QUESTIONS
1. Find the relationship between ‘ a ’ and ‘ b ’ so that the function ‘ f ’ defined by
ax + 1, if x  3
f ( x) =  is continuous at x = 3. (Easy)
bx + 3, if x  3
 ( x 2 − 2 x ) , if x  0

2. For what value of  is the function defined by f ( x ) =  is
4 x + 1,
 if x  0
continuous at x = 0? What about continuity at x = 1? (Average)
 sin x
 , if x  0
3. Find all the points of discontinuity of f , where f ( x ) =  x . (Average)

 x + 1, if x  0
 2 1
 x sin , if x  0
4. Determine if f defined by f ( x ) =  x is continuous a function? (Average)

0, if x = 0
sin x − cos x, if x  0
5. Examine the continuity of f , where f is defined by f ( x ) =  . (Easy)
−1, if x = 0
 k cos x 
  − 2 x , if x  2 
6. Determine the value of k, if f ( x) =  is continuous at x = . (Easy)
 3,  2
if x =
 2
kx 2 , if x  2
7. Find the value of k if ( ) 
f x = is continuous at x = 2 . (Easy)
3, if x  2
kx + 1, if x  
8. Find the value of k so that the function f ( x ) =  ,is continuous at x =  . (Easy)
cos x, if x  
kx + 1, if x  5
9. Find the value of k so that the function f ( x ) =  , is a continuous at x = 5 . (Easy)
3x − 5, if x  5
 5, if x  2
10. Find the values of a and b such that f ( x) = ax + b, if 2  x  10
is a continuous function.(Average)

 21, if x  10

ADDITIONAL QUESTIONS:
2 x + 3, if x  2
11. Find all points of discontinuity of f , where f is defined by: f ( x ) =  (Easy)
2 x − 3, if x  2
 x + 3, if x  −3

12. Find all points of discontinuity of f ,where f is defined by: f ( x ) = −2 x, if − 3  x  3 (Average)
6 x + 2, if x  3

 x + 1, if x  1
13. Find all points of discontinuity of f , where f is defined by: f ( x ) =  2 (Easy)
 x + 1, if x  1
x
 , if x  0
14. Find all points of discontinuity of f ,where f is defined by: f ( x ) =  x (Average)
−1, if x  0


SUBJECT: 35 – MATHEMATICS Page 48 of 164

x
 , if x  0
15. Find all points of discontinuity of f , where f is defined by: f ( x ) =  x (Average)
0, if x = 0

 x3 − 3, if x  2
16. Find all points of discontinuity of f , where f is defined by: f ( x ) =  2 (Easy)
 x + 1, if x  2
 x10 − 1, if x  1
17. Find all points of discontinuity of f , where f is defined by: f ( x) =  2 . (Easy)
 x , if x  1
 x + 5, if x  1
18. Is the function defined by f ( x ) =  a continuous function? (Easy)
 x − 5, if x  1
3, if 0  x  1

19. Discuss the continuity of the function f , where f is defined by: f ( x ) = 4, if 1  x  3
5, if 3  x  10

(Average)
 2 x, if x  0

20. Discuss the continuity of the function f ,where f is defined by: f ( x ) = 0, if 0  x  1
 4 x, if x  1

(Average)
−2, if x  −1

21. Discuss the continuity of the function f ,where f is defined by: f ( x ) = 2 x, if − 1  x  1
2, if x  1

(Average)
dy 1
22. If x 1 + y + y 1 + x = 0, for − 1  x  1 and 𝑥 ≠ 𝑦. Prove that =− . (Average)
dx (1 + x) 2
dy x2 −3 x2
23. Find , if y = ( x ) + ( x − 3) , for x  3 . (Average)
dx
dy
, if x + y = 1 .
y x
24. Find (Average)
dx
dy cos ( a + y )
2

25. If cos y = x cos ( a + y ) with cos a  1 prove that = (Easy)
dx sin a
dy y
26. If x = a cos3 and y = a sin3 , prove that =−3 . (Average)
dx x

, if x = a  cos t + log tan  , y = a sin t .
dy t
27. Find (Average)
dx  2

SUBJECT: 35 – MATHEMATICS Page 49 of 164

FIVE MARKS QUESTIONS
d2y dy
1. If y = 3 e + 2 e3 x , then prove that
2x
2
− 5 + 6 y = 0. (Easy)
dx dx
2
−1 d y dy
2. If y = sin x , then prove that (1 − x 2 ) 2 − x = 0. (Easy)
dx dx
d2y
3. If y = 5cos x − 3sin x , then prove that + y = 0. (Easy)
dx 2
d2y
4. If y = cos −1 x, find in terms if y alone (Average)
dx 2
5. If y = 3cos(log x) + 4sin(log x), show that x 2 y2 + xy1 + y = 0. (Easy)
2
6. If y = Ae
mx
+ Benx , prove that d 2y − (m + n) dy + (mn) y = 0. (Easy)
dx dx
7. If y = 500 e
7x
+ 600 e−7 x , show that y2 = 49 y. (Easy)
2 2
d y  dy 
8. If e ( x + 1) = 1 , Prove that
dy
y
= −e y hence prove that =  . (Average)
dx dx2  dx 
( )
2
9. If y = tan −1 x , show that (1 + x ) y2 + 2 x(1 + x ) y1 = 2.
2 2 2
(Easy)
ADDITIONAL QUESTIONS:
a cos−1 x d2y dy
10. If y = e , −1  x  1 , show that (1 − x 2 )
2
−x − a 2 y = 0. (Difficult)
dx dx
d2y
12.If x = a(cos t + t sin t ) and y = a(sin t − t cos t ), find . (Difficult)
dx 2
3

13. If ( x − a ) + ( y − b ) = c2 , for some c  0, prove that
2 2 (
1 + y12 2) (Difficult)
y2
is a constant independent of a and b.

SUBJECT: 35 – MATHEMATICS Page 50 of 164

CHAPTER -06
APPLICATION OF DERIVATIVES
MCQ /FB questions:
1. The rate of change of the area of a circle with respect to its radius r when (Average)
2
r = 5cm is.... c𝑚 /cm.
A)10𝜋 B)12 𝜋 C)8 𝜋 D)11 𝜋
2. The rate of change of the area of a circle with respect to its radius r when r = 4 cm is....
....𝜋c𝑚2 /cm. (Average)
A)10 B)12 C)8 D)11
3. The total revenue in Rupees received from the sale of x units of a product is given by
R (x)=13𝑥 2 + 26𝑥 − 15 , then the marginal revenue when x = 1 is ....... Rupees (Difficult)
A)26 B)13 C)52 D)104
4. The total revenue in rupees received from the sale of x units of a product is
R (x) = 3𝑥 2 + 36𝑥 + 5, then the marginal revenue when x = 15 is ....... Rupees (Difficult)
A)116 B)96 C)90 D)126
5. The radius of circle is increasing at the rate of 0.7 cm/sec, then the rate of increase of its
circumference is _____ 𝜋 cm/sec (Easy)
A)2 B)1.4 C)0.7 D)4.9
1
6. The radius of an air bubble is increasing at the rate of 2cm/s, then the volume of the bubble
increasing when the radius is 1 cm is ____cc/sec (Average)
A)2 𝜋 B) 2 C) 2 𝜋 D) 8 𝜋
7. The function f ( x) = cos x is increasing in the interval…… (Average)
π π
A)(0, 2 ) B)(0, π) C)( 2 , π) D)(π, 2π)
8. The function f ( x) = 3x + 17 is strictly increasing on ……. (Average)
A)(−∞, ∞) B)(0, ∞) C)(−∞, 0) D)(0, 3).
2 −x
9. The interval in which y = x e is increasing is…….. (Difficult)
A)(−∞, 0) B)(−∞, 0) ∪ (2, ∞) C)(2, ∞) D)(0, 2)
2 −x
10. The interval in which y = x e is decreasing is…….. (Difficult)
a)(−∞, 0) B)(−∞, 0) ∪ (2, ∞) C)(2, ∞) D)(0, 2)
11. The minimum value of x in R is…… (Easy)
A) 0 B) 1 C) 2 D) does not exist.
12. The maximum value of the function f (x) = |x| is.... in R. (Average)
A) 0 B) 1 C) 2 D) does not exist.
13. The maximum values of the function given by f(x) = x, x ∈ [0, 1] is ...... (Easy)
A) 0 B) 1 C) 2 D) does not exist.
14. The minimum values of the function given by f(x) = x, x ∈ [0, 1] is ...... (Easy)
A) 0 B) 1 C) 2 D) does not exist.
15. For a function ‘ f ’ defined on an interval I if f (c) = 0 and f (c)  0 for some c  I then at ‘c’ the
function f attains ___ (Average)
A) the absolute maximum value B) the absolute minimum value
C) a local maximum value. D) a local minimum value.

SUBJECT: 35 – MATHEMATICS Page 51 of 164

16. A stone is dropped into a quiet lake and waves move in circles at a speed of 4cm per
second. At the instant, when the radius of the circular wave is 10 cm, the enclosed area is
increasing at the rate of _____ 𝑐𝑚2 /sec (Average)
A)20𝜋 B)40 𝜋 C)80 𝜋 D)60 𝜋
17. Point of minimum value of the function given by f (x) = |x| is x = _____ (Average)
A)1 B)2 C)0 D) does not exist
18. The number of points of local maxima and local minima of the function f given by
f(x) = 𝑥 3 – 3x +3 is ______ (Average)
A)0 B)1 C)2 D) 3.
3
19. The absolute maximum value of the function f given by f (x) = 𝑥 ,x ∈ [-2, 2] is = ___ (Average)
A)-2 B)2 C)0 D)8.
2
20. The function f given by f (x) = 𝑥 – x + 1 is. (Average)
1
A) neither strictly increasing nor decreasing on (– 1, 1) B) decreasing on ( , ∞)
2
1
C) increasing on (–∞,∞) D) increasing on (−∞, 2)
𝜋
21. Which of the following functions is decreasing on (0, 2 ) (Average)
A) cosx B) sinx C) cos 3x D) tan x
22. The function f given by f (x) = 𝑥 2 + ax + 1 is increasing on [1, 2], then (Difficult)
A) a < -2 B) 𝑎 < 2 C) 𝑎 > −2 D) 𝑎 < 0
23. The function f given by 𝑓(𝑥) = 𝑙𝑜𝑔|𝑠𝑖𝑛𝑥|is increasing on (Difficult)
π π 3π
A) (0, 2 ) B)(0, π) C)( 2 , π) D)( 2 , 2π)
24. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314cubic
meter per hour. Then the depth of the wheat is increasing at the rate of (Average)
A)1 m/h B) 0.1 m/h C) 1.1 m/h D) 0.5 m/h
25. The maximum and minimum values of the function |𝑠𝑖𝑛4𝑥 + 3| are respectively (Difficult)
A)1,2 B)4,2 C)2,4 D)-1,1
26. On which of the following intervals is the function f given by f (x) = 𝑥 100 + sin x –1
decreasing? (Difficult)
π π
A)(0, 2 ) B)(0,1) C)( 2 , π) D) (0, π)
27. If a function f is such that 𝑓 / (c) = 0 and 𝑓 // (c) < 0 for some ‘c’ on an interval ‘I’, then
at c the function f attains (Easy)
A) the absolute maximum value B) the absolute minimum value
C) a local maximum value D) a local minimum value
28. The edge of a cube is increasing at the rate of 5cm / sec. How fast is the volume of the cube
increasing when the edge is 12cm long (Average)
3
A) 432 cm / sec 3
B) 2160 cm / sec 3
C) 180 cm / sec D) 5 12 cm / sec
3 3

29. The radius of an air bubble is increasing at the rate of 1cm/s, then the volume of the
bubble increasing when the radius is 1cm is _______ πcc/sec (Average)
30. Minimum value of f(𝑥) = |𝑥 + 2| − 1 is ________ (Average)
31. Local maxima of f(𝑥) = sin 𝑥 + cos𝑥, 0 <𝑥<𝜋 is ________ (Average)
2
32. If 𝑥 is a real, the minimum value of 𝑥 − 8𝑥 + 17 is _______ (Difficult)
33. The maximum value of 𝑠𝑖𝑛𝑥.𝑐𝑜𝑠𝑥 is _________ (Average)
34. If x + y = 10 , then the maximum value of 𝑥𝑦 ________ (Average)

SUBJECT: 35 – MATHEMATICS Page 52 of 164

35. Assertion (A) : The function f(x) = x2 is decreasing in the interval (0, ∞)
Reason (R) : Any function y = f(x) is decreasing, if dy  0 (Difficult)
dx
A) A is false and R is true B) A is false and R is false
C) A is true and R is false D) A is true and R is true.
36. Statement-1 : Intervals in which the function f ( x ) = x2 − 4 x + 6 is increasing is (2, ∞).
Statement-2: Let f be continuous on [a, b] and differentiable on the open interval (a, b).
Then f is an increasing function in [a, b] if f ′(x) > 0 for each x  (a, b). (Difficult)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false.
37. Assertion (A): A particle moving in a straight line covers a distance of x cm in t
seconds, where x = t 3 + 3t 2 − 6t + 18 . The velocity of particle at the end of 3 seconds is 9cm/s
𝑑𝑥
Reason (R): Velocity of the particle at the end of 3 seconds is
𝑑𝑡
𝑎𝑡 𝑡 = 3 (Difficult)
A) A is false and R is true B) A is true and R is false
C) A is true and R is true D) A is false and R is false
38. Statement-1: The function f (x) = x3 – 12x is increasing in (– ∞, – 2) ∪ (2, ∞).
Statement-2: For increasing function f in an open interval I , f ′(x) > 0 for all 𝑥 ∈ 𝐼. (Difficult)
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Statement 1 is true and Statement 2 is true
D) Statement 1 is false and Statement 2 is false
39. Assertion (A): The maximum value of the function f(x) = x5 , x ∈ [–1, 1], is attained at its critical
point, x = 0.
Reason (R): The local maximum or local minimum values of a function can only occur at
turning points. (Difficult)
A) A is false and R is true B) A is false and R is false
C)A is true and R is false D) A is true and R is true.
40. Statement-1: The rate of change of area of a circle with respect to its radius r when
r = 6 cm is 12π cm2 /cm.
dA
Statement-2 : Rate of change of area of a circle with respect to its radius r is , where A is
dr
the area of the circle. (Difficult)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
D) Statement 1 is false and Statement 2 is false.
41. The point of inflection for the following graph is
(Difficult)

𝜋 𝜋
(A) − 2 B) 2

C) 0 D) point of inflection does not exist

SUBJECT: 35 – MATHEMATICS Page 53 of 164

42. The value of x for local maxima, local minima and inflection
for the following graph of y = f(x) respectively are (Difficult)
A) −3, 3, 1
B) −3, 3, 0
C) -1, 0, 1
D) -3, 1, 3
43. Assertion (A): The maximum value of the function f(x) = 𝑥 3 , x ∈[-1, 1], is attained at its end
point, x = 1.
Reason (R): The function f(x) = 𝑥 3 is an increasing function in [-1, 1]. (Difficult)
A) Both A and R are true and R is the correct explanation for A
B) Both A and R are true but R is not the correct explanation for A.
C) A is false and R is true. D) Both A and R are false.
3
44. The absolute maximum value of y = 𝑥 – 3x + 2 in 0 ≤ x ≤ 2 is (Average)
A) 0 B) 2 C) 4 D) 6
45. The function f(x) = x + cos x ( for x≠n𝜋 ) is (Difficult)
A) Always increasing B) Always decreasing
C) Increasing for a certain range of x D) Decreasing for a certain range of x

46. Let the f: R → R be defined by f(x) = 2x + cos x , ( for x≠n𝜋 ) then (Difficult)
1. f has a maximum value at x = 0 2. f is an increasing function 3. f is a decreasing function
Which of the above statements are correct?
A) 1 only B) 2 only C) 3 only D) All 1, 2 and 3.
47. Consider the function
1.f(x) = 𝑒 −𝑥 2. f(x) = 𝑥 2 − 𝑠𝑖𝑛𝑥 3.f(x) = √𝑥 3 + 1
Which of the above functions is/ are increasing in [ 0,1 ] (Difficult)
A) 2 only B) 2 and 3 only C) 3 only D) 1 and 3 only.
48. The point(s) on the curve y = x², at which y-coordinate is changing four times as fast as
x- coordinate is/are (Average)
A) (4, 16) B) (2, 4) C) (-2, 4) D) (2, 4), (-2, 4).
49.The function f(x) = x – 5x + 5x – 1 has
5 4 3 (Average)
A) 1 critical point B) 2 critical points C) 3 critical points D) 4 critical points
50. The volume of a sphere is increasing at the rate of 𝜋 𝑐𝑚3 /𝑠𝑒𝑐. The rate at which the radius
is increasing is_____, when the radius is 3cm. (Average)
1
A) 36 𝑐𝑚/𝑠𝑒𝑐 B) 36𝑐𝑚/𝑠𝑒𝑐 C) 9 𝑐𝑚/𝑠𝑒𝑐 D) 27𝑐𝑚/𝑠𝑒𝑐
51. The function f(x) = 𝑥 𝑥 decreasing in (Average)
1
A) (0, e) B) (0, ∞) C) (0, 𝑒) D) (0, 1).
52. For the figure given below, consider the following statements 1,2 and 3 (Difficult)
Statement 1: f has local maximum values at x = 𝐶, 𝐸, G
Statement 2: f has local minimum values at x = 𝐷, 𝐹, H
Statement 3: f has neither local maximum nor local
minimum value at x = 𝐷, I.
A) Statement 1 is true and Statement 2 and 3 are false.
B) Statement 1 and 2 are true but Statement 3 is false.
C) Statement 1and 3 are true but Statement 2 is false.
D) Statement 2 and 3 are true but Statement 1 is false.

SUBJECT: 35 – MATHEMATICS Page 54 of 164

53. For the figure given below, consider the following statements 1 and 2 (Difficult)

Statement 1: The absolute maximum value of the function y= 𝑓(𝑥), x ∈ [a, b], is f(e)
Statement 2: The absolute minimum value of the function y= 𝑓(𝑥), x ∈ [a, b], is f(d)
A) Statement 1 is true and 2 is false. B) Statement 1 is true and 2 is true
C) Statement 1 and 2 are false. D) Statement 1 is false and 2 is true.
54. For the figure given below, consider the following statements 1 and 2 (Difficult)

Statement 1: The function y = 𝑓(𝑥) is decreasing in the interval(𝑏, 𝑐).
Statement 2: The absolute minimum value of the function y = 𝑓(𝑥), x ∈ [a, f], is f(a).
A) Statement 1 is true and 2 is false B) Statement 1 and 2 are true
C) Statement 1 and 2 are false D) Statement 1 is false and 2 is true
55. The function 𝑓(𝑥) = 𝑡𝑎𝑛𝑥 − 𝑥 (Average)
A) always increases B) always decreases
C) never increases D) sometimes increases and sometimes decreases.
56. The maximum value of 3𝑠𝑖𝑛𝑥 + 4 𝑐𝑜𝑠𝑥 is (Average)
A) 25 B) 5 C)√5 D) √7.
−𝑥
57. Statement 1 : The function 𝑦 = 𝑒 is ever increasing in the set of real numbers 𝑅.
Statement 2 : The function 𝑦 = 𝑙𝑜𝑔𝑒 𝑥 is ever decreasing in (0, ∞). (Difficult)
A) Statement 1 is false and Statement 2 is true
B) Statement 1 is true and Statement 2 is false
C) Both Statements 1 and 2 are true D) Both Statements 1 and 2 are false.
2
58. Consider the function 𝑓 given by 𝑓(𝑥) = −𝑥
Statement 1: The function 𝑓 is strictly increasing in (−∞, 0).
Statement 2: The function 𝑓 is neither increasing nor decreasing in (−∞, ∞). (Difficult)
A) Statement 1 is false and Statement 2 is true
B) Statement 1 is true and Statement 2 is false
C) Both Statements 1 and 2 are true D) Both Statements 1 and 2 are false.
59. Consider the function 𝑓 given by 𝑓(𝑥) = (𝑥 − 1)3
Statement 1:𝑥 = 1 is a point of inflection of 𝑓.
Statement 2: An interior critical point which is neither a point of local maxima nor a point
of local minima is a point of inflection. (Difficult)
A) Statement 1 is false and Statement 2 is true
B) Statement 1 is true and Statement 2 is false
C) Statement 1 is true, statement 2 is true and statement 2 is a correct explanation for
statement 1
D) Statement 1 is true, statement 2 is true and statement 2 is not a correct
explanation for statement 1.
SUBJECT: 35 – MATHEMATICS Page 55 of 164

60. Consider the function 𝑓 given by 𝑓(𝑥) = |𝑥| + 3
Statement 1: The critical point of 𝑓 is 0.
Statement 2: Local minimum value of 𝑓 𝑎𝑡 𝑥 = 0 𝑖𝑠 3. (Difficult)
A) Statement 1 is false and Statement 2 is true
B) Statement 1 is true and Statement 2 is false
C) Both Statements 1 and 2 are true
D) Both Statements 1 and 2 are false

TWO MARK QUESTIONS
1. The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its
circumference? (Average)
2. The total cost C ( x ) in Rupees, associated with the production of x units of an item is given by
C ( x) = 0.005x3 – 0.02 x 2 + 30 x + 5000 . Find the marginal cost when 3 units are produced, where by
marginal cost we mean the instantaneous rate of change of total cost at any level of output.
(Average)
3. The total revenue in Rupees received from the sale of x units of a product is given by
R( x) = 3x2 + 36 x + 5 . Find the marginal revenue, when x = 5 , where by marginal revenue
we mean the rate of change of total revenue with respect to the number of items sold at an
instant. (Average)
4. The total cost C ( x ) in Rupees associated with the production of x units of an item is given by
C ( x) = 0.007 x3 – 0.003x2 + 15 x + 4000 .Find the marginal cost when 17 units are produced.
(Average)
5. The total revenue in Rupees received from the sale of x units of a product is
given by R( x) = 13x +26 x + 15 . Find the marginal revenue when x = 7 .
2
(Average)
6. The total revenue in Rupees received from the sale of x units of a product is given by
R( x) = 3x2 + 36 x + 5 . Find the marginal revenue, when x = 15 . (Average)
7. The total cost C(x) in Rupees, associated with the production of x units of an item is
given by C(x) = 0.005 𝑥 3 – 0.02 𝑥 2 + 30x + 5000. Then the marginal cost when 3 units are
produced, where by marginal cost we mean the instantaneous rate of change of total cost at
any level of output, is _____ (Difficult)
8. The radius of a circle is increasing uniformly at the rate of 3 cm / s . Find the rate at which the
area of the circle is increasing when the radius is 10 cm . (Easy)
9. Find the rate of change of the area of a circle with respect to its radius r when r = 6 cm .(Easy)
10. A stone is dropped into a quiet lake and waves move in circles at a speed of 4cm per second.
At the instant, when the radius of the circular wave is 10 cm , how fast is the enclosed area
increasing? (Average)
11. Show that the function given by f ( x) = 7 x – 3 is increasing on R. (Easy)
12. Show that the function f given by f ( x) = x − 3x + 4 x , x  R is increasing on R.
3 2
(Average)
13. Find the interval in which the function f given by f ( x) = 2 x 2 − 3x is increasing. (Easy)
14. Show that the function given by f ( x) = cos x is
(i) decreasing in (0,  ) (ii) increasing in ( , 2 )
(iii) neither increasing nor decreasing in (0, 2 ) (Each sub question carries 2 marks) (Easy)

SUBJECT: 35 – MATHEMATICS Page 56 of 164

15. Show that the function given by f ( x) = 3 x + 17 is increasing on R. (Easy)
16. Show that the function given by f ( x) = e2 x is increasing on R. (Average)
17. Show that the function given by f ( x) = sin x is
 
(i) increasing in (0, 2 )
𝜋
(ii) decreasing in  , 
2 
(iii) neither increasing nor decreasing in ( 0,  ) (Each sub question carries 2 marks) (Easy)
18. Find the intervals in which the function f given by f ( x) = 2 x 2 − 3x is
(i) strictly increasing. (ii) strictly decreasing (Each sub question carries 2 marks) (Average)
19. Prove that the logarithmic function is increasing on (0, ) . (Easy)
𝜋
20. Show that the function given by f ( x) = cos 2 x is decreasing in (0, 2 ). (Easy)
𝜋
21. Show that the function given by f ( x) = cos 3 x is decreasing in (0, ). (Easy)
3
𝜋
22. Show that the function given by f ( x) = tan x is increasing in (0, 2 ). (Easy)
3 2
23. Prove that the function f given by 𝑓(𝑥) = 𝑥 − 3𝑥 + 3𝑥 − 100 is increasing in R. (Average)
24. Prove that the function f given by f ( x) = x 2e− x is increasing in ( 0, 2 ) . (Average)
25. Find the maximum and the minimum values of the following functions f given by
(i) f ( x) = x , x  R . (ii) f ( x) = x , x  R (iii) f ( x) = x, x  (0,1)
2
(Average)
(Each sub question carries 2 marks)
26. Prove that the following functions do not have maxima or minima.
(ii) f ( x) = e (iii) h( x) = x + x + x + 1
x 3 2
(i) g ( x) = log x (Average)
(Each sub question carries 2 marks)
27. It is given that at x = 1 , the function f ( x) = x – 62 x + ax + 9 attains its maximum value, on the
4 2

interval [0, 2] . Find the value of ‘a’. (Difficult)
28. Find all points of local maxima and local minima of the following functions f given by
(i) f ( x) = x – 3x + 3 . (ii) f ( x) = 2 x – 6 x + 6 x + 5
3 3 2

(Each sub question carries 2 marks) (Average)
29. Find local minimum value of the function f given by f ( x) = 3 + x , x  R. (Easy)
log x
30. Show that the function given by f ( x) = has maximum at x = e . (Average)
x
31. At what points in the interval [0, 2 ] , does the function sin 2x attain its maximum value?
(Average)
THREE MARK QUESTIONS
1. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900
cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases
when the radius is 15 cm. (Average)
1
2. The radius of an air bubble is increasing at the rate of cm / s . At what rate is the volume of the
2
bubble increasing when the radius is 1cm ? (Average)
3
3. A balloon, which always remains spherical, has a variable diameter ( 2 x + 1) .
2
Find the rate of change of its volume with respect to x . (Average)
SUBJECT: 35 – MATHEMATICS Page 57 of 164

4. The length x of a rectangle is decreasing at the rate of 3 cm / minute and the width y is increasing
at the rate of 2 cm / minute . When x = 10 cm and y = 6 cm , find the rate of change of the
perimeter. (Average)
5. The length x of a rectangle is decreasing at the rate of 3 cm / minute and the width y is increasing
at the rate of 2 cm / minute . When x = 10 cm and y = 6 cm , find the rate of change of the area of
the rectangle. (Average)
6. The length x of a rectangle is decreasing at the rate of 5 cm / minute and the width y is
increasing at the rate of 4 cm / minute . When x = 8 cm and y = 6 cm , find the rates of
change of the area of the rectangle. (Average)
7. The length x of a rectangle is decreasing at the rate of 5 cm / minute and the width y is
increasing at the rate of 4 cm / minute . When x = 8 cm and y = 6 cm , find the rates of
change of the perimeter. (Average)
3
8. The volume of a cube is increasing at the rate of 8 cm / s . How fast is the surface area
increasing when the length of an edge is 12 cm ? (Difficult)
9. The volume of a cube is increasing at a rate of 9 cubic centimeters per second. How fast is the
surface area increasing when the length of an edge is 10 centimeter? Average)
10. An edge of a variable cube is increasing at the rate of 3 cm / s . How fast is the
volume of the cube increasing when the edge is 10 cm long? (Average)
11. A car starts from a point P at time t = 0 seconds and stops at point Q. The distance x , in

meters, covered by it, in t seconds is given by x = t 2  2 −  .
t
 3
Find the time taken by it to reach Q (Difficult)
12. Find the intervals in which the function f given by f ( x) = x − 4 x + 6 is
2

(i) strictly increasing (ii) strictly decreasing (Easy)
13. Find the intervals in which the function f given by f ( x) = 4 x − 6 x − 72 x + 30 is
3 2

(i) strictly increasing (ii) strictly decreasing (Average)

14. Find intervals in which the function given by f ( x) = sin 3x, x  0,  is
 2
(i) increasing (ii) decreasing (Easy)
15. Find the intervals in which the function f given by f ( x) = 2 x3 − 3x 2 − 36 x + 7 is
decreasing. (Average)
16. Find the intervals in which the function f given by 10 – 6 x – 2 x 2 is
(i) increasing (ii) decreasing (Average)
17. Find the intervals in which the function f given by 6 – 9 x – x 2 is increasing. (Average)
18. Prove that the function f given by f ( x) = x 2 − x + 1 is neither increasing nor
decreasing on (–1,1) . (Average)
19. Show that the function given by f ( x) = x 100
+ sin x − 1 is
 
(i) increasing in (0,1) (ii) increasing in  ,  𝜋
(iii) increasing in (0, 2 ) (Average)
2 
20. Find the least value of ‘ a ’such that the function f given by f ( x) = x 2 + ax + 1 is
increasing on (1, 2) . (Average)

SUBJECT: 35 – MATHEMATICS Page 58 of 164

21. Prove that the function f given by f ( x) = log(sin x) is
 
𝜋
increasing on (0, 2 ) and decreasing on  ,  . (Average)
2 
22. Prove that the function f given by f ( x) = log(cos x) is
 
𝜋
decreasing on (0, 2 ) and increasing on  ,  . (Average)
2 
23. Find the intervals in which the function f given by f ( x) = sin x + cos x , 0  x  2 is
increasing or decreasing. (Average)
24. Find the intervals in which the following functions are increasing or decreasing:
(i) –2 x3 – 9 x 2 –12 x + 1 (ii) ( x + 1)3 ( x − 3)3 . (Difficult)
25. Show that y = log(1 + x) − 2 x , x  −1, is an increasing function of x throughout its domain.
2+ x
(Difficult)
26. Find the values of x for which y =  x( x − 2)  is an increasing function.
2
(Average)
4sin   
27. Prove that y = −  is an increasing function of  in 0,  . (Difficult)
(2 + cos  )  2
3 4 4 3 36
28. Find intervals in which the function given by f ( x) = x − x − 3x 2 + x + 11 is (Difficult)
10 5 5
(a) increasing (b) decreasing.

−1
29. Show that the function f given by f ( x) = tan (sin x + cos x), x  0 is always an

increasing function in  0,  . (Difficult)
 4
4sin x − 2 x − x cos x
30. Find the intervals in which the function f given by f ( x) = is
2 + cos x
(i) increasing (ii) decreasing. (Difficult)
1
31. Find the intervals in which the function f given by f ( x) = x3 + , x  0 is
x3
(i) increasing (ii) decreasing. (Average)
32. Find local maximum and local minimum values of the function f given by
f ( x) = 3x 4 + 4 x3 –12 x 2 + 12 . (Average)
33. Find the maximum and minimum values, if any, of the function given by (Easy)
(i) f ( x) = (2 x –1)2 + 3 (ii) f ( x) = 9 x2 + 12 x + 2 (iii) f ( x) = −( x –1)2 + 10
(iv) g ( x) = x + 1 (v) f ( x) = x + 2 –1 (vi) g ( x) = − x + 1 + 3
3

(vii) h( x) = sin(2 x) + 5 (viii) h( x) = sin 4 x + 3 + 5 (ix) h( x) = x + 1, x  (−1,1)
(Each sub question carries 3 marks)
34. Find both the maximum value and the minimum value of 3x – 8 x + 12 x – 48 x + 25 on
4 3 2

the interval [0,3] . (Difficult)
35. Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.
(Average)
36. Find two numbers whose sum is 24 and whose product is as large as possible. (Average)
37. Find two positive numbers x and y such that x + y = 60 and xy is maximum.
3
(Difficult)

SUBJECT: 35 – MATHEMATICS Page 59 of 164

38. Find two positive numbers x and y such that their sum is 35 and the product x 2 y5 is a
maximum. (Difficult)
39. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum
(Difficult)
40. Find the local maxima and local minima of the function g ( x) = x − 3x . Also find the local
3

maximum and the local minimum values. (Average)
41. Find the local maxima and local minima of the function f ( x) = x . Also find the local maximum
2

and the local minimum values. (Average)
x 2
42. Find the local maxima and local minima of the function g ( x) = + , x0.
2 x
Also find the local maximum and the local minimum values. (Average)
43. Find the local maxima and local minima of the function f ( x) = sin x − cos x, 0  x  2 .
Also find the local maximum and the local minimum values. (Average)
44. Find the local maxima and local minima of the function 𝑓(𝑥) = 𝑥√1 − 𝑥, 0 < 𝑥 < 1.
Also find the local maximum and the local minimum values. (Difficult)
1
45. Find the local maxima and local minima of the function g ( x ) = .
x +22

Also find the local maximum and the local minimum values. (Average)
46. Find the absolute maximum value and the absolute minimum value of the function
f ( x) = ( x − 1)2 + 3, x [– 3,1] . (Average)
47. Find the absolute maximum value and the absolute minimum value of the function
f ( x) = x3 , x [– 2, 2] . (Average)

48. Find the absolute maximum and minimum values of a function f ( x) = 2 x –15 x + 36 x + 1 on the
3 2

interval [1,5] . (Average)
49. Find the absolute maximum value and the absolute minimum value of the function
f ( x) = sin x + cos x, x  [0,  ] . (Average)
50. Find the absolute maximum and the absolute minimum values of the function f given by

f ( x) = cos2 x + sin x, x [0,  ] . (Difficult)
51. Find the maximum profit that a company can make, if the profit function is given by
p( x) = 41– 72 x –18x 2 . (Average)

52. Find the maximum value of 2 x – 24 x + 107 in the interval [1,3] .
3
(Average)
53. Find the maximum and minimum values of x + sin 2x on [0, 2 ] . (Average)
1 − x + x2
54. For all real values of x , find the minimum value of . (Difficult)
1 + x + x2

SUBJECT: 35 – MATHEMATICS Page 60 of 164

ADDITIONAL QUESTIONS
1. A water tank has the shape of an inverted right circular cone with its axis vertical and vertex
−1
lowermost. Its semi-vertical angle is tan (0.5) . Water is poured into it at a constant rate of 5
cubic meter per hour. Find the rate at which the level of the water is rising at the instant
when the depth of water in the tank is 4m. (Difficult)
2. A sand is pouring from a pipe at the rate of 12cm3/s. The falling sand form a cone on the
ground in such a way that the height of the cone is always one-sixth of the radius of the base.
How fast is the height of the sand cone increasing when the height is 4cm ? (Difficult)
3. A ladder 5 m long is leaning against a wall. The bottom of the ladder is Pulled along the ground,
Away from the wall at the rate of 2 m / sec .How fast is its height on the wall decreasing when
the foot of the ladder is 4 m away from the wall? (Difficult)
4. A ladder 24 feet long leans against a vertical wall. The lower end is moving away at the rate of
3feet/sec. find the rate at which the top of the ladder is moving downwards, if its foot is 8feet
from the wall. (Difficult)
5. A man of height 2 meters walks at a uniform speed of 5 km/hour, away from a lamp post which
is 6 meters high. Find the rate at which the length of the his shadow increases. (Difficult)
6. A particle moves along the curve 6 y = x3 + 2 . Find the points on the curve at which the
y − coordinate is changing 8 times as fast as the x − coordinate. (Average)
7. If length of three sides of a trapezium other than base are equal to 10cm, then find the area of
the trapezium when it is maximum. (Difficult)
8. Prove that the radius of the right circular cylinder of greatest curved surface area which
can be inscribed in a given cone is half of that of the cone. (Difficult)
9. An Apache helicopter of enemy is flying along the curve given by y = x + 7 . A soldier,
2

placed at (3, 7) , wants to shoot down the helicopter when it is nearest to him.
Find the nearest distance. (Difficult)
10. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square
from each corner and folding up the flaps to form the box. What should be the side of the
square to be cut off so that the volume of the box is the maximum possible. (Difficult)
11. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off
square from each corner and folding up the flaps. What should be the side of the square to be
cut off so that the volume of the box is maximum ? (Difficult)
12. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum
area. (Difficult)
13. Show that the right circular cylinder of given surface and maximum volume is such that its
height is equal to the diameter of the base. (Difficult)
14. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square
and the other into a circle. What should be the length of the two pieces so that the combined
area of the square and the circle is minimum? (Difficult)
8
15. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of
27
the volume of the sphere. (Difficult)
16. Show that the right circular cone of least curved surface area and given volume has an altitude
equal to 2 time the radius of the base. (Difficult)

SUBJECT: 35 – MATHEMATICS Page 61 of 164

17. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height
−1
is tan 2. (Difficult)
18. An open topped box is to be constructed by removing equal squares from each corner of a 3
meter by 8 meter rectangular sheet of aluminum and folding up the sides. Find the volume of
the largest such box. (Difficult)
2 2
19. Find the maximum area of an isosceles triangle inscribed in the ellipse x 2 + y2 = 1 with its
a b
vertex at one end of the major axis. (Difficult)
20. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the
sum of their areas is least when the side of square is double the radius of the circle. (Difficult)
21. A window is in the form of a rectangle surmounted by a semicircular opening. The total
perimeter of the window is 10 m . Find the dimensions of the window to admit maximum light
through the whole opening. (Difficult)
22. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a
sphere of radius r is 4 r . (Difficult)
3
23. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of
2R
radius R is . Also find the maximum volume. (Difficult)
3
24. Find the point on the curve x 2 = 2 y which is nearest to the point (0,5) . (Average)

……………………………………………………………………

SUBJECT: 35 – MATHEMATICS Page 62 of 164

CHAPTER -7
INTEGRALS
MCQ /FB questions:

1. If ∫ 𝑓 (𝑥) 𝑑𝑥 = 𝐹(𝑥)+C, then (Easy)
𝑑
A) f (x) is called primitive or anti derivative B) 𝑑𝑥 (𝐹(𝑥)) = 𝑓(𝑥)
C) F(x) is called Integrand D) C is any integer.
1
2. The anti-derivative of (√𝑥 + 𝑥) is equal to (Easy)
√
1 1 1
2 2
1
A) 3 𝑥 3 + 2𝑥 2 + 𝑐 B) 3 𝑥 + 2 𝑥 2 + 𝑐
3

2 3 1
3 3
1 1
C) 3 𝑥 2 + 2𝑥 2 + 𝑐 D) 2 𝑥 2 − 2 𝑥 2 + 𝑐
2 𝑥3
3. ∫ 𝑥 𝑒 𝑑𝑥 = (Average)
1 3 1 2 1 3 1 2
A) 3 𝑒 𝑥 + 𝑐 B) 3 𝑒 𝑥 + 𝑐 C) 2 𝑒 𝑥 + 𝑐 D) 2 𝑒 𝑥 + 𝑐
3⁄2 𝑥 1
4. ∫ (𝑥 + 2𝑒 − 𝑥) 𝑑𝑥= (Average)
2x5/2 2x5/2
A) 5
+ 2𝑒 𝑥 − log|x| + C B) 5 + 2𝑒 𝑥 + log|x| + C
5x5/2 5x5/2
C) 2 + 2𝑒 𝑥 − log|x| + C D) 2 + 2𝑒 𝑥 + log|x| + C.
2⁄3
5. ∫(𝑥 + 5) 𝑑𝑥= (Easy)
3x5/3 3x5/3 5x5/2 5x5/2
A) 5 + C B) 5 + 5x + C C) 3 + 5𝑥 + C D) 3 + C.
6. ∫ (4𝑒 𝑥 + 1) 𝑑𝑥 (Easy)
𝑒𝑥
𝐴) 4𝑒 𝑥 + 1 + 𝐶 𝐵) 4𝑒 𝑥 + 𝐶 𝐶) 4𝑒 𝑥 + 𝑥 + 𝐶 𝐷) +𝑥+ 𝐶
4
7. ∫(cos 𝑥 − sin 𝑥)𝑑𝑥 = (Easy)
A) sin 𝑥 − cos 𝑥 + 𝐶 B) sin 𝑥 + cos 𝑥 + 𝐶 C) − sin 𝑥 − cos 𝑥 + 𝐶 D) −sin 𝑥 + cos 𝑥 + 𝐶

 (2 x − 3cos x + e )dx is
x
8. (Average)
A) 𝑥 2 + 3𝑠𝑖𝑛𝑥 + 𝑒 𝑥 + 𝐶 B) 𝑥 2 − 3𝑠𝑖𝑛𝑥 + 𝑒 𝑥 + 𝐶
C)𝑥 2 + 𝑠𝑖𝑛𝑥 + 𝑒 𝑥 + 𝐶 D) 2𝑥 2 − 3𝑠𝑖𝑛𝑥 + 𝑒 𝑥 + 𝐶

 (ax + bx + c)dx is
2
9. (Average)

𝑎𝑥 3 𝑏𝑥 2 𝑎𝑥 3 𝑏𝑥 2 𝑎𝑥 3 𝑏𝑥 2 𝑎𝑥 3 𝑏𝑥 2
(A)
3
+ 2
+𝑐+𝐶 (B) 3
+ 2
+𝐶 (C) 3
+ 2
+ 𝑐𝑥 + 𝐶 (D) 3
+ 2
+ 𝑐𝑥

10.  (1 − x) xdx (Average)
3 5 3 5 3 5 3 5
3x2 2x2 2x2 2x2 2x2 5x2 3x2 5x2
A) 2 − 5 + C B) 3 − 5 + C C) 3 − 2 + C D) 2 − 2 + C
1
11. The anti-derivative of w .r .t x is equal to (Average)
√𝑥 2 −𝑎2
1 𝑎+𝑥 1 𝑥−𝑎 𝑥
A)2𝑎 𝑙𝑜𝑔 |𝑎−𝑥| + 𝐶 B)2 𝑙𝑜𝑔 |𝑥+𝑎| + 𝐶 C)𝑙𝑜𝑔|𝑥 + √𝑥 2 − 𝑎2 | + 𝐶 D)sin−1 𝑎 + 𝐶
1
12. The anti-derivative of (Easy)
√𝑎 2 −𝑥 2
𝑥 𝑥 1 𝑥 𝑥
A) sin−1 𝑎 + 𝐶 B) cos−1 𝑎 + 𝐶 C)a sin−1 𝑎 + 𝐶 + 𝐶 D) 𝑐𝑜𝑠𝑒𝑐 −1 𝑎 + 𝐶

SUBJECT: 35 – MATHEMATICS Page 63 of 164

1
13. The anti-derivative of , x  1 with respect to x (Easy)
x x2 −1
A)sin−1 𝑥 + 𝐶 B)cos−1 𝑥 + 𝐶 C)cosec −1 𝑥 + 𝐶 D) 𝑠𝑒𝑐 −1 𝑥 + 𝐶

14. ∫ ( x 2  1 − 1  ) 𝑑𝑥 = (Average)
 x2 
𝑥3 𝑥3 𝑥2 𝑥3
𝐴) −1+𝐶 𝐵) +𝑥+𝐶 𝐶) +𝑥+𝐶 𝐷) −𝑥+𝐶
3 3 2 3
𝑑𝑥
15. ∫ 2
𝑥 +2𝑥+2
= (Difficult)
−1 (𝑥 −1 (𝑥
A) 𝑥 tan + 1) + 𝑐 B) tan + 1) + 𝑐
−1 (𝑥 −1 (𝑥)
C) (𝑥 + 1) tan + 1) + 𝑐 D) tan +𝑐
16. The anti-derivative of sin2x with respect to x (Easy)
1 1
A) 𝑐𝑜𝑠 2𝑥 + 𝐶 B) 𝑐𝑜𝑠2𝑥 +𝐶 C)−𝑐𝑜𝑠 2𝑥 + 𝐶 D) − 2 𝑐𝑜𝑠2𝑥 +𝐶
2
17. ∫ cosec 𝑥 (cosec 𝑥 + cot 𝑥) 𝑑𝑥 = (Average)
A) – cot x – cosec x + C B)cot 𝑥 − cosec 𝑥 + C
C)−cot 𝑥 + cosec 𝑥 + C D)cot 𝑥 + cosec 𝑥 + C

 tan 2 x dx is
2
18. (Average)
𝑠𝑒𝑐2𝑥 𝑡𝑎𝑛2𝑥 𝑡𝑎𝑛(2𝑥)
A) sec2x - x +C B)
2
+ x+C C) 2
– x +C D) 2
+x+C
1− x
19.  dx is (Average)
x
3 3 3 3
3x2 2x2 3x2 3x2
A) 2√𝑥 + +C B) 2√𝑥 − 3 + C C)2√𝑥 + +C D)2√𝑥 − +C
2 2 2
2
20. The anti derivative of 𝑥 2 (3 + ) with respect to x. (Average)
𝑥
𝑥3 3𝑥 3 2𝑥 2
A) 𝑥 3 + 𝑥 2 + 𝐶 B) 3 + 𝑥 + 𝐶 C) 3 𝑥 3 + 2𝑥 2 + 𝐶 D) 3 + 2 + 𝑥 + 𝐶
x3 + 5 x 2 − 4
21.  dx = (Average)
x2
x2 x2 4 x2 4 x2 8
A) 2 + 5𝑥 − 4log|x 2 | + C B) 2 + 5𝑥 − x + C C) 2 + 5𝑥 + x + C D) 2 + 5𝑥 − x3 + C.
x3 − x 2 + x − 1
22.  dx is (Difficult)
x −1
𝑥3 𝑥3 𝑥3 𝑥2
A) 𝑥 2 + 𝑥 + 𝐶 B) 3 + 𝑥 + 𝐶 C) 3 − 𝑥 + 𝐶 D) 3 + 2 + 𝑥 + 𝐶

 x (3x + 2 x + 3)dx =
2
23. (Average)
6x5/2 4x3/2 6x1/2 6x7/2 4x5/2 6x3/2
A) + + +C B) + + +C
5 3 1 7 5 3
6x 7/2 4x 5/2 6x 3/2 21x 7/2 10x 5/2 9x 3/2
C) 2 + 2 + 2 + C D) 2 + 2 + 2 + C

 (2 x − 3sin x + 5 x )dx =
2
24. (Average)
2x3 10x3/2 2x3 10x3/2
A) 3
− 3cosx + 3 + C B) 3 + 3cosx + 3
+C
2x3 5x3/2 2x3
C) 3 − 3cosx + 3 + C D) 3 + 3cosx + 10√x + C
(1−sinx)dx
25. ∫ cos2 x
= (Average)
𝐴)𝑠𝑒𝑐𝑥 − 𝑡𝑎𝑛𝑥 + 𝐶 𝐵)𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 + 𝐶 𝐶)𝑡𝑎𝑛𝑥 − 𝑠𝑒𝑐𝑥 + 𝐶 𝐷) − (𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥) + 𝐶

SUBJECT: 35 – MATHEMATICS Page 64 of 164

𝑠𝑖𝑛2 𝑥−𝑐𝑜𝑠2 𝑥
26. ∫ 𝑑𝑥 is equal to (Average)
𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠2 𝑥
A) tanx + cotx + c B) tanx + cosecx + c
C) - tanx + cotx + c D) tanx + secx + c
𝑥 3 −1
27. ∫( 𝑥 2 )𝑑𝑥 equals (Average)
x2 1 x2 1 x2 2 x2 2
A) 2 − X + C B) 2 + X + C C) 2 + X + C D) 2 − X + C

28.  sec x(sec x + tan x)dx = (Average)

𝐴)𝑠𝑒𝑐𝑥 − 𝑡𝑎𝑛𝑥 + 𝐶 𝐵) 𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥 + 𝐶 𝐶)𝑡𝑎𝑛𝑥 − 𝑠𝑒𝑐𝑥 + 𝐶 𝐷) − (𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥) + 𝐶
29. The anti-derivative of sin 2x – 4 𝑒 3𝑥 w .r. t x (Average)
−𝑐𝑜𝑠2𝑥 𝑒 3𝑥 𝑐𝑜𝑠2𝑥 𝑒 3𝑥 𝑠𝑖𝑛(2𝑥) 𝑒 3𝑥
A) 2
-43 +C B) 2
-43 +C C)2𝑐𝑜𝑠2𝑥 − 12𝑒 3𝑥 +C D) 2
-43 +C

30.  1 + sin 2xdx = (Average)

𝐴)𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛𝑥 + 𝐶 𝐵)𝑠𝑖𝑛𝑥 − 𝑐𝑜𝑠𝑥 + 𝐶 𝐶)𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 + 𝐶 𝐷) − (𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥) + 𝐶
sec2 x
31.  dx = (Difficult)
cos ec 2 x
A)secx − x + C B)tanx + x + C C)secx + x + C D)tanx − x + C
𝒆𝒙 (𝟏+𝒙)
32. ∫ 𝒅𝒙 = (Difficult)
𝒔𝒊𝒏𝟐 (𝒙𝒆𝒙 )

A)𝑐𝑜𝑡(𝑥𝑒 𝑥 ) + 𝐶 B)−𝑐𝑜𝑠𝑒𝑐(𝑥𝑒 𝑥 ) + 𝐶 C)−𝑡𝑎𝑛(𝑥𝑒 𝑥 ) + 𝐶 D)−𝑐𝑜𝑡(𝑥𝑒 𝑥 ) + 𝐶
𝒅𝒙
33. ∫ 𝒙𝟐+ 𝒙−𝟐 = (Difficult)
1 𝑥+2 1 𝑥−1 1 𝑥−2 1 𝑥+1
A)3 𝑙𝑜𝑔 |𝑥−1| + 𝐶 B)3 𝑙𝑜𝑔 |𝑥+2| + 𝐶 C)3 𝑙𝑜𝑔 |𝑥+1| + 𝐶 D)3 𝑙𝑜𝑔 |𝑥−2| + 𝐶.
sin 2 x
34.  dx = (Average)
1 + cos x
A) 𝑥 − 𝑠𝑖𝑛𝑥 + 𝐶 B) 𝑥 + 𝑠𝑖𝑛𝑥 + 𝐶 C) 𝑥 + 𝑐𝑜𝑠𝑥 + 𝐶 D) 𝑥 − 𝑐𝑜𝑠𝑥 + 𝐶
𝒄𝒐𝒔𝟐 𝒙
35. ∫ 𝟏+𝒔𝒊𝒏𝒙 𝒅𝒙 = (Average)
A) 𝑥 − 𝑠𝑖𝑛𝑥 + 𝐶 B) 𝑥 + 𝑠𝑖𝑛𝑥 + 𝐶 C)𝑥 + 𝑐𝑜𝑠𝑥 + 𝐶 D)𝑥 − 𝑐𝑜𝑠𝑥 + 𝐶
1
36.  sin x cos xdx =
2 2
(Average)

𝐴) 𝑡𝑎𝑛𝑥 − 𝑐𝑜𝑡𝑥 + 𝐶 𝐵) 𝑡𝑎𝑛𝑥 + 𝑐𝑜𝑡𝑥 + 𝐶 𝐶)𝑠𝑒𝑐𝑥 − 𝑐𝑜𝑠𝑒𝑐𝑥 + 𝐶 𝐷) − 𝑐𝑜𝑡𝑥 − 𝑡𝑎𝑛𝑥 + 𝐶
𝟐𝒔𝒊𝒏𝒙−𝟑𝒄𝒐𝒔𝒙
37. ∫ 𝟐𝒄𝒐𝒔𝒙+𝟑𝒔𝒊𝒏𝒙 𝒅𝒙= (Difficult)
A) log(2cosx+3sinx)+C B) log(2cosx-3sinx)+C
C)-log(2cosx+3sinx)+C D) - log(2sinx -3cosx)+C
dx
38.  x − 16 =
2
(Easy)
𝑥 1 𝑥−4 1 𝑥−4 1 4+𝑥
A) tan-1( 4) +C B) 8log|𝑥+4| +C C)4log|𝑥+4| +C D)8log|4−𝑥| +C

39.  ax + bdx = (Average)
3 3
1 (𝑎𝑥+𝑏)2 2(𝑎𝑥+𝑏)2
A)2√𝑎𝑥 + 𝑏 + 𝐶 B)2 𝑎𝑥+𝑏 + 𝐶 C) 𝑎
+𝐶 D) 3𝑎 + 𝐶.
√

SUBJECT: 35 – MATHEMATICS Page 65 of 164

sin(tan −1 x)
40.  dx = (Average)
1 + x2
A)− cos(𝑥 2 + 1) + 𝐶 B) cos(𝑥 2 + 1) + 𝐶 C)−𝑐𝑜𝑠(tan−1 𝑥) + 𝐶 D)𝑐𝑜𝑠(tan−1 𝑥) + 𝐶
10 x9 + 10 x log e 10
41.  dx = (Average)
x10 + 10 x
1
A)10 𝑙𝑜𝑔|𝑥 10 + 10𝑥 | + 𝐶 B)10 𝑙𝑜𝑔|𝑥 10 + 10𝑥 | + 𝐶 C)𝑙𝑜𝑔|𝑥 9 + 10𝑥 | + 𝐶 D)log|𝑥 10 + 10𝑥 | + C
2 − 3sin x
42.  cos x dx = 2
(Average)

A)2tanx - 3 secx +C B)2tanx+3secx + C C)2tanx - 2secx+C D) 2secx -3tanx+C
2
 1 
43.   x −  dx = (Difficult)
 x
𝑥2 𝑥2
A) 2 + log|𝑥| + 2𝑥 + 𝐶 B) 2 + log|𝑥| − 2𝑥 + 𝐶
𝑥2 𝑥2
C) 2 − log|𝑥| + 2𝑥 + 𝐶 D) 2 − log|𝑥| − 2𝑥 + 𝐶

 1 + x dx =
2
44. (Average)
𝑥 1 𝑥 1
A) 2 √1 + 𝑥 2 + 2 log|𝑥 + √1 + 𝑥 2 | + 𝐶 B)2 √1 + 𝑥 2 − log|𝑥 + √1 + 𝑥 2 | + 𝐶
2
𝑥 1 𝑥 1
𝐶) √1 + 𝑥 2 + log|𝑥 − √1 + 𝑥 2 | + 𝐶 D)2 √1 + 𝑥 2 + sin−1 𝑥 + 𝐶
2 2 2
dx
45.  x − 6 x + 13 =
2
(Difficult)
𝑥−3 1 𝑥−3 1 𝑥−3 1 3+𝑥
A) tan-1(
2
) +C B)4log|𝑥+3| +C C)2tan-1( 2 ) +C D)4log|3−𝑥| +C
2x
46.  1 + x dx = 2
(Average)

A) 2tan-1𝑥 +C B)log|1 + 𝑥 2 | +C C)tan-1(𝑥) +C D)2 log|1 + 𝑥 2 | +C
(log x)2
47.  dx = (Average)
x
(𝑙𝑜𝑔𝑥)3 (𝑙𝑜𝑔𝑥)2 (𝑙𝑜𝑔𝑥)2 1
A) 3
+C B) 3𝑥
+logx + C C) 2
+C D) 2(𝑙𝑜𝑔𝑥)3 + 𝑥 +C
−1
e tan x
48.  dx = (Average)
1 + x2
−1
𝑒 𝑡𝑎𝑛 𝑥 1 −1 𝑥 2 −1 𝑥 1
A) 1+𝑥 2
+C B)2 (𝑒 𝑡𝑎𝑛 ) +C C)𝑒 𝑡𝑎𝑛 +C D)1+x2 +C
sin −1 x
49.  1 − x dx = 2
(Average)

2
1 1 𝑠𝑖𝑛−1 𝑥 (𝑠𝑖𝑛−1 𝑥)
A) 2
+C B)2 (𝑠𝑖𝑛−1 𝑥) + C C) 2
+C D) 2
+ 𝐶.
√1−𝑥 √1−𝑥
2 (7
50. ∫ 𝑠𝑒𝑐 − 4𝑥)𝑑𝑥= (Average)
1 1 1
A) tan(7 − 4𝑥)+C B)− 4tan(7 − 4𝑥)+C C)4tan(7 − 4𝑥) +C D)− 4 sec(7 − 4𝑥)tan(7 − 4𝑥) +C
𝑐𝑜𝑠𝑥
51. ∫ 𝑑𝑥= (Average)
√1+𝑠𝑖𝑛𝑥
cos2 x 1
A) 2 +C B)2√1 + sinx+C C)2 √1 + sinx +C D√1 + sinx +C
1
52.  1 − cos x dx = (Difficult)

A) – cot x – cosec x + C B)cot 𝑥 − cosec 𝑥 + C C)−cot 𝑥 + cosec 𝑥 + C D)cot 𝑥 + cosec 𝑥 + C
SUBJECT: 35 – MATHEMATICS Page 66 of 164

53.  sin 2 x cos 3xdx = (Difficult)
𝑠𝑖𝑛5𝑥 1 cos5x
A) 5
+ 𝑠𝑖𝑛𝑥 + C B)2 (− 5
+ cosx) + C
cos5x 1 cos5x
C) 5 + cosx + C D)2 [− 5 − cosx] + 𝐶.
x −1
54.  dx = (Difficult)
x2 −1
A) √𝑥 2 − 1 + log|x + √𝑥 2 − 1| + 𝐶 B)√𝑥 2 − 1 – log|x + √𝑥 2 − 1| + 𝐶
3
2
C)3 (𝑥 2 − 1)2 +log|x + √𝑥 2 − 1| + 𝐶 D)2√𝑥 2 − 1 – log|x + √𝑥 2 − 1| + 𝐶

55.  x sin xdx = (Average)
A) –x cos x – sin x + C B) xcos 𝑥 − sin 𝑥 + C C)−xcos 𝑥 + sin 𝑥 + C D) xcos 𝑥 + sin𝑥 + C
56. ∫ ex (x 5 + 5x 4 + 1) dx (Average)
A)𝑒 𝑥+1 . 𝑥 5 + 𝐶 B) 5𝑥 4 𝑒 𝑥 +C C)𝑒 𝑥 𝑥 5 + 𝐶 D) 𝑒 𝑥 (𝒙𝟓 + 1) + 𝐶.

 xe dx is
x
57. (Average)

A) xex + ex + C B) xex − ex + C C) x+ex + C D)x 2 +ex + C
58. ∫x cos x dx = (Average)
A) –x sinx + cos x + C B) xcos 𝑥 − sin 𝑥 + C C)xsin 𝑥 − cos 𝑥 + C D)xsin 𝑥 + cos 𝑥 + C
3cos x + 4
59.  sin 2 x
dx = (Average)

A) -3cosecx – 4cotx +C B)4cotx+3cosecx + C
C) -3cosecx + 4cotx +C D) 3cosecx – 4cotx +C
x3 + 3x + 4
60.  dx is (Average)
x
3 5
3x2 6x2 2x7/2 6x3/2
A) + + 4√x + C B) 7 + + 8√x + C
2 5 3
5
7x7/2 9x3/2 2 2x2 3x3/2
C) 2 + 2 + x + C D) 5 + 2 + 8√x + C
√
61. ∫log x dx = (Average)
x2
A) xlogx + x + C B) logx + x + C C) xlogx − x + C D) xlogx − 2 + C

 e sec x(1 + tan x)dx =
x
62. (Average)
A)𝑒 𝑥 𝑠𝑒𝑐𝑥 + 𝐶 B) 𝑒 𝑥 𝑠𝑒𝑐 2 𝑥 + 𝐶 C)𝑒 𝑥 𝑡𝑎𝑛𝑥 + 𝐶 D) 𝑒 𝑥 (𝟏 + 𝑡𝑎𝑛𝑥) + 𝐶.
63. ∫ 𝑒 𝑥 (𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥)𝑑𝑥= (Easy)
A)𝑒 𝑥 𝑐𝑜𝑠𝑥 + 𝐶 B) 𝑒 𝑥 𝑠𝑖𝑛𝑥 + 𝐶 𝑥
C)𝑒 𝑡𝑎𝑛𝑥 + 𝐶 𝑥
D) 𝑒 (𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥) + 𝐶.
1 1
64. ∫ 𝑒 𝑥 (𝑥 − 2 ) 𝑑𝑥 (Easy)
𝑥
𝑒𝑥 𝑒𝑥
A) 2 + 𝐶
𝑥
B) 𝑥𝑒 𝑥 + 𝐶 C) 𝑥 + 𝐶 D) 𝑥 2 𝑒 𝑥 + 𝐶.

65.  x2e x dx =
3
(Average)
1 x2 1 x3 1 x2 1 x3
A) e +C B) e +C C) e +C D) e +C
3 3 2 2

SUBJECT: 35 – MATHEMATICS Page 67 of 164

1
66. ∫ 𝑒 𝑥 (tan−1 𝑥 + ) 𝑑𝑥 = (Easy)
1+𝑥 2
A) tan−1 𝑥 + 𝑐 B) 𝑒 𝑥 tan−1 𝑥 + 𝑐
𝑒𝑥 1
C) 1+𝑥2 + 𝑐 D)
1 + x2
67. ∫ 𝑒 𝑥 (𝑠𝑖𝑛𝑥 − 𝑐𝑜𝑠𝑥)𝑑𝑥 = (Easy)
A) 𝑒 𝑥 𝑠𝑖𝑛𝑥 + 𝑐 𝑥
B) 𝑒 𝑐𝑜𝑠𝑥 + 𝑐
C) −𝑒 𝑥 𝑠𝑖𝑛𝑥 + 𝑐 D) −𝑒 𝑥 𝑐𝑜𝑠𝑥 + 𝑐
e2 x − 1
68.  2 x dx = (Difficult)
e +1
A)log(𝑒 2𝑥 + 1) + 𝐶 B)log(𝑒 2𝑥 − 1) + 𝐶 C)log(𝑒 𝑥 − 𝑒 −𝑥 ) + 𝐶 D)log(𝑒 𝑥 + 𝑒 −𝑥 ) + 𝐶
1
69.  dx = (Difficult)
x− x
1
A)2 log(√𝑥 − 1) + 𝐶 B)2 log(√𝑥 + 1) + 𝐶 C)2log(√𝑥 + 1) + 𝐶 D)2 log(1 − √𝑥) + 𝐶.
1
70. ∫ 𝑥+𝑥𝑙𝑜𝑔𝑥 𝑑𝑥 = (Average)
A)2 log(1 + 𝑥) + 𝐶 B)log(𝑥 + 𝑙𝑜𝑔𝑥) + 𝐶 C)log(1 + 𝑙𝑜𝑔𝑥) + 𝐶 D)2 log(1 + 𝑙𝑜𝑔𝑥) + 𝐶.
sin 𝑥
71. ∫ (1+cos 𝑥)2 𝑑𝑥 = (Average)
−3 1 −1
A)− log (1 + 𝑐𝑜𝑠𝑥) + 𝐶 B) (1+𝑐𝑜𝑠𝑥)3 + 𝐶 C)1+𝑐𝑜𝑠𝑥 + 𝐶 D) 1+𝑐𝑜𝑠𝑥 + 𝐶.
𝑥
72. If f(x) =∫0 t sint dt then 𝑓 /(x) is (Difficult)
(𝐴)𝑐𝑜𝑠𝑥 + 𝑥 𝑠𝑖𝑛 𝑥 (𝐵)𝑥 𝑠𝑖𝑛𝑥 (𝐶)𝑥 𝑐𝑜𝑠𝑥 (𝐷) 𝑠𝑖𝑛𝑥 + 𝑥 𝑐𝑜𝑠𝑥.
4
dx
73.  16 + x =
0
2 (Easy)
𝜋 𝜋 𝜋 𝜋
(A) 2 (B) 4 (C) 8 (D) 16
√3 1
74. ∫1 (1+𝑥2 ) 𝑑𝑥 = (Average)
𝜋 2𝜋 𝜋 𝜋
(A) 3
(B) 3 (C) 6 (D) 12
2
1
75. ∫03 ( 4+9𝑥2 ) 𝑑𝑥 = (Difficult)

𝜋 𝜋 𝜋 𝜋
(A) 4 (B) 24 (C) 6 (D) 12
3
xdx
76.  x +1 =
2
2
(Average)
1 1 4 1
A)2𝑙𝑜𝑔2 B)2 log 2 C)2 log 3 D)2 log 50.
1
77. ∫0 𝑥𝑒 𝑥 𝑑𝑥 = (Average)
(A) 𝑒 − 1 (B) 1 (C) −1 (D) 2𝑒 − 1

x x
 (sin 2 − cos 2 )dx =
2 2
78. (Average)
0
A)0 B) 1 C)−1 D) −2.
1
79. ∫−1 𝑥 17 𝑐𝑜𝑠 4 𝑥 𝑑𝑥 = (Easy)
(A) 1 (B) −1 (C) 0 (D) 17
1 3 5
80. ∫−1(𝑥 + 𝑥𝑐𝑜𝑠𝑥 + 𝑡𝑎𝑛 𝑥)𝑑𝑥 = (Average)
(A) 1 (B) −1 (C) 0 (D) 8

SUBJECT: 35 – MATHEMATICS Page 68 of 164

1
81. ∫−1 1 𝑑𝑥 = (Average)
(A) 1 (B) −1 (C) 0 (D) 2
1
82. ∫−1 𝑠𝑖𝑛5 𝑥𝑐𝑜𝑠 4 𝑥 𝑑𝑥 = (Average)
(A) 1 (B) −1 (C) 0 (D) 2
 /2

 sin xdx =
7
83. (Easy)

− /2

(A) 1 (B) −1 (C) 0 (D) 7
2
84. 𝑇ℎ𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑜𝑓 2𝑥𝑠𝑖𝑛(𝑥 + 1) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑥 𝑖𝑠 (Average)
(𝐴)2𝑐𝑜𝑠(𝑥 2 + 1) + 𝐶 (𝐵) 𝑐𝑜𝑠(𝑥 2 + 1) + 𝐶 (𝐶) − 𝑐𝑜𝑠(𝑥 2 + 1) + 𝐶 (𝐷) 2𝑐𝑜𝑠(𝑥 2 + 1) + 𝑥 2 + 𝐶
85. The integral of sin−1(cos 𝑥) is (Average)
1 𝜋 𝑥2 𝜋 𝑥2
A)𝑠𝑖𝑛𝑥 + 𝐶 B)cos −1 (cos 𝑥) + 𝐶 C) 2 𝑥 − 2 + 𝐶 D) 2 − 2 + 𝐶
86. ∫ 𝑥√1 + 2𝑥 2 dx = (Average)
3 3
3 3 3
x2 2(1+2x2 )2 1 1 8
A)
2
− 3
+C B)4 (1 + 2x 2 )2 + C C)6 (1 + 2x 2 )2 + C D)3 (1 + 2x 2 )2 + C
87. ∫(4𝑥 + 2)√𝑥 2 + 𝑥 + 1dx = (Average)
3 3
4 2 1
A) 3 (𝑥 2 + 𝑥 + 1) + C 2 B)3 (𝑥 2 + 𝑥 + 1) + C C) 4√𝑥 2 + 𝑥 + 1 + C
2 D) +C
√𝑥 2 +𝑥+1
𝑥
88. ∫ 9−4𝑥2dx = (Average)
1 3+2𝑥 1 3−2𝑥 1
A) 6log|3−2𝑥| +C B) 6log|3+2𝑥| +C C)−8log|9 − 4𝑥 2 | +C D)− 8log|9 − 4𝑥 2 | +C
1
89. Assertion(A): ∫−1(𝑥 3 + 𝑠𝑖𝑛 𝑥)𝑑𝑥 = 0 5
(Average)
5
Reason(R): f(x) = x³ + 𝑠𝑖𝑛 𝑥 is an odd function.
A) A is false and R is true B) A is true and R is false
C) A is true and R is true D)A is false and R is false.
90. The equation of the family of curves in the
given figure is (Difficult)
A) 𝑦 = ∫ 𝑑𝑥 B) 𝑦 = ∫ 2𝑑𝑥
2
𝐶)𝑦 = ∫ 𝑥 𝑑𝑥 D) 𝑦 = ∫ 𝑥𝑑𝑥

91. The equation of the family of curves in the
given figure is (Difficult)

A) 𝑦 = ∫ 𝑑𝑥 B) 𝑦 = ∫ √3𝑑𝑥
1
𝐶)𝑦 = ∫ √3𝑥𝑑𝑥 D) 𝑦 = ∫ 3 𝑑𝑥
√

92. Statement 1 : e ( cos x + sin x ) dx = e cos x + c

x x

Statement 2 :  e ( f ( x ) + f ( x ) ) dx = e f ( x ) + c
x 1 x
(Difficult)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true
C) Statement 1 is false and Statement 2 is true
D) Statement 1 is false and Statement 2 is false.

SUBJECT: 35 – MATHEMATICS Page 69 of 164

2x f 1 ( x)
 1+ x = + +  f ( x ) dx = log f ( x ) + C
2
93. Statement 1: 2
dx log x 1 C, Statement 2: (Difficult)

A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
D) Statement 1 is false and Statement 2 is false.
1
94. Assertion(A): ∫−1 (2 )𝑑𝑥 = 4
𝑎 𝑎
Reason(R): ∫−𝑎 𝑓(𝑥 )𝑑𝑥 = 2 ∫0 𝑓(𝑥 )𝑑𝑥, 𝑖𝑓 f(x) is an even function (Difficult)
A) A is false and R is true B) A is true and R is false
C) A is true and R is true D) A is false and R is false.
95. Statement 1: The anti-derivative of 𝑠𝑒𝑐𝑥 w. r. t x is 𝑙𝑜𝑔|𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥|.
Statement 2: The derivative of 𝑙𝑜𝑔|𝑐𝑜𝑠𝑒𝑐𝑥 − 𝑐𝑜𝑡𝑥| w. r. t is cosecx (Difficult)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false
3
96. ∫−3 1 𝑑𝑥 =______________ (Average)
2
97. ∫0 [𝑥] 𝑑𝑥 =___________________ (Average)
98. ∫ ex (x3 + kx 2 + 10) dx = ex (x 3 + 10) + c, then k = ____________ (Average)
99. Match Column I with Column II (Difficult)
Column I Column II
a) ∫ e−x dx i)
−e−2x
+C
2
b) ∫ e−2x dx x 2
ii) e + C
x2
c) ∫ 2xe dx iii) −e−x + C
Choose the correct answer from the options given below:
(A) a-i , b-ii, c-iii (B) a-iii, b-ii, c-i (C) a-ii, b-iii, c-i (D) a-iii, b-i, c-ii.
100. Statement 1: The anti-derivative of (√1 + 𝑥 2 ) with respect to x is
𝑥 1
2
√1 + 𝑥 2 + 2 log|𝑥 + √1 + 𝑥 2 | + 𝐶.
𝑥 1
Statement 2: The derivative of 2 √1 + 𝑥 2 + 2 log|𝑥 + √1 + 𝑥 2 | + 𝐶
with respect to x is √1 + 𝑥 2 (Difficult)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false
101. Statement 1 : ∫ ex (1 − cotx + cot 2 x) dx = −ex cotx + c (Difficult)
 ( )
Statement 2 : e f ( x ) + f ( x ) dx = e f ( x ) + c
x 1 x

A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true.
C) Statement 1 is false and Statement 2 is true.
D) Statement 1 is false and Statement 2 is false.

SUBJECT: 35 – MATHEMATICS Page 70 of 164

1
102. Statement 1 : ∫−1 𝑥 𝑑𝑥 = 0.
𝑎
Statement 2 : ∫−𝑎 𝑓(𝑥 )𝑑𝑥 = 0, 𝑖𝑓 f(x) is an odd function
𝑑𝑒 𝑥
Statement 3: ∫ ex dx = ex ∵ 𝑑𝑥 = ex (Difficult)
Which of the above statements are correct?
(A) 1and 3 only (B) 2 and 3 only (C) 3 only (D) All 1, 2 and 3.
1
103. ∫−1{𝑥 − [𝑥]} 𝑑𝑥 =…….. (Average)
𝜋 𝜋
𝜋 𝜋
104. 𝐼𝑓 ∫0 𝑠𝑖𝑛2 𝑥 𝑑𝑥 =
2 and ∫0 𝑐𝑜𝑠 2 𝑥 𝑑𝑥 = , then k =……
2 (Average)
4 𝑘
1 1 −1 𝑥
105. Statement 1:∫ 𝑑𝑥 = 𝑎 sec (𝑎) + 𝐶.
𝑥√𝑥 2 −𝑎 2
𝑥 𝑎
Statement 2: The derivative of sec −1 (𝑎) with respect to x is (Difficult)
𝑥√𝑥 2 −𝑎 2
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false
106. Statement 1 : The process of differentiation and integration are inverses of each other
Statement 2 : Two indefinite integrals with the same derivative lead to the same family of
Curves. (Difficult)
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Statement 1 is true and Statement 2 is true
D) Statement 1 is false and Statement 2 is false

TWO MARKS QUESTIONS

1. Integrate with respect to x : tan2(2x – 3). (Average)
sin 2 x x3 − x 2 + x − 1
2. Find  1 + cos xdx (Difficult) 13. Find:  x − 1 dx (Average)

dx 14. Find:  log xdx
3. Find:  2
(Average)
(Easy)
x − 16 15. Evaluate ∫ 𝑥 𝑙𝑜𝑔𝑥 𝑑𝑥2
(Difficult)
3 1
4. Evaluate ∫ ( √𝑥 + 3 ) 𝑑𝑥
√𝑥
(Easy) 16. Find:  ax + bdx (Average)
1 2
17. Find:  e dx
2 x +3
5. Find ∫ (√𝑥 − 𝑥) 𝑑𝑥. (Average) (Easy)
√

 xe dx 18. Find  x (3 x + 2 x + 3)dx (Average)
x 2
6. Find: (Average)
2
7. Evaluate ∫ 𝑥𝑒 (𝑥 +1) 𝑑𝑥 (Average) 19. Find  (2 x − 3cos x + e )dx
x
(Average)
 (4e + 1)dx
3x
8. Find: (Easy)
20. Find  (2 x − 3sin x + 5 x )dx (Average)
2

9. Find:  (ax + bx + c )dx
2
(Easy) 21. Find ∫ sec 𝑥 (sec 𝑥 + tan 𝑥)𝑑𝑥 (Average)
6 3𝑥 1
10. Find: ∫ (7𝑥 + 𝑒 + ) 𝑑𝑥 (Average) sec2 x
x3 + 5 x 2 − 4
√1−𝑥 2 22. Find  cos ec2 x dx (Average)
11. Find:  dx (Average)
23. Find:  sin (cos x) dx
−1
x2 (Difficult)
x3 + 3x + 4
12. Find:  dx (Average) 24. Find  1 + cos 2xdx (Difficult)
x
SUBJECT: 35 – MATHEMATICS Page 71 of 164

25. Find  1 − cos 2xdx (Average) 37. Find  cos ecxdx (Difficult)

26. Find  1 + sin 2xdx (Average) 10 x9 + 10 x log e 10
38. Find :  dx (Average)
27. Evaluate: ∫(𝑎 𝑥 𝑒 𝑥 ) 𝑑𝑥 (Average) x10 + 10 x
28. Evaluate:∫(𝑒 2 log sec 𝑥 ) 𝑑𝑥 dx
39. Find:  2
(Average)
𝑥 𝑒 𝑒
(Average)
29. Evaluate: ∫(𝑒 − 𝑥 + 𝑒 ) 𝑑 (Average) x − 6 x + 13
cos 2𝑥
30. Find ∫ 𝑠𝑖𝑛2 𝑥+𝑐𝑜𝑠2 𝑥 𝑑𝑥 40. Find:  x e dx
(Difficult) 2 x3
(Average)
cos2 𝑥−sin 2𝛼
31. Find∫ cos 𝑥−sin 𝛼 𝑑𝑥 (Difficult) 2 − 3sin x
32. Find  2 x sin( x + 1) dx
2
(Average)
41. Find  cos x dx 2
(Average)

dx
sin(tan −1 x) 42. Find:  3x + 13x − 10 (Difficult)
33. Find  dx (Average) 2
1 + x2
1 − cos 2 x
34. Find  tan xdx (Average) 43. Find  1 + cos 2 xdx (Difficult)


35. Find cot xdx (Average)
44. Find  5x − 2 x
dx
2
(Difficult)
36. Find  sec xdx (Difficult)

 sin 2 x  x
45. Find  tan  1 + cos 2 x  dx (Average)
−1
60. Find  9 − 4 x dx 2
(Difficult)

x
 e dx
2
 1  61. Find (Difficult)
46. Find:   x −  dx (Difficult) x2

 x

62. Find x 1 + 2 x dx
2
(Difficult)
1 − sin x
47. Find  dx (Difficult) −1
cos 2 x e tan x
63. Find  dx (Easy)
3cos x + 4 1 + x2
48. Find  dx (Average)
sin 2 x e2 x − 1
1 64. Find  2 x dx (Difficult)
49. Find  dx (Difficult) e +1
1 − cos x 65. Find  tan (2 x − 3) dx
2
(Difficult)
2x
50. Find  dx
 sec (7 − 4 x)dx
(Average) 2
1 + x2 66. Find
(log x)2 (Difficult)
51. Find  dx (Average)
sin −1 x
x
52. Find  sin x sin(cos x)dx (Difficult)
67. Find  1 − x dx 2
(Easy)

1

53. Find sin(ax + b) cos(ax + b)dx (Difficult) 68. Find  cos x(1 − tan x) dx
2 2
(Difficult)

54. Find  (4 x + 2) x + x + 1dx (Difficult) 2 cos x − 3sin x
2

x
69. Find  6 cos x + 4sin xdx (Difficult)
55. Find  x+4
dx (Difficult)
cos x
𝑥+1
56. Find ∫ 𝑥+5 (Difficult)
70. Find x
dx (Average)
√
𝒅𝒙
57. Find ∫ (𝟐𝒙+𝟏)𝟐−𝟏𝟔 (Average) 71. Find  sin 2 x cos 2 xdx (Difficult)

x2 cos x
58. Find  (2 + 3x3 )3 dx (Difficult) 72. Find  1 + sin x dx (Difficult)

59. Find 
dx
, x  0, m  1 (Difficult)
73. Find  cot x log sin xdx (Average)
x(log x)m
SUBJECT: 35 – MATHEMATICS Page 72 of 164

 dx
2
74. Find cos xdx (Average) 97. Find  9x − 4x 2
(Difficult)
75. Find  sin 2 x cos 3 xdx (Difficult)
76. Find  sin xdx 3
(Difficult)
98. Find  x sin xdx (Average)

99. Find  x sin 3 xdx (Average)
77. Find  log xdx (Average)
100. Find  x e dx
2 x
(Average)
78. Find  e sec x(1 + tan x)  dx (Difficult)
x

101. Find  x log xdx (Average)
cos 2 x
79. Find  (sin x + cos x)2 dx (Difficult)
102. Find  x log 2 xdx (Difficult)


−1
80. Find tan xdx 103. Find  x log xdx
2
(Average) (Average)
81. Find  sin x cos xdx 104. Find  x sec xdx
3 2 2
(Difficult) (Difficult)
2
3x 105. Find  x + 2 x + 5dx
 x6 + 1dx
2
82. Find (Average) (Difficult)

1 106. Find  3 − 2x − x dx . 2
(Difficult)
83. Find  1+ 4x dx (Difficult)
107. Find  4 − x dx
2 2
(Average)
1
84. Find  dx (Difficult) 108. Find  1 − 4x dx 2
(Difficult)
(2 − x) 2 + 1
1 109. Find  x + 4 x + 6 dx
2
(Difficult)
85. Find  9 − 25 x 2
dx (Average)
110. Find  x + 4 x + 1 dx
2
(Difficult)
3x
86. Find  dx
111. Find  1 − 4x − x dx
(Difficult) 2
1 + 2 x4 (Difficult)
𝑥2
87. Find ∫ 1−𝑥6 𝑑𝑥 (Difficult) 112. Find  1 + 3x − x dx
2
(Difficult)
x −1
88. Find  x − 1dx 2
(Difficult) 113. Find  x + 3xdx
2
(Difficult)

114. Find  1 + x dx
2
x2 (Average)
89. Find  x + a dx (Difficult)
115. Find  7 − 8x + x dx
6 6 2
(Difficult)
sec 2 x
90. Find  tan x + 4 dx 2
(Average) x2
116. Find  1 + dx (Difficult)
9
dx
91. Find  x + 2x + 2 2
(Average)
117. Evaluate:
 /2

 sin xdx
7
(Easy)

− /2
dx
92. Find  9x + 6x + 5
2
(Difficult) 2𝜋
118. Evaluate:∫0 𝑐𝑜𝑠 5 𝑥𝑑𝑥 (Difficult)
 /2

 sin xdx
dx
 7 − 6x − x
3
93. Find (Difficult) 119. Evaluate: (Easy)
2

− /2
b
dx
94. Find  (Difficult) 120. Evaluate:  xdx (Easy)
( x − 1)( x − 2) a
5
dx
95. Find  8 − 3x − x 2
(Difficult) 121. Evaluate:  ( x + 1) dx
0
(Average)

dx 3
96. Find  x + 2x + 2
2
(Average) 122. Evaluate:  x dx
2
(Easy)
2

SUBJECT: 35 – MATHEMATICS Page 73 of 164

4  /2
136. Evaluate  cos 2xdx
 ( x − x)dx
2
123. Evaluate: (Average) (Difficult)
1 0
1  /2
137. Evaluate  cos 2 xdx
 e dx
x
124. Evaluate: (Easy) (Difficult)
−1 0
4 3
xdx
125. Evaluate:  ( x + e 2 x ) dx
0
(Average) 138. Evaluate  x +1
2
2
(Difficult)

4 1
dx
126. Evaluate:  139. Evaluate  xe x dx
2
(Average) (Average)
0
16 + x 2 0
2/3
3 dx
127. Evaluate : 
dx
(Difficult) 140. Evaluate  (Difficult)
1 + x2 0
4 + 9 x2
1

3
xdx
128. Evaluate: (sin 2 x − cos 2 x ) dx (Difficult)
 0
2 2
141. Evaluate  x +1
2
2
(Average)
3  /4
dx
 x −1  sin xdx
2
129. Evaluate: 2
(Average) 142. Evaluate (Difficult)
2 − /4 
 /4 1
130. Evaluate:  sin 2xdx (Average)

143. Evaluate sin x cos xdx
5 4
(Easy)
0 −1
 /2  /2
131. Evaluate:  cos 2xdx (Average) 144. Evaluate  sin xdx
7
(Easy)
0 − /2 
 /4 2
132. Evaluate:  tan xdx
0
(Average) 145. Evaluate  cos5 xdx (Difficult)
0
1  /2
dx 146. Evaluate
133. Evaluate:   ( x + x cos x + tan x + 1)dx
3 5
(Average)
0 1− x 2 − /2

2 (Difficult)

134. Evaluate (4 x3 − 5 x 2 + 6 x + 9)dx (Average)
1
147. Find  cos 6 x 1 + sin 6 xdx (Difficult)
 /4
135. Evaluate  sin 2xdx
0
(Average)

THREE MARKS QUESTION
3
1. Find the antiderivative of f(x)=4x 3 − such that f(2)=0 (Difficult)
x4
2. Find the antiderivative of F of f defined by f(x)=4x3-6, where F(0)=3 (Average)
dx

8. Find tan (2 x − 3) dx
2
3. Find  (Average)
(Difficult)
x + x log x 1
tan 4
x sec x 2 9. Find  cos x(1 − tan x)2 dx
2 (Average)
4. Find  x
dx (Difficult)
1
10. Find  dx (Difficult)
sin x 1 − tan x
5. Find  dx (Difficult)
sin( x + a) 1
11. Find  dx (Difficult)
1 1 + cot x
6. Find  dx (Average)
1 + tan x ( x + 1)( x + log x)2
12. Find  dx (Average)
e2 x − 1 x
7. Find  2 x dx (Difficult)
e +1 tan x
13. Find  dx (Difficult)
sin x cos x
SUBJECT: 35 – MATHEMATICS Page 74 of 164

x3 sin(tan −1 x 4 ) (3sin  − 2) cos 
14. Find  1 + x8 dx (Difficult) 35. Find  5 − cos  − 4sin  d (Difficult)
2

(1 + log x ) dx x2 + x + 1
2

15. Find  (Average) 36. Find  dx (Difficult)
x ( x + 2)( x 2 + 1)
16. Find  sin 2 x cos 2 xdx (Average) 37. Find 
x
dx (Average)
sin x
( x + 1)( x + 2)
17. Find  1 + cos x dx (Average)
38. Find  2
dx
(Average)
sin x x −9
18. Find  (1 + cos x) dx 2
(Average)
39. Find 
3x − 1
dx (Difficult)
1
( x − 1)( x − 2)( x − 3)
19. Find  dx (Difficult) 3x − 1
1 + cot x 40. Find  dx (Difficult)
(sin 2 x − cos 2 x) ( x − 1)( x − 2)( x − 3)
20. Find  dx (Average) 2x
sin 2 x cos 2 x 41. Find  2 dx (Average)
x + 3x + 2
sin 3 x + cos3 x
21. Find  dx (Difficult) 1 − x2
sin 2 x cos 2 x 42. Find  dx (Difficult)
x(1 − 2 x)
22. Find  x x + 2dx (Difficult)
x
x+2 43. Find  2 dx (Difficult)
23. Find  2 x2 + 6 x + 5 dx (Difficult) ( x + 1)( x − 1)
3x + 5
x+3 44. Find  3 dx (Difficult)
24. Find  dx (Difficult) x − x2 − x + 1
5 − 4 x + x2 2x − 3
dx 45. Find  2 dx (Difficult)
25. Find  ( x − a)( x − b) (Difficult) ( x − 1)(2 x + 3)
5x
4x +1 46. Find  dx (Difficult)
( x + 1)( x 2 − 4)
26. Find  2x + x − 32
dx (Difficult)
x3 + x + 1
x+2 47. Find  dx (Difficult)
27. Find  4 x − x dx 2
(Difficult) x2 −1
2
x+2 48. Find  dx (Difficult)
(1 − x)(1 + x 2 )
28. Find  x + 2 x + 3 dx
2
(Difficult)
3x − 1
x+3 49. Find  dx (Difficult)
29. Find  x − 2 x − 5dx
2
(Difficult) ( x + 2)2
dx
5x + 3 50. Find  4 (Difficult)
30. Find  x + 4 x + 10 dx
2
(Difficult) x −1
1
dx 51. Find  dx (Difficult)
x( x + 1)
 ( x + 1)( x + 2)
n
31. Find (Average)
cos x
52. Find  dx (Difficult)
x2 + 1 (1 − sin x)(2 − sin x)
32. Find  2 dx (Difficult)
x − 5x + 6 ( x 2 + 1)( x 2 + 2)
3x − 2 53. Find  2 dx (Difficult)
33. Find  dx (Difficult) ( x + 3)( x 2 + 4)
( x + 1)2 ( x + 3) 2x
x2 54. Find  2 dx (Average)
34. Find  2 dx (Difficult) ( x + 1)( x 2 + 3)
( x + 1)( x 2 + 4) dx
55. Find  (Difficult)
x( x 4 − 1)
SUBJECT: 35 – MATHEMATICS Page 75 of 164

dx x2 + x + 1
56. Find  e −1 x
(Difficult) 79. Find  ( x + 1)2 ( x + 2)dx (Difficult)

57. Find 
xdx 2 + sin 2 x
(Average) 80. Find  dx (Difficult)
( x − 1)( x − 2) 1 + cos 2 x
dx 1− x
58. Find 

(Difficult)
x( x 2 + 1) 81. Find dx (Difficult)
1+ x
x sin −1 x
 1 − x dx
9
59. Find (Average) x
2 82. Evaluate  dx (Difficult)
4
(30 − x3/2 ) 2

x
60. Find e sin xdx (Average) 2
xdx
61. Prove that (Easy) 83. Evaluate  9( x + 1)( x + 2)
1
(Average)
𝑥 | (𝑥)) 𝑥
∫ 𝑒 (𝑓(𝑥) + 𝑓 𝑑𝑥 = 𝑒 𝑓(𝑥) = 𝑐  /4
84. Evaluate  sin 3 2t cos 2tdt (Difficult)
 x sin xdx
−1
62. Find (Difficult)
0


−1
2x + 3
1
63. Find x tan xdx (Difficult)
85. Evaluate  dx (Difficult)
5x2 + 1
64. Find  x cos xdx
−1
(Difficult) 0
2
5x2
65. Find  x(log x) dx 1 x 2 + 4 x + 3dx
2
(Difficult) 86. Evaluate (Difficult)

66. Find  ( x + 1) log xdx
2
(Difficult)  /4
87. Evaluate  (2sec 2 x + x 3 + 2) dx (Difficult)
67. Find  e (sin x + cos x)dx
x
(Difficult) 0

6x + 3
2
 1 + sin x  88. Evaluate 

x
68. Find e  dx dx (Difficult)
 1 + cos x  0
x2 + 4
(Average) 1
x
x 89. Evaluate  ( xe x + sin ) dx (Difficult)
xe
69. Find  dx (Average) 0
4
(1 + x)2 1

 x5 + 1dx
4
x1 1  90. Evaluate 5 x (Average)
70. Find  e  − 2 dx (Average) −1
x x 
tan −1 x
1

( x − 3)e x 91. Evaluate  dx (Average)
71. Find  dx (Difficult) 1 + x2
( x − 1)3
0
 /2
92. Evaluate  sin  cos5  d (Difficult)

2x
72. Find e sin xdx (Difficult)
0
 2x 
73. Find  sin 
−1
 2x 
1
2 
dx (Difficult) 93. Evaluate  sin −1 
 1+ x  2 
dx (Difficult)
0  1+ x 
74. Find  e sec x(1 + tan x)dx
x
(Average) 2
94. Evaluate  x x + 2dx (Difficult)
cos 2 x
75. Find  dx (Difficult) 0
(sin x + cos x)2  /2
sin x
dx 95. Evaluate  dx (Average)
76. Find  x (Difficult) 0
1 + cos 2 x
e + e− x 2
dx
1− x 96. Evaluate  (Difficult)
77. Find  tan −1 dx (Difficult) x + 4 − x2
1+ x
0
1
dx
( x + 1) log( x + 1) − 2 log x 
2 2 97. Evaluate  x + 2x + 5
2
(Difficult)
78. Find  x4
dx −1

(Difficult)
SUBJECT: 35 – MATHEMATICS Page 76 of 164

1 1 
2
ex
98. Evaluate   − 2 e2 x dx (Difficult) 116. Find (1 + e x )(2 + e x )dx (Difficult)
0
x 2x 
( x − x3 )1/3 1
1
99. Evaluate  dx (Difficult) 117. Find  2 dx (Difficult)
1/3
x4 ( x + 1)( x 2 + 4)
 cos xe dx
x 3 logsin x
118. Find (Difficult)
100. If f(x)=  t sin tdt , then find the value of f '(x)
119. Find  e
−1
0 ( x + 1) dx
3log x 4
(Difficult)
(Difficult) 1

101. Prove that 120. Prove that  sin −1 xdx = − 1 (Difficult)
 a 2
2 f ( x)dx if f(2a-x) =f(x)
2a 0

0 f ( x)dx =  0 1− x
0 if f(2a-x) =-f(x)

121. Find  1 + x dx (Difficult)
(Difficult)
 2x −1 
1
122. Evaluate  tan −1 
2a a a
2 
dx (Difficult)
102. Prove that
 f ( x)dx =  f ( x)dx +  f (2a − x)dx
0 0 0 0  1+ x − x 
a
(Difficult)

123. If f(a+b-x)=f(x), then find xf ( x)dx (Average)
( x 4 − x)1/4
103. Find 
b
dx . (Difficult) 124. Evaluate Prove that
x5  /4
x 4 dx  2 tan xdx = 1 − log 2
3
(Difficult)
104. Find  . (Difficult)
( x − 1)( x 2 + 1) 0

 1   1 − sin x 
 e  1 − cos x dx
x
105. Find  log(log x) +
125. Find (Difficult)
(log x)2 
dx (Difficult)
 
 /4
/2

sin x cos x
106. Find   cot x + tan x dx . (Difficult) 126. Find  cos x + sin xdx (Difficult)
  4 4
0
sin 2 x cos 2 x
107. Find  dx (Difficult)  /2
cos 2 x
9 − cos 4 (2 x) 127. Find  dx (Difficult)
0
cos 2 x + 4sin 2 x
1
108. Find  dx (Difficult)  /3
sin x + cos x
x − x3 128. Find

 sin 2 x
dx (Difficult)
1 /6
109. Find  dx (Difficult) 1
dx
x+a + x+b 129. Find  (Difficult)
5x 1+ x − x
110. Find 
0
dx (Difficult)  /4
( x + 1)( x 2 + 9) sin x + cos x
sin x
130. Find  9 + 16sin 2 xdx (Difficult)
111. Find 
0
dx (Difficult)
sin( x − a)
4

e5log x − e4log x
131. Find   x − 1 + x − 2 + x − 3  dx (Difficult)
112. Find  3log x
1
dx (Average)
− e2log x
3
e 132. Prove that  dx 2 2
= + log (Difficult)
sin 8 x − cos8 x x ( x + 1) 3
2
3
113. Find 
1
dx (Difficult) 1
1 − 2sin 2 x cos 2 x 133. Prove that  xe x dx = 1 (Difficult)
1
114. Find 
0
dx (Difficult)
cos( x + a) cos( x + b)
x3
115. Find  dx (Difficult)
1 − x8

SUBJECT: 35 – MATHEMATICS Page 77 of 164

FIVE MARKS QUESTION
1 dx
1. Find the integral of with respect to x and evaluate  (Difficult)
a −x
2 2
2x − x2
1 dx
2. Find the integral of with respect to x and evaluate  (Average)
a2 − x2 9 − 25 x 2
1 dx
3. Find the integral of with respect to x and evaluate  (Difficult)
a2 − x2 9x − 4x2
1 dx
4. Find the integral of 2 with respect to x and evaluate  2 (Average)
x −a 2
x − 16
1 dx
5. Find the integral of 2 with respect to x and evaluate  2 (Average)
x −a 2
x − 16
1 x 2 dx
6. Find the integral of 2
a − x2
with respect to x and evaluate  1 − x6 (Difficult)

1 sin xdx
7. Find the integral of 2 with respect to x and evaluate  (Difficult)
a −x 2
1 − 4 cos 2 x
1 dx
8. Find the integral of 2 with respect to x and evaluate  2 (Average)
x +a 2
x + 16
1 dx
9. Find the integral of 2 with respect to x and evaluate  2 (Average)
x +a 2
9x + 4
1 3x 2 dx
10. Find the integral of 2
x + a2
with respect to x and evaluate  x6 + 1 (Average)

1 3xdx
11. Find the integral of 2 with respect to x and evaluate  (Difficult)
x +a 2
1 + 2x4
1 sin xdx
12. Find the integral of 2 with respect to x and evaluate  (Difficult)
x +a 2
1 + cos 2 x
1 cos xdx
13. Find the integral of 2 with respect to x and evaluate  (Average)
x +a 2
1 + sin 2 x
1 dx
14. Find the integral of with respect to x and evaluate  (Difficult)
x2 − a2 5x2 − 2 x
1 dx
15. Find the integral of with respect to x and evaluate  (Average)
x +a
2 2
1 + 4x2
𝑑𝑥 𝑥2
16. Prove that ∫ = 𝑙𝑜𝑔|𝑥 + √𝑥 2 + 𝑎2 | + 𝑐 and hence find ∫ 𝑑𝑥. (Average)
√𝑥 2 +𝑎2 √𝑥 6 +𝑎6
𝑑𝑥 𝑠𝑒𝑐 2 𝑥
17. Prove that ∫ 2 2 = 𝑙𝑜𝑔|𝑥 + √𝑥 2 + 𝑎2 | + 𝑐 and hence find ∫ 𝑑𝑥. (Average)
√𝑥 +𝑎 √𝑡𝑎𝑛2 𝑥+4
ADDITONAL QUESTIONS:
1 dx
1. Find the integral of with respect to x and evaluate  (Difficult)
a2 − x2 8 + 3x − x 2
1 dx
2. Find the integral of with respect to x and evaluate  (Difficult)
a2 − x2 7 − 6x − x2
1 dx
3. Find the integral of 2 with respect to x and evaluate  2 (Average)
x +a 2
x + 2x + 2
4.
1 dx
5. Find the integral of with respect to x and evaluate  2 (Difficult)
x +a
2 2
x − 6 x + 13

SUBJECT: 35 – MATHEMATICS Page 78 of 164

1 dx
6. Find the integral of with respect to x and evaluate  2 (Difficult)
x +a 2 2
9x + 6x + 5
1 dx
7. Find the integral of 2 with respect to x and evaluate  2 (Difficult)
x +a 2
a cos x + b 2 sin 2 x
2

1 dx
8. Find the integral of with respect to x and evaluate  (Difficult)
x2 − a2 ( x − 1)( x − 2)
1 dx
9. Find the integral of with respect to x and evaluate  (Difficult)
x −a
2 2
( x − a)( x − b)
1 dx
10. Find the integral of 2 with respect to x and evaluate  2 (Difficult)
x −a 2
3x + 13x − 10
𝑑𝑥 1
11. Prove that ∫ = 𝑙𝑜𝑔|𝑥 + √𝑥 2 + 𝑎2 | + 𝑐 and hence find ∫ 𝑑𝑥. (Difficult)
√𝑥 2 +𝑎2 √(2−𝑥)6 +1
𝑑𝑥 1
12. Prove that ∫ = 𝑙𝑜𝑔|𝑥 + √𝑥 2 + 𝑎2 | + 𝑐 and hence find ∫ 𝑑𝑥. (Difficult)
√𝑥 2 +𝑎2 √𝑥 2 +2𝑥+2

SIX MARKS QUESTION
a a  /4
1. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  log(1 + tan x )dx (Average)
0 0 0
a a  /2
sin x
2. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Average)
0 0 0 sin x + cos x
a a  /2
sin 3/2 x
3. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Average)
0 0 0
sin 3/2 x + cos3/2 x
a a  /2
cos5 x
4. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Average)
0 0 0
sin 5 x + cos5 x
a a  /2
sin 4 x
5. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Average)
0 0 0
sin 4 x + cos 4 x
a a a
x
6. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Difficult)
0 0 0 x + a−x
 /2
sin x − cos x
a a
7. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Difficult)
0 0 0
1 + sin x cos x
a a 
x sin x
8. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Difficult)
0 0 0
1 + cos 2 x
a a 
x
9. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Difficult)
0 0 0
1 + sin x
a a  /4
10. Prove that  f ( x)dx =  f (a − x )dx and hence evaluate  log(1 + tan x )dx (Difficult)
0 0 0
 /2
 4 + 3sin x 
a a
11. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  log   dx (Difficult)
0 0 0  4 + 3cos x 
b b  /3
dx
12. Prove that  f ( x)dx =  f (a + b − x)dx and hence evaluate 
a a  /6 1 + tan x
. (Average)

SUBJECT: 35 – MATHEMATICS Page 79 of 164

 a
13. Prove that  f ( x)dx =  0
 /2
2 f ( x)dx if f(x) is even
a
hence evaluate  sin 7 xdx . (Average)
−a 0 if f(x) is odd − / 2

ADDITIONAL QUESTIONS:
a a  /2
14. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  log sin xdx (Difficult)
0 0 0
a a 
15. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  log(1 + cos x )dx (Difficult))
0 0 0
a a  /2
16. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  (2 log sin x − log sin 2 x )dx (Difficult)
0 0 0
a a 
x tan x
17. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Difficult)
0 0 0
sec x + tan x
a a 
x
18. Prove that  f ( x)dx =  f (a − x)dx and hence evaluate  dx (Difficult)
0 0 0
a cos x + b 2 sin 2 x
2 2

 a
19. Prove that  f ( x)dx =  0
 /4
2 f ( x)dx if f(x) is even
a
hence evaluate  sin 2 xdx (Difficult)
−a 0 if f(x) is odd − / 4

 a
2 f ( x)dx if f(x) is even
a 1

20. Prove that  f ( x)dx =  0 hence evaluate  x cos xdx .
17 4
(Difficult)
−a 0 if f(x) is odd −1

 a
2 f ( x)dx if f(x) is even
a 1

21. Prove that  f ( x)dx =  0 hence evaluate  sin x cos xdx .
5 4
(Difficult)
−a 0 if f(x) is odd −1

 a
22. Prove that  f ( x)dx =  0
 /2
2 f ( x)dx if f(x) is even
a

 ( x + x cos x + tan x)dx .
3 5
hence evaluate
−a 0 if f(x) is odd 
− /2

(Difficult)
 a
23. Prove that  f ( x)dx =  0
 /2
2 f ( x)dx if f(x) is even
a
hence evaluate  sin 2 xdx . (Difficult)
−a 0 if f(x) is odd − /2

a c b 2

24. Prove that  f ( x) dx =  f ( x) dx +  f ( x) dx hence evaluate  x − x dx .
3
(Difficult)
b a c −1
a c b 5
25. Prove that  f ( x) dx =  f ( x) dx +  f ( x) dx hence evaluate  x + 2 dx . (Difficult)
b a c −5
a c b 8

26. Prove that  f ( x) dx =  f ( x) dx +  f ( x) dx hence evaluate
b a c
 x − 5 dx .
2
(Difficult)

a c b 3/2

27. Prove that  f ( x) dx =  f ( x) dx +  f ( x) dx hence evaluate
b a c
 x sin( x) dx
−1
(Difficult)

SUBJECT: 35 – MATHEMATICS Page 80 of 164

 a
2 f ( x)dx if f(2a-x)=f(x)
2a
𝜋 𝑥
28. Prove that 0 f ( x ) dx =  0 hence evaluate ∫0 𝑎2 cos2 𝑥+𝑏2 sin2 𝑥 𝑑𝑥 (Difficult)
0 if f(2a-x)=-f(x)

 a
29. Prove that  f ( x)dx =  0

2 f ( x)dx if f(2a-x)=f(x)
2a
x tan x
hence evaluate  sec x + tan x dx . (Difficult)
0 0 if f(2a-x)=-f(x) 0

 a
30. Prove that  f ( x)dx =  0

2 f ( x)dx if f(2a-x)=f(x)
2a
hence evaluate  log(1 + cos x)dx (Difficult)
0 0 if f(2a-x)=-f(x) 0


ADDITIONAL QUESTIONS:
1. Find the integral of x 2 − a 2 with respect to x and evaluate  x 2 + 4 x + 1dx (Difficult)

2. Find the integral of x 2 − a 2 with respect to x and evaluate  x 2 + 3xdx (Difficult)

3. Find the integral of x 2 − a 2 with respect to x and evaluate  x 2 − 8x + 7dx (Difficult)

4. Find the integral of x 2 + a 2 with respect to x and evaluate  x 2 + 4 x + 6dx (Difficult)

5. Find the integral of x 2 + a 2 with respect to x and evaluate  x 2 + 2 x + 5dx (Difficult)

x2
6. Find the integral of x + a with respect to x and evaluate 
2 2
1 + dx (Difficult)
9
7. Find the integral of a 2 − x 2 with respect to x and evaluate  4 − x 2 dx (Difficult)

8. Find the integral of a 2 − x 2 with respect to x and evaluate  3 − 2 x − x 2 dx (Difficult)

9. Find the integral of a 2 − x 2 with respect to x and evaluate  1 − 4 x − x 2 dx (Difficult)

10. Find the integral of a 2 − x 2 with respect to x and evaluate  1 + 3x − x 2 dx (Difficult)

11. Find the integral of a 2 − x 2 with respect to x and evaluate  1 − 4 x 2 dx (Difficult)

SUBJECT: 35 – MATHEMATICS Page 81 of 164

CHAPTER -08
APPLICATION OF INTEGRALS
FIVE MARK QUESTIONS
1. Find the area enclosed by the circle x + y = a .
2 2 2
(Easy)
2. Find the area lying in the first quadrant and bounded by the circle x + y = 4 and the
2 2

lines x = 0 and x = 2 (Average)
2 2
x y
3. Find the area enclosed by the ellipse 2 + =1. (Easy)
a b2
x2 y 2
4. Find the area of the region bounded by the ellipse + =1 (Easy)
4 9
x2 y 2
5. Find the area of the region bounded by the ellipse + =1 (Easy)
16 9
6. Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates
x = –1 and x = 1. (Average)
7. Find the area of the region bounded by the curve y2=4x , y-axis and the line y=3. (Easy)
8. Find the area bounded by the curve y=cosx between x=0 and x=2  . (Average)
9. Find the area bounded by the curve y=sinx between x=0 and x=2  . (Average)

ADDITIONAL QUESTIONS:
10. Find the area of the region bounded by the curve y = x and the lines x=1, x=4
2

and the x-axis in the first quadrant. (Average)
11. Find the area of the region bounded by y = 9 x and the lines x=2, x=4
2

and the x-axis in the first quadrant. (Average)
12. Find the area of the region bounded by x = 4 y , y=2, y=4 and the y-axis
2

in the first quadrant. (Average)
13. Find the area of the region bounded by the curve y=x2 and the line y=4. and the y-axis
in the first quadrant. (Average)
14. Find the area of the region bounded by the curve y2=4x and the line x=3. (Easy)
15. Find the area of the region bounded by the curve y = 4x, y-axis and the line y= 3. (Easy)
2

16. Find the area of the parabola y = 4ax bounded by its latus rectum.
2
(Average)
17. Find the area under the given curves and given lines y=x , x=1,x=2 and x-axis. (Easy)
2

18. Find the area under the given curves and given lines y=x4, x=1,x=5 and x-axis. (Easy)
19. Find the area bounded by the curve y=x3, the x-axis and the ordinates x=-2 and x=1.
(Easy)
2 2
x y
20. Find the area bounded by the ellipse 2 + = 1 and the ordinates x=0 and x=ae,
a b2
where b = a (1 − e ) and e<1.
2 2 2
(Average)
21. Find the area of the region lying in the first quadrant and bounded by y=4x . x=0, y=1
2

and y=4. (Average)
22. Find the area bounded by the curve y = x | x | , x-axis and
the ordinates x = – 1 and x = 1 (Difficult)
0
23. Sketch the graph of y = x + 3 and evaluate  x + 3 dx . (Difficult)
−6

SUBJECT: 35 – MATHEMATICS Page 82 of 164

CHAPTER -09
DIFFERENTIAL EQUATIONS
MCQ /FB questions.
dy
1. The Order and Degree of the differential equation dx − cosx = 0 is (Easy)
(a) 1,2 (b) 2,1 (c) 1,1 (d) 1,0
d2 y dy 2 dy
2. The Order and Degree of the differential equation xydx2 + x ( ) − y = 0 is (Easy)
dx dx
(a) 1,2 (b) 2,2 (c) 2,1 (d) 1,1
dy
d3 y d2 y
3. The Order and Degree of the differential equation dx 3 + dx 2 + e dx = 0 is (Average)
(a) 3,1 (b) 3,2 (c) 1,3 (d) 3, not defined
d4 y d2 y
4. Find the Order and Degree of the differential equation dx4 + sin (dx2 ) = 0 is (Average)
(a) 4,1 (b) 4,2 (c) 2,4 (d) 4, not defined
dy
5. Find the order and degree of the differential equation dx + 5y = 0 is (Easy)
(a) 1,2 (b) 2,1 (c) 1,1 (d) 1,5
ds 4 d2 s
6. Find the order and degree of the differential equation ( dt ) + 3s dt2 = 0 is (Easy)
(a) 1,4 (b) 2,1 (c) 2,4 (d) 4,2
2
d2 y dy
7. Find the order and degree of the differential equation (dx2 ) + cos ( ) = 0 is (Easy)
dx
(a) 2,2 (b) 1,2 (c) 2,2 (d) 2, not defined
d2 y
8. Find the order and degree of the differential equation 2 = cos3x + sin3x is (Average)
dx
(a) 2,1 (b) 1,2 (c) 2,2 (d) 2, not defined
9. Find the order and degree of the differential equation (Easy)
111 2 11 3 1 4 5
(y ) + (𝑦 ) + (𝑦 ) + 𝑦 = 0 is
(a) 3,5 (b) 3,2 (c) 3,3 (d) 3,4
d3 y d2 y dy
10. Find the order and degree of the differential equation dx3 + 2 dx2 + dx = 0 is (Easy)
(a) 3,1 (b) 3,2 (c) 1,3 (d) 3, 3
dy x
11. Find the order and Degree of the differential equation + y = e is (Easy)
dx
(a) 1,2 (b) 2,1 (c) 1,1 (d) 1, not defined.
d2 y dy 2
12. Find the order and degree of the differential equation dx2 + (dx) + 2y = 0 is (Easy)
(a) 2,1 (b) 1,2 (c) 2,2 (d) 2, 0
13. Find the order and degree of the differential equation y ′′ + 2 y ′ + siny = 0 is (Easy)
(a) 1,2 (b) 2,1 (c) 2,2 (d) 2, not defined.
3
d2 y dy 2 dy
14. Find the order and degree of the differential equation (dx2 ) + (dx) + sin (dx) + 1 = 0 is
(Difficult)
(a) 2,3 (b) 3,2 (c) 3, not defined (d) 2, not defined
d2 y dy
15. Find the order and degree of the differential equation 2x 2 dx2 − 3 dx + y = 0 is (Easy)
(a) 2,1 (b) 1,2 (c) 2,3 (d) 2, 0.
d2 y dy 2
16. Find the order and degree of the differential equation dx2 + 5x (dx) − 6y = logx is (Easy)
(a) 1,2 (b) 2,1 (c) 2,2 (d) 2, not defined.
𝑑𝑦 3 dy 2
17. Find the order and degree of the differential equation (𝑑𝑥 ) − 4 (dx) + 7y = sinx is (Easy)
(a) 1,3 (b) 2,1 (c) 2,2 (d) 2, not defined
𝑑𝑦 2𝑥
18. The order of the differential equation 𝑑𝑥 = 𝑒 is (Easy)
(a) 1,1 (b) 1,2 (c) not defined, 1 (d) 1, not defined.
d2 y
19. The order of the differential equationdx2 + 2𝑦 = 0 is (Easy)
(a) 2,1 (b) 1,2 (c) 2, 2 (d) 2, not defined.
SUBJECT: 35 – MATHEMATICS Page 83 of 164

3
d3 y d2 y
20. The order of the differential equation( 3 ) + x 2 ( 2 ) = 0 is (Easy)
dx dx
(a) 3 (b) 2 (c) 1 (d) not defined.
𝑑𝑦 2 dy 3
21. The degree of the differential equation (𝑑𝑥 ) + sin (dx) = 0 is (Average)
(a) 3 (b) 2 (c) 1 (d) not defined.
2
d3 y d2 y 𝑑𝑦
22. The degree of the differential equation (dx3 ) + 2 (dx2 ) − 𝑑𝑥 + 𝑦 = 0 is (Easy)
(a) 3 (b) 2 (c) 1 (d) not defined.
𝑑𝑦 2 𝑑𝑦
23. The degree of the differential equation (𝑑𝑥 ) + 𝑑𝑥 − 𝑠𝑖𝑛3 𝑦 = 0 is (Easy)
(a) 3 (b) 2 (c) 1 (d) not defined.
𝑑𝑦
24. A differential equation of the form 𝑑𝑥= F (x, y) is said to be homogenous if
F(x, y) is a homogenous function of degree (Average)
(a) 1 (b) 0 (c) 2 (d) 2, not defined
25. What is the order of differential equation y’’ + 5(𝒚’)𝟑 + 6 = 0 (Easy)
(a) 0 (b) 1 (c) 2 (d) 3
( ) +(y ) +(y ) + y = 0
4 11 3 1 2
26. What is the degree of differential equation y111 5
(Easy)
(a) 2 (b) 3 (c) 4 (d) 5
27. Find the number of arbitrary constants in the particular solution of a differential equation
of third order is: (Average)
(a) 3 (b) 2 (c) 1 (d) 0
28. Which of the following is a homogeneous differential equation? (Average)
(a) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0 (b) (xy) dx – (x3 + y3 )dy = 0
(c) (x3 + 2y2 ) dx + 2xy dy = 0 (d) y2 dx + (x2 – xy – y2 ) dy = 0
dy
29. Find the Integrating Factor of the differential equation x + 2 y = x2 ( x  0) (Easy)
dx
(a) log x (b) x 2 (c) 2 log x (d) x
ydx − xdy
30. The general solution of the differential equation = 0 is (Average)
y
(a) xy = C (b) x = Cy2 (c) y = Cx (d) y = Cx2
31. The number of arbitrary constants in the general solution of a differential equation of fourth
order are … (Easy)
𝑑𝑦 𝑥
32. A homogeneous differential equation of the from 𝑑𝑥 = ℎ (𝑦) can be solved by making the
substitution …….. (Average)
(a) y=vx (b) v=yx (c) x=vy (d) x=v
𝑑𝑦 k
33. The Integrating Factor of the differential equation x − 𝑦 = 2𝑥 2 is , then k is… (Easy)
𝑑𝑥 x
𝑑𝑥
34. The Integrating Factor of the differential equation (1 − 𝑦 2 ) 𝑑𝑦 + 𝑦𝑥 = 𝑎𝑦 is …… (Average)
1 1 1 1
(a) (b) (c) (d)
y −1
2
y2 −1 1− y2 1− y2
35. The Integrating Factor of the differential equation 𝑒 𝑥 dy + (y 𝑒 𝑥 + 2x) dx = 0 is e , then k is
kx

… (Average)
𝑑𝑦 𝑘𝑥
36. The Integrating Factor of the differential equation − 𝑦 = 𝑐𝑜𝑠𝑥 is 𝑒 , 𝑡ℎ𝑒𝑛 𝑘 𝑖𝑠 …. (Average)
𝑑𝑥
𝑑𝑦
37. The Integrating Factor of the differential equation 𝑥 𝑑𝑥 + 2𝑦 = 𝑥 2 is 𝑥 𝑘 , 𝑡ℎ𝑒𝑛 𝑘 𝑖𝑠 …… (Difficult)
k
38. The Integrating Factor of the differential equation 𝑦dx - (x + 2𝑦 2 ) dy = 0 is , then k is ……
y
(Average)
𝑑𝑦 𝑘𝑥 2
39. The Integrating Factor of the differential equation 𝑑𝑥 = 𝑥 + 𝑥𝑦 is 𝑒 , 𝑡ℎ𝑒𝑛 𝑘 𝑖𝑠 …. (Difficult)
SUBJECT: 35 – MATHEMATICS Page 84 of 164

40. The order of a differential equation whose general solution is y =Asinx + Bcosx is _____
(A, B are arbitrary constants) (Easy)
(a)4 (b) 2 (c) 0 (d) 3
𝑑𝑦 𝑥+𝑦
41. The differential equation 𝑑𝑥 = 1+𝑥2 is (Easy)
a) of variable separable form (b) homogeneous
(c) linear (d) Exact differential equation
𝑑𝑦
42. Statement 1: The Integrating factor of the differential equation 𝑑𝑥 + (𝑡𝑎𝑛𝑥)𝑦 = 𝑠𝑒𝑐𝑥 is 𝑠𝑒𝑐𝑥.
𝑑𝑦
Statement 2: The Integrating factor of the differential equation 𝑑𝑥 + 𝑃𝑦 = 𝑄 is 𝑒 ∫ 𝑝𝑑𝑥 .(Easy)
(a) Statement 1 is false and Statement 2 is true
(b) Statement 1 is true and Statement 2 is false
(c) Statement 1 is true, statement 2 is true and statement 2 is a correct explanation for
statement 1
(d) Statement 1 is true, statement 2 is true and statement 2 is not a correct explanation for
statement 1.
𝑑𝑦
43. Statement 1: The general solution of the differential equation 𝑑𝑥 + 𝑃𝑦 = 𝑄 is
y(IF)=∫ 𝑄(𝐼𝐹)𝑑𝑥 + 𝐶
𝑑𝑥
Statement 2: The Integrating factor of the differential equation 𝑑𝑌 + 𝑃𝑥 = 𝑄 is 𝑒 ∫ 𝑝𝑑𝑦 . (Easy)
(a) Statement 1 is false and Statement 2 is true
(b) Statement 1 is true and Statement 2 is false
(c) Statement 1 is true, statement 2 is true and statement 2 is a correct explanation for
statement 1
(d) Statement 1 is true, statement 2 is true and statement 2 is not a correct explanation for
statement 1
44. Statement 1: The order of the differential equation is the order of the highest order
derivative present in the equation.
dy 2 dy 3
Statement 2: The order of the differential equation ( ) + sin ( ) = 0 is 1. (Easy)
dx dx
a) Statement 1 is true and 2 is false. b) Statement 1 is true and 2 is true
c) Statement 1 and 2 are false. d) Statement 1 is false and 2 is true.
45. Statement 1: A function that satisfies the given differential equation is called its solution.
Statement 2: Order and degree (if defined) of a differential equation are always positive
integers. (Easy)
a) Statement 1 is true and 2 is false. b) Statement 1 is true and 2 is true
c) Statement 1 and 2 are false. d) Statement 1 is false and 2 is true.
46. Statement 1: The degree of the differential equation is represented by the power of the
highest order derivative in the given differential equation.
Statement 2: The degree of any differential equation can be found when it is in the form of
a polynomial; otherwise, the degree cannot be defined.
𝑑𝑦
Statement 3: The degree of tan (𝑑𝑥 ) = x + y is 1. (Easy)
a) Statement 1 is true and Statement 2 and 3 are false.
b) Statement 1 and 2 are true but Statement 3 is false.
c) Statement 1 and 3 are true but Statement 2 is false.
d) Statement 2 and 3 are true but Statement 1 is false
47. The order of the differential equation is always (Easy)
(a) Rational number (b) Whole number (c) Negative integer (d) Positive integer
48. The equation y = mx +c is general solution of (Easy)
𝑑𝑦 𝑑𝑦 𝑑2 𝑦 𝑑2 𝑥
(a) x 𝑑𝑥 = 𝑦 (b) y =𝑥 (c) =0 (d) =0
𝑑𝑥 𝑑𝑥 2 𝑑𝑦 2

SUBJECT: 35 – MATHEMATICS Page 85 of 164

TWO MARKS QUESTIONS
1. Find order and degree (if defined) of the following differential equations
(Each sub question carries 2 marks)
2
dy d2y  dy  dy
(i) − cos x = 0 (Easy) (ii) xy + x  − y =0 (Easy)
 dx 
2
dx dx dx
2
 d2y   d3y 
(iii) y + y + e = o
''' 2 y'
(Easy) (iv)  2  + sin  3  = 0 (Average)
 dx   dx 
2
 ds 
4
d 2s  d2y   dy 
(v)   + 3s 2 = 0 (Easy) (vi)  2  + cos   = 0 (Average)
 dt  dt  dx   dx 
d2y
(vii) y + 5 y = 0 = cos3x + sin 3x
'
(Easy) (viii) (Easy)
dx 2
(ix) ( y ''' ) + ( y '' ) + ( y ' ) + y = 0 (x) y '+ y = e
2 3 5 4 x
(Easy) (Easy)
(xi) y '''+ ( y ') + 2 y = 0 (xii) y ''+ 2 y '+ sin y = 0
2
(Easy) (Easy)
 d y   dy 
2
 dy 
2
d2y
(xiii) + 5 x   − 6 y = log x (Easy) (xiv) 2 x 2  2 
− 3  + y = 0 (Easy)
 dx   dx   dx 
2
dx
 d3y 
3 2
 dy   dy  d4y
(xv)   − 4   + 7 y = sin x (Easy) (xvi) − sin  3 =0 (Average)
 dx   dx  dx 4  dx 
3
 d 2 y   dy 
2
 dy 
(xvii)  2  +   + sin   + 1 = 0 (Average)
 dx   dx   dx 
2. Find the number of arbitrary constants in the general solution of differential equation of
fourth order also find the number of arbitrary constants in the particular solution of
differential equation of third order. (Average)
dx
3. Find the general solution of a differential equation: + P1 x = Q1 . (Easy)
dy
4. Verify that the given functions (explicit or implicit) is a solution of the corresponding
differential equation: (Each sub question carries 2 marks)
−3x d 2 y dy
(i) y = e : + − 6y = 0 (Easy)
dx 2 dx
d2y
(ii) y = a cos x + b sin x , where a,b  R : +y=0 (Easy)
dx 2
(iii) y= e x + 1 : y ''− y ' = 0 (Easy)
(iv) y = x + 2 x + c
2
: y'-2x-2=0 (Easy)
(v) y=cosx+C : y’+sinx=0 (Easy)
(vi) y = 1 + x 2 : y'= xy 2 (Easy)
1+ x
(vii) y = x sin x : xy'=y+x x − y ( x  0 and x>y or x<-y)
2 2
(Average)
(viii) y=Ax : xy’=y(x  0). (Easy)
y2
(ix) xy=logy+C : y'= (xy  1) (Easy)
1 − xy
(x) y- cosy =x : (ysiny+cosy+x)y’=y (Average)
−1
: y y + y +1 = 0
2 ' 2
(xi) x+y=tan y (Easy)

SUBJECT: 35 – MATHEMATICS Page 86 of 164

dy
(xii) y= a 2 − x2 x  (-a,a) : x+y = 0( y  0) (Easy)
dx
x −x 2 d2y dy
(xiii) y=ae +be +x : x 2 +2 -xy+x 2 -2=0 (Average)
dx dx
2
d y
(xiv) y=xsin3x : + 9 y − 6 cos 3x = 0 (Average)
dx 2
x d 2 y dy
(xv) y=e (acosx+bsinx) : -2 +2y=0 (Average)
dx 2 dx
dy
(xvi) x = 2 y log y : (x 2 + y 2 ) − xy = 0
2 2
(Average)
dx
THREE MARKS QUESTIONS
1. Find the general solution of the following differential equations
(Each sub question carries 3 marks)
dy x + 1 −x −x
(vii) (e + e )dy − (e − e )dx = 0
x x
(i) = , ( y  2) (Easy)
dx 2 − y (Average)
dy 1 + y 2 (viii) sec x tan ydx + sec y tan xdy = 0 (Average)
2 2
(ii) = (Easy)
dx 1 + x 2 (ix) y log ydx − xdy = 0 (Average)
dy 1 − cos x dy
(iii) = (Average) (x) x 5 = − y5 (Easy)
dx 1 + cos x dx
dy dy
(iv) = 4 − y 2 (−2  y  2) (Easy) (xi) = sin −1 x (Easy)
dx dx
dy
(xii) e tan ydx + (1 − e )sec ydy = 0
x x 2
(v) + y = 1( y  1) (Easy) (Average)
dx
dy dy 1− y 2

(vi) = (1 + x 2 )(1 + y 2 ) (Easy) (xiii) + =0 (Average)
dx dx 1 − x2
dy
(xiv) = e x+ y (Average)
dx
2. Find the particular solution satisfying the given condition of the following differential
equations (Each sub question carries 3 marks)
dy
(i) = −4 xy 2 given that y=1, when x=0. (Easy)
dx
dy
(ii) ( x3 + x 2 + x + 1) = 2 x 2 + x; y = 1 when x=0 (Difficult)
dx
dy
(iii) x(x 2 − 1) = 1; y = 0 when x=2 (Difficult)
dx
 dy 
(iv) cos   = a(a  R); y = 2 when x=0 (Average)
 dx 
dy
(v) = y tan x; y = 1 when x=0 (Average)
dx
(vi) (1 + e )dy + (1 + y )e dx, given that y=1 when x=0
2x 2 x
(Difficult)
(vii) ( x − y )(dx + dy ) = dx − dy , given that y=-1, when x=0. (Difficult)
3. Find the equation of the curve passing through the point (1, 1) whose differential equation
is xdy = (2 x + 1)dx( x  0) .
2
(Average)
4. Find the equation of a curve passing through the point (-2, 3), given that the slope of the
2x
tangent to the curve at any point (x, y) is . (Average)
y2
SUBJECT: 35 – MATHEMATICS Page 87 of 164

5. In a bank, principal increases continuously at the rate of 5% per year. In how many years
Rs.1000 double itself? (Difficult)
6. Find the equation of a curve passing through the point(0,0) and whose differential equation
is y ' = e sin x .
x
(Difficult)
dy
7. For the differential equation xy = ( x + 2)( y + 2) , find the solution curve passing through
dx
the point (1,-1). (Difficult)
8. Find the equation of curve passing through the point (0,-2) given that at any point (x,y) on
the curve, the product of the slope of its tangent and y coordinate of the point is equal to
the x coordinate of the point. (Difficult)
9. At any point (x,y) of a curve, the slope of the tangent is twice the slope of the line segment
joining the point of contact to the point (-4,-3). Find the equation of the curve given that it
passes through (-2,1). (Difficult)
10. The volume of spherical balloon being inflated changes at a constant rate. If initially its
radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t
seconds. (Difficult)
11. In a bank, principal increases continuously at the rate of r% per year. Find the value of r if
Rs.100 if Rs.100 double itself in 10 years (loge2=0.6931). (Difficult)
12. In a bank principal increases continuously at the rate of 5% per year. An amount of
Rs.1000 is deposited with this bank, how much will it worth after 10 years (e 0.5=1.648).
(Difficult)
13. In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours.
In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is
proportional to the number present? (Difficult)

FIVE MARKS QUESTIONS
1. Find the general solution of the following differential equations
(Each sub question carries 5 marks)
dy dy
(i) − y = cos x . (Easy) (ii) x + 2 y = x 2 ( x  0) (Average)
dx dx
dy
(iii) ydx − ( x + 2 y )dy = 0 + 2 y = sin x
2
(Average) (iv) (Easy)
dx
dy dy y
(v) + 3 y = e −2 x (Easy) (vi) + = x2 (Easy)
dx dx x
dy dy
(vii) x + 2 y = x 2 log x (Average) (viii) x − y = 2x2 (Average)
dx dx
dy 2 dy
(ix) x log x + y = log x (Average) (x) + (sec x) y = tan x(0  x   / 2) (Difficult)
dx x dx
(xi) (1 + x )dy + 2 xydx = cot xdx( x  0)
2
(Difficult)
(xii) x dy + y − x + xy cot x = 0( x  0) (Difficult)
dx
dy
(xiii) ( x + y ) =1 (Average)
dx
(xiv) ydx + ( x − y )dy = 0
2
(Difficult)
dy
(xv) ( x + 3 y 2 ) = y ( y  0) (Difficult)
dx
dy
(xvi) cos 2 x + y = tan x(0  x   / 2) (Difficult)
dx
dx
(xvii) (1 − y 2 ) + yx = ay (−1  y  1) (Difficult)
dy
SUBJECT: 35 – MATHEMATICS Page 88 of 164

2. Find the particular solution satisfying the given condition of the following differential
equations (Each sub question carries 5 marks)
dy 
(i) + 2 y tan x = sin x , y=0 when x= (Average)
dx 3
dy 1
(ii) (1 + x 2 ) + 2 xy = , y=0 when x=1 (Difficult)
dx 1 + x2
dy 
(iii) − 3 y cot x = sin 2 x , y=2 when x= (Average)
dx 2
3. Find the equation of a curve passing through the origin given that the slope of the tangent
to the curve at any point (x,y) is equal to the sum of the coordinates of the point. (Average)
4. Find the equation of a curve passing through the point (0,2) given that the sum of the
coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to
the curve at that point by 5. (Average)
ADDITIONAL QUESTIONS
 y  y
1. Solve the differential equation: ( xdy − ydx) y sin   = ( ydx + xdy ) x cos   . (Difficult)
x  x
−1
2. Solve the differential equation: (tan y − x)dy = (1 + y 2 )dx . (Difficult)
3. Find the particular solution of the differential equation: (1 + e )dy + (1 + y )e dx = 0
2x 2 x

given that y=1 when x=0. (Difficult)
4. Find a particular solution of the differential equation (x-y)(dx+dy)=dx-dy, given that y=-1,
when x=0. (Average)
5. Find a particular solution of the differential equation :
dy 
+ y cot x = 4 x cos ecx( x  0), given that y=0 when x= . (Average)
dx 2
dy
6. Show that the differential equation ( x − y ) = x + 2 y is homogeneous and solve it.
dx
(Difficult)
 y  dy  y
7. Show that the differential equation x cos   = y cos   + x is homogeneous and solve it.
 x  dx x
(Difficult)
x x

8. Show that the differential equation 2 ye dx + ( y − 2 xe )dy = 0 is homogeneous and find its
y y

particular solution, given that, x=0 when y=1. (Difficult)
9. Show that the family of curves for which the slope of the tangent at any point (x,y) on it is
x2 + y 2
, is given by x − y = cx
2 2
(Difficult)
2 xy
10. Show that the differential equation ( x + xy)dy = ( x + y )dx is homogeneous and solve it.
2 2 2

(Difficult)
x+ y
11. Show that the differential equation y ' = is homogeneous and solve it. (Average)
x
12. Show that the differential equation ( x − y )dy − ( x + y )dx = 0 is homogeneous and solve it.
(Easy)
13. Show that the differential equation ( x − y )dx + 2 xydy = 0 is homogeneous and solve it.
2 2

(Easy)
dy
14. Show that the differential equation x 2 = x 2 − 2 y 2 + xy is homogeneous and solve it.
dx
(Easy)
15. Show that the differential equation xdy − ydx = x + y dx is homogeneous and solve it.
2 2

(Easy)

  y  y    y  y 
16. Show that the differential equation  x cos   + y sin    ydx =  y sin   − x cos    xdy is
 x  x    x  x 
homogeneous and solve it. (Average)
dy  y
17. Show that the differential equation x − y + x sin   = 0 is homogeneous and solve it.
dx x
(Average)
 y
18. Show that the differential equation ydx + x log   dy − 2 xdy = 0 is homogeneous and solve it.
x
(Average)
x x
x
19. Show that the differential equation (1 + e )dx + e (1 − )dy = 0 is homogeneous and solve it.
y y

y
(Difficult)
20. For, (x+y)dy+(x-y)dx=0, find the particular solution satisfying the given condition, y=1 when
x=1. (Average)
21. For x dy + ( xy + y )dx ,find the particular solution satisfying the given condition, y=1 when
2 2

x=1. (Average)
  y 
 − y  dx + xdy = 0; find the particular solution satisfying the given condition,
2
22. For,  x sin 
 x 
y =  / 4 when x=1. (Difficult)
dy y  y
23. For, − + cos ec   = 0; find the particular solution satisfying the given condition, y=0
dx x x
when x=1. (Difficult)
dy
24. For, 2 xy + y 2 − 2 x 2 = 0; find the particular solution satisfying the given condition, y=2
dx
when x=1. (Difficult)

SUBJECT: 35 – MATHEMATICS Page 90 of 164

CHAPTER-10
VECTOR ALGEBRA
MCQ /FB questions.
1. Which of the following measures as vectors. (Easy)
(A)1000𝑐𝑚3 (B) 30 km/hr (C) 10 g/𝑐𝑚3 (D) 20 m/s towards north.
2. Which of the following measures as scalar. (Easy)
(A) 10 Newton (B) force (C) work done (D) velocity.
3. Which of the following is not true: (Easy)
(A) Time − scalar. (B) Density-scalar (C) Speed-scalar (D) Force- scalar
4. Which of the following is true: (Difficult)
(A) 𝑎 ⃗⃗⃗ and −𝑎 are collinear.
(B) Two collinear vectors are always equal in magnitude.
(C) Two collinear vectors are always same direction.
(D) Two collinear vectors having the same magnitude are equal.
5. If 𝑎 and 𝑏⃗ are two collinear vectors, then which of the following is incorrect: (Easy)
⃗
(A) 𝑏 = 𝜆𝑎, where 𝜆 ∈ 𝑅.. (B) 𝑎 = ±𝑏. ⃗
⃗
(C) The direction ratios of 𝑎 and 𝑏 are not proportional.
(D) Both the vectors 𝑎 and 𝑏⃗ have same direction, but different magnitudes.
6. The values of x, y and z. so that the vectors 𝑎 = x 𝑖̂ + 2𝑗̂ + z 𝑘̂ and 𝑏⃗ = 2𝑖̂ + y𝑗̂ + 𝑘̂ are
equal (Easy)
(A) 1, 2, 2 (B) 2, 1, 2 (C) 2, 2, 1 (D) 1, 1, 2
7. In vector addition, which of the following is not true: (Average)
⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
(A) 𝐴𝐵 + 𝐵𝐶 + 𝐶𝐴 = 0 ⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
(B) 𝐴𝐵 + 𝐵𝐶 − 𝐴𝐶 = 0 ⃗⃗⃗⃗⃗ ⃗
⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
(C) 𝐴𝐵 + 𝐵𝐶 − 𝐶𝐴 = 0 ⃗ (D) ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ + 𝐶𝐴
𝐴𝐵 − 𝐶𝐵 ⃗⃗⃗⃗⃗ = 0⃗
8. If 𝑎 is a nonzero vector of magnitude ‘a’ and 𝜆 a nonzero scalar, then 𝜆𝑎 is unit vector if (Easy)
1
(A) 𝜆 = 1 (B) 𝜆 = −1 (C) 𝑎 = |𝜆| (D) 𝑎 = |𝜆|
9. A unit vector in the direction of vector 𝑎 = 2 𝑖̂ + 3𝑗̂ + 𝑘̂ is (Easy)
̂
2 𝑖̂ + 3𝑗̂ + 𝑘 ̂
2 𝑖̂ + 3𝑗̂ + 𝑘 ̂
2 𝑖̂ + 3𝑗̂ + 𝑘 ̂
2 𝑖̂ + 3𝑗̂ + 𝑘
(A) (B) − (C) (D) −
√14 √14 √6 √6
10. The direction ratio’s of the vector 𝑎 = 𝑖̂ + 𝑗̂ - 2𝑘̂ are (Easy)
(A) 1,1,2 (B) -1,1,-2 (C)−1,1, −2 (D) -1,-1,2
̂
11. The direction cosin’s of the vector 𝑎 = 𝑖̂ + 2𝑗̂ +3𝑘 are (Easy)
1 2 3 −1 2 −3 −1 −2 −3 1 2 3
(A) 6 , 6 , 6 (B) 14 , 14 , 14 (C) 6 , 6 , 6 (D) 14 , 14 , 14
√ √ √ √ √ √ √ √ √ √ √ √
12. The direction ratio’s of the line joining the points A(1,2,-3) and B(-1,-2,1),
directed from A to B are (Easy)
(A) (2,4,-4) (B) (0,0,2) (C) (-2,-4,4 ) (D) (0,0,-2).
1 1 1
13. The magnitude of the vector 3 𝑖̂ + 3 𝑗̂ − 3 𝑘̂ is (Easy)
√ √ √
1
(A) 3 (B) √3 (C) (D) 1.
√3
14. The vector with initial point P (2, 3, 0) and terminal point Q (−1, −2, −4) is (Easy)
̂
(A) 3𝑖̂ + 5𝑗̂ + 4𝑘 (B) −3𝑖̂ − 5𝑗̂ − 4𝑘 ̂ (C) 𝑖̂ + 𝑗̂ − 4𝑘̂ ̂
(D) −3𝑖̂ − 5𝑗̂ + 4𝑘
⃗⃗⃗⃗⃗
15. The unit vector in the direction of 𝑃𝑄 where P and Q are the points (1, 2, 3) and
(4, 5, 6), respectively is (Easy)
̂
(A) 3𝑖̂ + 3𝑗̂ + 3𝑘 ̂
(B) 𝑖̂ + 𝑗̂ − 𝑘 ̂
(C) −𝑖̂ − 𝑗̂ − 𝑘 (D) 1
i+
1
j+
1
k
3 3 3
16. The value of x for which x(𝑖̂ + 𝑗̂ + 𝑘̂) is a unit vector. (Average)
1
(A) 3 (B) √3 (C) 3 (D) 1
√
17. The unit vector in the direction of 𝑎 +𝑏⃗ where 𝑎 = 2𝑖
̂ − 𝑗̂ + 2𝑘̂ and 𝑏⃗ = −𝑖̂ + 𝑗̂ − 𝑘̂ is (Easy)
1 1
(A) 𝑖̂ + 𝑘̂ (B) 2 (𝑖̂ + 𝑘̂ ) (C) −3𝑖̂ + 2𝑗̂ − 3𝑘̂ (D) 22 (−3𝑖̂ + 2𝑗̂ − 3𝑘̂ )
√ √

SUBJECT: 35 – MATHEMATICS Page 91 of 164

18. The position vector of the mid-point of the vector joining the points P(2, 3, 4) and
Q(4, 1, –2) is (Easy)
̂
(A) 3𝑖̂ + 2𝑗̂ + 𝑘 ̂
(B) 3𝑖̂ + 2𝑗̂ − 𝑘 ̂
(C) 𝑖̂ − 𝑗̂ − 3𝑘 (D)- 𝑖̂ + 𝑗̂ + 3𝑘̂
19. The position vector of a point which divides the join of points with position vectors
3𝑎 − 2𝑏⃗ and 𝑎 + 𝑏⃗ externally in the ratio 2:1 is (Easy)
5𝑎⃗
(A) (B) 4𝑏⃗ − 𝑎 ⃗⃗⃗⃗ + 𝑏⃗
(C)4𝑎 (D) 2𝑎⃗⃗⃗⃗ + 𝑏⃗
3
20. The value of 𝜆 for which the vectors 2𝑖̂ − 3𝑗̂ + 4𝑘̂ and −4𝑖̂ + 𝜆𝑗̂ − 8𝑘̂ are collinear is (Easy)
(A) 3 (B) 6 (C)−3 (D)−6
21. If 𝑖̂ , 𝑗̂ 𝑎𝑛𝑑 𝑘̂ are unit vectors, then which of the following is incorrect (Average)
̂ ̂
(A) 𝑖̂ ∙ 𝑖 = 𝑗̂ ∙ 𝑗̂ = 𝑘 ∙ 𝑘 = 1 ̂ ̂
(B) 𝑖̂ ∙ 𝑗̂ = 𝑗̂ ∙ 𝑘 = 𝑘 ∙ 𝑖̂ = 0
̂ ̂
(C) 𝑖̂ × 𝑖 = 𝑗̂ × 𝑗̂ = 𝑘 × 𝑘 = 0 ⃗ (D) 𝑖̂ × 𝑗̂ = 𝑗̂ × 𝑘̂ = 𝑘̂ × 𝑖̂ = 1
22. The projection vector of ⃗⃗⃗⃗⃗ AB on the directed line l, if angle θ = π will be. (Average)
(A) 𝑍𝑒𝑟𝑜 vector (B)𝐴𝐵 ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
(C)𝐵𝐴 (D) 𝑈𝑛𝑖𝑡 vector.
𝜋 3𝜋
⃗⃗⃗⃗⃗
23. The projection vector of 𝐴𝐵 on the directed line l, if angle 𝜃 = 2 𝑜𝑟 2 will be. (Easy)
(A) 𝑍𝑒𝑟𝑜 vector. (B)𝐴𝐵 ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
(C)𝐵𝐴 (D) 𝑈𝑛𝑖𝑡 vector.
24. The projection of 𝑎 = 2𝑖 + 3𝑗 + 2𝑘 𝑜𝑛 𝑏⃗ = 𝑖 + 2𝑗 + 𝑘 𝑖𝑠 (Easy)
10 5 10 10
(A) (B) 3 (C) (D) 102.
√6 √ √17 √
25. The projection of a vector 𝑎 = 𝑖̂ + 3𝑗̂ + 7𝑘̂ along 𝑏⃗ = 7𝑖̂ − 𝑗̂ + 8𝑘̂ is (Easy)
60 60 63 63
(A) (B) 114 (C) 59 (D) 114
√59 √ √ √
26. The projection of a vector 𝑖̂ + 𝑗̂ along 𝑖̂ − 𝑗̂ is (Easy)
1
(A) 2 (B) √2 (C) 0 (D) 2.
√
27. Projection vector of 𝑎 on 𝑏⃗ is (Easy)
⃗
𝑎⃗ ∙𝑏 ⃗
𝑎⃗ ∙𝑏 ⃗
𝑎⃗ ∙𝑏 ⃗
𝑎⃗ ∙𝑏
(A) 2 (B) 2 𝑏⃗ (C) |𝑎⃗| (D) |𝑎⃗|2
⃗|
|𝑏 ⃗|
|𝑏
28. If 𝜃 is the angle between two vectors 𝑎 and 𝑏⃗, then 𝑎 ∙ 𝑏⃗ ≥ 0 only when (Difficult)
 
(A) 0    (B) 0    (C) 0     (D) 0    
2 2
29. The angle between two vectors 𝑎 and 𝑏⃗ with magnitudes 1 and 2
respectively and 𝑎 ∙ 𝑏⃗ =1 is (Average)
𝜋 𝜋 𝜋 𝜋
(A) 6 (B) 4 (C) 3 (D) 2
30. The angle between two vectors 𝑎 and 𝑏⃗ with magnitudes √3 and 2 respectively and
𝑎 ∙ 𝑏⃗ =√6 is (Easy)
𝜋 𝜋 𝜋 𝜋
(A) 6 (B) 4 (C) 3 (D) 2
31. The magnitude of two vectors a and b, having the same magnitude and such that the angle
𝜋 1
between them is 3 and their scalar product is 2. (Average)
(A) 2 (B) −1 (C) 1 (D) −2
̂ ̂
32. The value of 𝜆 for which the two vectors 2𝑖̂ − 𝑗̂ + 2𝑘 and 3𝑖̂ + 𝜆𝑗̂ + 𝑘 are perpendicular is (Easy)
(A) 2 (B) 4 (C) 6 (D) 8
33. Let 𝑎 and 𝑏 be two unit vectors and 𝜃 is the angle between them. Then 𝑎 + 𝑏⃗ is a unit vector if
⃗
(Average)
   2
(A)  = (B)  = (C)  = (D)  =
4 3 2 3
34. The value of 𝑖̂ ∙ (𝑗̂ × 𝑘̂ ) + 𝑗̂ ∙ (𝑖̂ × 𝑘̂ ) + 𝑘̂ ∙ (𝑖̂ × 𝑗̂) is (Average)
(A) 0 (B) -1 (C) 1 (D) 3
35. A unit vector perpendicular to the both the vectors i + j and j + k is (Difficult)
−i−j+k i+j+k i+j−k i− j + k
(A) 3
(B) (C) 3 (D)
√ √ 3 √3

SUBJECT: 35 – MATHEMATICS Page 92 of 164

𝜋 𝜋
36. If a unit vector 𝑎 makes angles with 3 with 𝑖̂ and 4 with 𝑗̂ and an acute
angle 𝜃 with 𝑘̂, then 𝜃 is θ (Average)
𝜋 𝜋 𝜋 𝜋
(A) 6
(B) 4 (C) 3 (D) 2
37. If |𝑎 ∙ 𝑏⃗| = −|𝑎| |𝑏⃗| then the angle between 𝑎 & 𝑏⃗ 𝑖𝑠 (Easy)
  
(A) 4 (B) 2 (C) π (D) 3.
38. If 𝑎 and 𝑏⃗ are vectors such that a + b = a − b , then the angle between vectors a &b is- (Easy)
  
(A) 4 (B)2
(C) π 3
(D) .
  
39. If a and b are unit vectors and is the angle between them, then |𝑎 + 𝑏⃗| is
3
(Easy)
(A) √3 (B) 3 (C) 1 (D) √2.
40. If 𝜃 is the angle between any two vectors 𝑎 and 𝑏, then |𝑎 ∙ 𝑏⃗| = |𝑎 × 𝑏⃗| when 𝜃 is equal to (Easy)
⃗
 
(A) 0 (B) (C) (D) 
4 2
41. If |𝑎| = 8, |𝑏⃗| = 3 and |𝑎 × 𝑏⃗| = 12, then value of 𝑎 ∙ 𝑏⃗ is (Easy)
(A) 6√3 (B) 8√3 (C)12√3 (D) 12 8
42. The unit vector perpendicular to the vectors 𝑖̂ − 𝑗̂ and 𝑖̂ + 𝑗̂ forming a right handed system is
(Easy)
𝑖̂−𝑗̂ 𝑖̂+𝑗̂
̂
(A) 𝑘 ̂
(B) −𝑘 (C) (D) 2
√2 √
43. The number of unit vectors perpendicular to the vectors a = 2i + j = 2k &b = j + k is (Easy)
(A) one (B) two (C) three (D) infinite
⃗ ⃗ ⃗
44. If 𝑎 ∙ 𝑏 =0 and 𝑎 × 𝑏 =0, then the vectors 𝑎 and 𝑏 are (Average)
(A) Perpendicular (B) collinear ⃗ ⃗
(C) either 𝑎 = 0 𝑜𝑟 𝑏 = 𝑜 (D) none of these.
√2
45. Let the vectors 𝑎 and 𝑏⃗ be such that |𝑎| = 3, |𝑏⃗| = 𝑎 × 𝑏⃗ is a unit vector.
3
The angle between 𝑎 and 𝑏⃗. (Easy)
  
(A) 4 (B) 2 (C) π (D) 3.
46. The scalar components of vector with initial point (2,1) and terminal point (−7,5). (Easy)
(A) -5, 6 (B) -9, 4 (C) 9, -4 (D) 5, 4
47. If (𝑎 − 𝑏⃗) × (𝑎 + 𝑏⃗) = 𝑘(𝑎 × 𝑏⃗),then k is (Average)
(A) 1 (B) 0 (C) 2 (D) -2
48. The value of |𝑥 |, if for a unit vector 𝑎, (𝑥 − 𝑎) ∙ (𝑥 + 𝑎) = 8 (Easy)
(A) -3 (B) 3 (C) -9 (D) 9
49. The vectors 2i − 3 j + 4k and − 4i + 6 j − 8k are (Easy)
(A) Perpendicular B) collinear (C) Equal (D) Negative of each other.
50. The value of 𝜆 ,if (2𝑖̂ + 6𝑗̂ + 27𝑘̂ ) × (𝑖̂ + 𝜆𝑗̂ + 𝜇𝑘̂ ) = 0⃗ (Average)
(A) -3 (B) 3 (C) -9 (D) 9
51. If (3𝑎 − 5 𝑏⃗) × (2𝑎 + 7𝑏⃗) = 𝑘(𝑎 × 𝑏⃗), then k is (Average)
(A) 11 (B) 10 (C) 21 (D)31
52. If 𝑎 ∙ 𝑎 = 0 and 𝑎 ∙ 𝑏⃗ = 0 , then the vector 𝑏⃗ is (Average)
⃗
(A) 0 ⃗
(B) 𝑏 ⊥ 𝑎 (C) collinear to 𝑎 (D) any vector
53. If 𝑎 × 𝑏 = 0 and 𝑎 ∙ 𝑏 = 0 .What can you conclude about the vectors 𝑎 and 𝑏⃗ ?
⃗ ⃗ (Average)
(A) |𝑎 | ⃗
= |𝑏| (B) 𝑎 ⊥ 𝑏 ⃗ ⃗
(C) 𝑎 and 𝑏 are collinear | ⃗
(D)|𝑎 = 0 𝑜𝑟 |𝑏| = 0

SUBJECT: 35 – MATHEMATICS Page 93 of 164

( ) ( )
54. Statement 1: The value of i . j  k + j . i  k + k .( i  j ) is 1

Statement 2: i .i = j . j = k .k = 0 (Average)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation
for Statement1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
D) Statement 1 is false and Statement 2 is false.
55. Assertion (A): The two vectors 𝑎 = 2𝑖̂ − 𝑗̂ + 2𝑘̂ and 𝑏⃗ = −6𝑖̂ + 3𝑗̂ − 6𝑘̂ are collinear vectors.
Reason (R): If two vectors 𝑎 and 𝑏⃗ are collinear, then 𝑎 =⋌ 𝑏⃗, where ⋌ 𝜖 𝑅. (Easy)
A) A is false and R is true B) A is true and R is true
C) A is true and R is false D) A is false and R is false.
1 1 1
56. Statement 1: The magnitude of vector 𝑎 = 3 𝑖̂ + 𝑗̂ − 𝑘̂ . is 1
√ 3 3 √ √
Statement 2: if r = xi + yj + zk then r = x + y + r
2 2 2
(Easy)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is true
C) Statement 1 is true and Statement 2 is true
D) Statement 1 is false and Statement 2 is false.
57. Assertion (A): The two vectors 𝑎 = 4𝑖̂ + 4𝑗̂ − 2𝑘̂ and 𝑏⃗ = 4𝑖̂ − 2𝑗̂ + 4𝑘̂ are perpendicular vectors.
Reason (R): If two vectors 𝑎 and 𝑏⃗ are perpendicular, then |𝑎| =|𝑏⃗|. (Easy)
A) A is false and R is true B) A is true and R is true
C) A is true and R is false D) A is false and R is false.
58. Statement 1: The vector joining the points A(1,0,-1) and B(2,1,0) in the directed from B to A is
𝑖̂ + 𝑗̂ + 𝑘̂
Statement 2: PQ = OQ − OP (Easy)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is true.
59. Statement 1: If either |𝑎| = 0 or |𝑏⃗| = 0 then a.b = 0
Statement 2: If 𝑎 × 𝑏⃗ = 0 ⃗ , 𝑡ℎ𝑒𝑛 𝑎 ⊥ 𝑡𝑜 𝑏⃗. (Average)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is true,
C) Statement 1 is true and Statement 2 is true,
D) Statement 1 is false and Statement 2 is false.
60. For the given figure, 𝑎 − 𝑏⃗ 𝑖𝑠 (Average)

A) ⃗⃗⃗⃗⃗
𝑂𝐶 ⃗⃗⃗⃗⃗
B) 𝐶𝑂 ⃗⃗⃗⃗⃗
C) 𝐵𝐴 D) ⃗⃗⃗⃗⃗
𝐴𝐵

61. In the figure if R is the mid-point of P and Q and position vectors
of R, Q are 𝑎, 𝑏⃗ respectively then the position vector of P is (Average)
⃗
𝑎⃗ −𝑏 ⃗
𝑎⃗ +𝑏
A) 2
B) 2
C) 2𝑎 − 𝑏⃗ D) 2𝑎 + 𝑏⃗

SUBJECT: 35 – MATHEMATICS Page 94 of 164

62. For the given figure, 𝑃𝑆 ⃗⃗⃗⃗ 𝑎𝑛𝑑 𝑆𝑄
⃗⃗⃗⃗⃗ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦 𝑎𝑟𝑒 (Difficult)
A) 𝑎 + 𝑏⃗ + 𝑐 , 𝑏⃗ − 𝑐
B) 𝑎 + 𝑏⃗ + 𝑐 , −𝑏⃗ − 𝑐
C) −𝑎 − 𝑏⃗ − 𝑐 , 𝑏⃗ + 𝑐
D) 𝑎 + 𝑏⃗ + 𝑐 , −𝑏
⃗⃗⃗⃗⃗ + 𝑐

63. In the figure If 𝑂𝐴 ⃗⃗⃗⃗⃗ = 𝑏⃗, 𝑂𝐶
⃗⃗⃗⃗⃗ = 𝑎 , 𝑂𝐵 ⃗⃗⃗⃗⃗ = 𝑐 and ⃗⃗⃗⃗⃗
𝐵𝐶 = 𝑥 then LM is (Difficult)
𝑎⃗ .𝑐
A) |𝑎⃗|
⃗
𝑎⃗ .𝑏
B) |𝑎⃗|
𝑎⃗ .(𝑏 ⃗ +𝑐)
C) |𝑎⃗ |
𝑎⃗ .(𝑐 −𝑏 ⃗)
D) |𝑎⃗|
64. If (2𝑎 − 3𝑏⃗) × (3𝑎 − 2𝑏⃗) = 𝜆(𝑎 × 𝑏⃗) , then the value of λ is _____ (Average)
65. The projection of the vector 𝑎 = 2𝑖̂ + 3𝑗̂ + 2𝑘̂ on y-axis is --------- (Average)
66. The value of 𝜇 ,if (2𝑖̂ + 6𝑗̂ + 3𝑘̂ ) × (𝑖̂ + 3𝑗̂ + 𝜇𝑘̂ ) = ⃗0 is --------- (Average)
67. If |𝑎| = 5, 𝑎 ∙ 𝑏⃗ = 8 and |𝑎 × 𝑏⃗| = 6, then value of |𝑏⃗| is --------- (Average)
68. The value of 𝑖̂ ∙ (𝑗̂ + 𝑘̂ ) + 𝑗̂ ∙ (𝑖̂ + 𝑘̂ ) + 𝑘̂ ∙ (𝑖̂ + 𝑗̂) is --------- (Average)
69. The value of |𝑖̂ × (𝑗̂ + 𝑘̂ ) + 𝑗̂ × (𝑖̂ + 𝑘̂ ) + 𝑘̂ × (𝑖̂ + 𝑗̂)| is________ (Average)
70. The value of |𝑎 × (𝑏⃗ + 𝑐) + 𝑏⃗ × (𝑐 + 𝑎) + 𝑐 × (𝑎 + 𝑏⃗)| is_________ (Difficult)
71. The projection on the y axis of the vector 3i+4k is_________ (Average)
72. The scalar product of λ i + j – 3k and 3i – 4j + 7k is -10, then the value of λ is_ (Average)
73. Which of the following vectors are collinear in the figure given below? (Average)
⃗
d

⃗a
⃗c
⃗a

⃗b
A) a⃗ , c⃗ and d⃗ B) a⃗ , b⃗ and d⃗ C) a⃗ and d⃗ D) b⃗ and d⃗
⃗⃗ ⃗
74. For the given figure 𝑀 𝑎𝑛𝑑 𝑁 𝑎𝑟𝑒 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑙𝑦 (Difficult)
A) 𝑎 × 𝑏⃗ and −⃗⃗⃗𝑎 × −𝑏⃗
B) 𝑎 + 𝑏⃗ and 𝑎 − 𝑏⃗
C) 𝑎 × 𝑏⃗ and 𝑎 ∙ 𝑏⃗
D) 𝑎 × 𝑏⃗ and 𝑏⃗ × 𝑎

Two mark questions:
1. Find the unit vector in the direction of the 𝑎 = 𝑖̂ + 𝑗̂ + 2𝑘̂ (Easy)
2. Find the unit vector in the direction of the 𝑎 = 2𝑖̂ + 3𝑗̂ + 𝑘̂ (Easy)
2 2
3. For what value of , is the vector 3 𝑖̂ − 𝜆𝑗̂ + 3 𝑘̂ a unit vector? (Easy)
4. Find the value of 𝜆 for which 𝜆(2𝑖̂ + 𝑗̂ − 2𝑘̂ ) is a unit vector. (Easy)
̂ ̂
5. Show that the vectors 2𝑖̂ − 3𝑗̂ + 4𝑘 and −4𝑖̂ + 6𝑗̂ − 8𝑘 are collinear. (Easy)
6. If 2𝑖̂ − 3𝑗̂ + 4𝑘̂ and 𝜆 𝑖̂ + 6𝑗̂ − 8𝑘̂ are collinear then find 𝜆. (Easy)
7. Find the angle between two vectors 𝑎 and 𝑏⃗ with magnitude √3 and 2 respectively and
𝑎 . 𝑏⃗ = √6. (Easy)
̂ ⃗ ̂
8. If ⃗⃗⃗𝑎 = 𝑥𝑖̂ + 2𝑗̂ − 𝑧𝑘 and 𝑏 = 3𝑖̂ − 𝑦𝑗̂ + 𝑘 are two equal vectors, then write the value of x+y+z. (Easy)
SUBJECT: 35 – MATHEMATICS Page 95 of 164

9. Find the direction cosines of the vector 𝑎 = 𝑖̂ + 2𝑗̂ + 3𝑘̂. (Easy)
10. Write the scalar components and vector components of the vector joining the points
𝐴 = (𝑥1 , 𝑦1 , 𝑧1 ) and 𝐵 = (𝑥2 , 𝑦2 , 𝑧2 ). (Easy)
⃗⃗⃗⃗⃗ ̂ ⃗⃗⃗⃗⃗ ̂
11. If vector 𝐴𝐵 = 2𝑖̂ − 𝑗̂ + 𝑘 and 𝑂𝐵 = 3𝑖̂ − 4𝑗̂ + 4𝑘, find the position vector OA . (Easy)
12. Find the scalar components of vector with initial point (2,1) and terminal point (−7,5) . (Easy)
13. Find the unit vector in the direction of a +b, where a = 2i - j + 2k and b = -i + j -k . (Easy)
14. Find the position vector of a point R which divides the line joining two points P and Q whose
position vectors are i + 2j - k and -i + j + k respectively, in the ratio 2 : 1 internally. (Easy)
15. Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and
Q (4, 1, −2). (Easy)
16. Show that (a -b)×(a +b) = 2(a ×b) (Average)
17. Show that the vector i + j + k is equally inclined to the axes OX , OY and OZ. (Average)
18. Find the angle between the vectors i - 2j + 3k and 3i - 2j + k . (Easy)
19. Find angle between the vectors a = i + j - k and b = i + j + k . (Easy)
20. If a = 5i - j - 3k and b = i + 3j - 5k, then show that the vectors a + b and a - b are perpendicular. (Easy)
21. Find the magnitude of two vectors a and b, having the same magnitude and such that the angle
1
between them is 60 and their scalar product is . (Average)
2
22. Find|x|, if for a unit vector a, (x - a).(x + a) = 12 (Average)
23. If a + b = a − b , prove that a and b, are perpendicular. (Average)

24. Find |b|, if (a +b).(a -b) = 8 and |a|= 8|b|. (Easy)
25. Find the projection of the vector i + 3j + 7k on the vector 7i - j + 8k . (Easy)
26. Find the projection of the vector a = 2i + 3j + 2k on the vector b = i + 2j + k . (Easy)
27. If two vectors a and b such that |a|= 2, |b|= 3 and a.b = 4, find |a - b|. (Easy)
28. If a = 4i + 3j + k and b = i -2k, then find |2b × a| (Average)
29. Find the area of the parallelogram whose adjacent sides are determined by the vectors a = i - j + 3k
and b = 2i - 7j + k . (Easy)
30. Find the area of a parallelogram whose adjacent sides are given by the vectors a = 3i + j + 4k
and b = i - j + k . (Easy)
31. Find the area of the parallelogram whose adjacent sides determine by the vectors a = i + j - k
and b = i - j + k . (Easy)
32. Find the area of the triangle whose adjacent sides are determined by the vectors a = -2i- 5k
and b = i - 2j - k . (Easy)
2
33. Let the vectors a and b be such that |a|= 3, |b|= and a × b is a unit vector. Find the angle
3
between a and b (Easy)
34. Find  and , if (2i + 6j + 27k) × (i + λj + μk) = 0 . (Average)
35. If a . a =0 and a.b =0, then what can be concluded about the vector b ? (Average)

SUBJECT: 35 – MATHEMATICS Page 96 of 164

36. Show that |a| b + |b| a is perpendicular to|a| b - |b| a ,for any two nonzero
vectors a and b . (Average)
37. If either vector a =0 or b =0, then a . b =0. But the converse need not be true. Justify your
answer with an example. (Average)
38. Find|a × b|, if a = i − 7 j + 7k and b = 3i - 2j + 2k (Easy)
39. Find the area of the triangle with vertices A(1,1,2), B(2,3,5) and C(1,5, 5) (Easy)
40. The two adjacent sides of a parallelogram are 2i - 4j + 5k and i - 2j − 3k .Find the unit vector parallel
to its diagonal. (Difficult)
41. Find the projection of a on b if a . b =8 and b = 2 ˆi + 6j + 3k (Average)

Three mark questions:

1. Show that the position vector of the point P, which divides the line joining the points A and B
mb + na
having the position vectors a and b internally in the ratio m:n is . (Average)
m+n
2. Find a unit vector perpendicular to each of the vectors a + b and a - b , where a = i + j + k, b = i + 2j + 3k
(Average)
3. If a,b,c are unit vectors such that a +b + c = 0, find the value of a  b + b  c + c  a . (Average)
( ) ( ) (
4. Show that the points A 2iˆ − ˆj + kˆ , B ˆi − 3ˆj − 5kˆ and C 3iˆ − 4ˆj − 4k )
ˆ are the vertices of a right
angled triangle (Difficult)
5. If the vertices A, B and C of a triangle are (1,2,3) , ( −1,0,0 ) and ( 0,1,2) respectively, then find
the ABC . (Difficult)
6. Three vectors 𝐚⃗, 𝐛 𝐚𝐧𝐝 𝐜 satisfy the condition a +b + c = 0, . Evaluate the quantity = 𝐚⃗. 𝐛 + 𝐛. 𝐜 + 𝐜. 𝐚⃗ ,
if |𝐚⃗| = 𝟏, |𝐛| = 𝟒&|𝐜| = 𝟐. (Average)
7. Find the vector of magnitude 5 units and parallel to the resultant of the vectors 𝒂 ̂
⃗ = 𝟐𝒊̂ + 𝟑𝒋̂ − 𝒌
and 𝒃 ⃗ = 𝒊̂ − 𝟐𝒋̂ + 𝒌̂ (Average)
8. If 𝒂 ⃗
⃗ ,𝒃&𝒄 ⃗ are three vectors such that |𝒂 ⃗
⃗ | = 𝟑, |𝒃| = 𝟒, |𝒄 ⃗ | = 𝟓 and each vector is orthogonal to sum
of the other two vectors then find |𝒂 ⃗
⃗ +𝒃+𝒄 ⃗ |. (Average)
̂ ̂ ̂
9. If 𝒊̂ + 𝒋̂ + 𝒌 , 𝟐𝒊̂ + 𝟓𝒋̂ , 𝟑𝒊̂ + 𝟐𝒋̂ − 𝟑𝒌 and 𝒊̂ − 𝟔𝒋̂ − 𝒌 are the position vectors of points A,B,C & D
respectively then find the cosine angle between 𝑨𝑩 ⃗⃗⃗⃗⃗⃗ and 𝑪𝑫⃗⃗⃗⃗⃗⃗ . (Difficult)
10. If 𝒂⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝒌̂,𝒃 ⃗ = 𝒊̂ + 𝒋̂ − 𝟐𝒌̂ and 𝒄 ⃗ = 𝒊̂ + 𝟑𝒋̂ − 𝒌 ̂ such that ⃗𝒂 is perpendicular to (𝝀𝒃 ⃗ +𝒄⃗ ) then
find 𝝀. (Average)
11. The two adjacent sides of a parallelogram are 𝟐𝒊̂ − 𝟒𝒋̂ + 𝟓𝒌 ̂ , 𝒊̂ − 𝟐𝒋̂ − 𝟑𝒌
̂ then find the unit vector
parallel to its diagonal. Also find area of the parallelogram. (Difficult)
12. The scalar product of the vector 𝒊̂ + 𝒋̂ + 𝒌 ̂ with a unit vector along the sum of vectors
𝟐𝒊̂ + 𝟒𝒋̂ − 𝟓𝒌̂ & 𝝀𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌 ̂ is equal to one. Find the value of 𝝀. (Difficult)
13. If ⃗𝒂 = 𝒊̂ + 𝒋̂ − 𝒌 ̂,𝒃 ⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝒌̂&𝒄 ⃗ = −𝒊̂ + 𝟐𝒋̂ − 𝒌 ̂ then find the unit vector perpendicular to both
⃗𝒂 + 𝒃 ⃗ and 𝒃 ⃗ +𝒄 ⃗. (Average)
14. If 𝒂 ̂ ⃗ ̂
⃗ = 𝒊̂ + 𝟒𝒋̂ + 𝟐𝒌 , 𝒃 = 𝟑𝒊̂ − 𝟐𝒋̂ + 𝟕𝒌 & 𝒄 ̂ ⃗
⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝟒𝒌 then find a vector 𝒅 which is perpendicular
to both 𝒂 ⃗
⃗ 𝐚𝐧𝐝 𝒃 and 𝒄 ⃗
⃗ . 𝒅 = 𝟏𝟓. (Difficult)
⃗
15. If ⃗𝒂,𝒃 and 𝒄 ⃗ are mutually perpendicular vectors of equal magnitudes then prove that the vector
⃗𝒂 + 𝒃 ⃗ +𝒄 ⃗ is equally inclined to 𝒂 ⃗ and 𝒄
⃗ ,𝒃 ⃗ . (Average)

SUBJECT: 35 – MATHEMATICS Page 97 of 164

CHAPTER-11
THREE DIMENSIONAL GEOMETRY
1. The direction cosines of x, -axis. (Easy)
(A) 0 , 1 , 0 (B) 0, 0 , 1 (C) 1 , 0 , 0 (D) 0 , 1 , 1
2. The direction cosines of y, -axis. (Easy)
(A) 0 , 1 , 0 (B) 0, 0 , 1 (C) 1, 0 , 0 (D) 1 , 0 , 1
3. The direction cosines of z, -axis. (Easy)
(A) 0 , 1 , 0 (B) 0 , 0 , 1 (C) 1 , 0 , 0 (D) 1 , 1 , 0
4. The direction cosines of negative Z - axis are (Easy)
(A) 0,0,1 (B) 0,0,-1 (C) -1,-1,0 (D) 0,1,0
5. If a line has the direction ratios 2, -1, –2, then its direction cosines (Easy)
2 1 2 2 1 2
(A) –2 , 1 , 2 (B) 3 , - 3 , − 3 or − 3 , 3 , 3
2 1 2 2 1 2 2 1 2 2 1 2
(C) − 3 , , or , , (D) 3 , - 3 , − 3 or 3 , 3 , − 3
3 3 3 3 3
6. The direction cosines of the line passing through the two points (– 2, 4, – 5) and (1, 2, 3)
(Average)
3 2 8 3 2 8
(A) 3 , −2 , 8 (B) , − 77 , 77 (C) −3 , 2 , −8 (D) 77 , − 77 , 77.
77 √ √ √
7. Find the direction ratios of a line joining the points (-2, 4, 5) and (1, 2, 3). (Average)
(A) 3 , 2 , 2 (B) 3 , 2 , −2 (C) -3 , 2 , 2 (D) -3 , -2 , −2
8. For any line, if a, b, c are direction ratios of a line, then number of sets of direction ratios (Easy)
(A) 0 (B) 1 (C) 2 (D) infinitely many sets.
9. Let a, b, c direction ratios and l, m and n be the direction cosines of a line, such that
𝑙 𝑚 𝑛
𝑎
= 𝑏 = 𝑐 = 𝑘,then k is equal to (Easy)
1 1
(A) ± (B) ± 𝑎2 +𝑏2 +𝑐 2 (C) ± 𝑎2 + 𝑏 2 + 𝑐 2 (D) ± √𝑎2 + 𝑏 2 + 𝑐 2 .
√𝑎 2 +𝑏2 +𝑐 2
3𝜋 𝜋 𝜋
10. If a line makes 2 , , 4 with x, y, z axes resply, then its direction cosines are (Easy)
4
1 1 1 1 1 1 1 1
(A) 0 , - , (B) 0 , - ,− (C) 1 , , (D) 0 , ,
√2 √2 √2 √2 √2 √2 √2 √2
𝜋 𝜋 𝜋
11. If a line makes 2 , 3 , 6 with x, y, z axes resply, then its direction cosines are (Easy)
√3 1 1 √3 √3 1 1 √3
(A) 0 , - 2 , 2 (B) 0 , 2 , 2 (C) 0, 2 , 2 (D) 1, 2 , 2
12. If the line makes angles 90o, 60o and 45o with the positive direction of x , y and z-axes
respectively. Find its direction cosines. (Easy)
1 1 1 1 1 1 √3 1
(A) 0 , 2 , (B) 0 , - 2 , − 2 (C) 1 , 2 , (D) 0 , 2 ,
√2 √ √2 √2
13. If the direction cosines of a line is k, k, k, then (Average)
1 1
(A) k> 0 (B) 0<k<1 (C) k = 1 (D) k = 3 or -
√ √3
14. The equation of x-axis is (Easy)
(A) x = 0 (B) y = 0 and z=0 (C) x = 0 and y=0 (D) y = 0
15. If 𝛼, 𝛽, 𝑎𝑛𝑑 𝛾 are direction angles of a directed line ⃗⃗⃗⃗⃗
𝑂𝑃,then direction angles of the directed line
⃗⃗⃗⃗⃗
𝑃𝑂 are (Easy)
𝜋 𝜋 𝜋
(A) 𝛼, 𝛽, 𝛾 (B) −𝛼, −𝛽, − 𝛾 (C) 𝜋 − 𝛼, 𝜋 − 𝛽, 𝜋 − 𝛾 (D) 2 − 𝛼, 2 − 𝛽, 2 − 𝛾
𝜋 3𝜋 𝜋
16. If ,
2 4
and are direction angles of a directed line ⃗⃗⃗⃗⃗
4
𝑂𝑃,then direction angles of the directed
⃗⃗⃗⃗⃗ are
line 𝑃𝑂 (Difficult)
𝜋 3𝜋 𝜋 𝜋 𝜋 3𝜋 𝜋 −3𝜋 −𝜋 𝜋 3𝜋 𝜋
(A) − 2 , − 4 , - 4 (B) , , 4
2 4
(C) 2
, 4 , 4 (D) 2
, 4 , 4.
17. If the directed line ⃗⃗⃗⃗⃗
𝑂𝑃 makes angles 90o, 60o and 45o with the positive direction of x , y and
z-axes respectively. Then its direction cosines of the directed line ⃗⃗⃗⃗⃗
𝑃𝑂 (Difficult)
1 1 1 1 1 1 1 1
(A) 0 ,2 , 2 (B) 1 , − 2 , − 2 (C) 1 , 2 , 2 (D) 0 , − 2 , − 2.
√ √ √ √
18. If l, m, n are the direction cosines of a line, then (Easy)
(A) l2+ m2+ n2 = 0 (B) l2+ m2+ n2 = 2 (C) 2l2+2 m2+2 n2 = 2 (D) l2+ m2+ n2 = -1

SUBJECT: 35 – MATHEMATICS Page 98 of 164

19. If a line makes an angles α, β, γ with the positive direction of the co-ordinate axes.
Then sin2α + sin2β + sin2γ = (Average)
(A) 2 (B) 1 (C) 0 (D) -1
20. If a line makes an angles α, β, γ with the positive direction of the co-ordinate axes.
Then cos2α + cos2β + cos2γ (Easy)
(A) 2 (B) 1 (C) 0 (D) -1
21. A line makes equal angles with co–ordinate axes, then direction cosines of the lines are (Easy)
1 1 1 1 1 1 1 ±1 1
(A) ±1, ±1, ±1 (B) ± , ± 3 , ± 3 (C) 3 , − 3 , - 3 (D) ± 3 , 3 , ± 3
√3 √ √ √ √ √
22. The direction ratios of x, -axis. (Easy)
(A) 0 , k , 0 (B) 0 , 0 , 𝑘 (C) 𝑘 , 0 , 0 (D) k , 𝑘 , 𝑘
23. The equation of a line parallel to x-axis and passing through the origin is (Average)
x y z x y z x+5 y−2 z+3 x−5 y+2 z−3
(A) =1=1 (B) 1 = 0 = 0 (C) 0 = 1 = 0 (D) 0 = 0 = 1
0
24. The equation of the line through the point (5, 2, -4) and which is parallel to the vector
3𝑖̂+ 2𝑗̂ - 8𝑘̂ is (Easy)
x−3 y−2 z+8
̂
(A) r = (5î + 2 ̂j − 4k) + λ(3𝑖̂ + 2𝑗̂ − 8𝑘 )̂ (B) 5 = = −4
2
x+5 y+2 z+4
(C) r = (3𝑖̂ + 2𝑗̂ − 8𝑘̂ ) + λ (5î + 2 ̂j − 4k̂) (D) = = 3 2 −8
𝜋 𝜋 3𝜋
25. The equation of the line passing through origin with direction angles If 2 , 4 and 4
is
(Average)
x y z x y z x y z x y z
(A) = 2 = − 2 (B) 0 = −1 = −1 (C) 0 = 1 = −1 (D) 0 = 1 = 1
1 √ √
x−5 y+4 z−6
26. The Cartesian equation of the line is 3 = 7 = 2 , then vector equation of the line is (Easy)
(A) r = (−5î + 4 ̂j − 6k̂) + λ(3𝑖̂ + 7𝑗̂ + 2𝑘̂ ) ̂ ̂
(B) r = (5î + 4 ̂j − 6k) + λ(3𝑖̂ + 7𝑗̂ + 2𝑘 )

(C) r = (5î − 4 ̂j + 6k̂) + λ(3𝑖̂ + 7𝑗̂ + 2𝑘̂ ) (D) r = (3𝑖̂ + 7𝑗̂ + 2𝑘̂ ) + λ(5î − 4 ̂j + 6k̂)
𝑥−5 𝑦−2 𝑧+4
27. Find the direction cosines of the line = 2 = −8 (Average)
3
3 2 8 5 2 4 5 2 4
(A) 3 , 2 , −8 (B) 77 , 77 , − 77 (C) 77 , 77 , − 77 (D) , , − 45
√ √ √ √ √ √ √45 √45 √
𝑥+1 𝑦−2 𝑧+3 𝑥−1 𝑦+2 𝑧−3
28. The angle between the straight lines 7 = −5 = 1 and 1 = 2 = 3 is (Easy)
0 0 0 0
(A) 45 (B) 30 (C) 60 (D) 90
𝑥−1 𝑦−2 𝑧−3 𝑥−1 𝑦−1 𝑧−6
29. Lines −3 = 2𝑘 = 2 and 3𝑘 = 1 = −5 are perpendicular, then 𝑘 = (Easy)
10 10 7 7
(A) − 7 (B) 7 (C) − 10 (D) 10
𝑥−1 𝑦−2 𝑧−3 𝑥−1 𝑦−1 𝑧−6
30. Lines 3 = = 2 and 3𝑝 = 1 = 5 are perpendicular, then 𝑝 = (Easy)
2𝑝
10 10 11 11
(A) − 11 (B) 11 (C) − 10 (D) 10
31. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b is
(Easy)
0 0 0 0
(A) 45 (B) 30 (C) 60 (D) 90
32. The angle between two diagonals of a cube is (Average)
−1 1 −1 2 −1 −1 1
(A) cos (3) (B) cos (3) (C) tan √2 (D) cos (2)
33. The measure of the angle between the lines x = k +1, y = 2k - 1, z = 2k + 3, k ∈R and
x−1 y−2 z−3
2
= 1 = −1 is (Difficult)
2 2 1
(A) cos−1 (3 6) (B)cos−1 (3) (C)tan−1 √2 (D)cos −1 (2)
√
𝑥−1 𝑦−1 𝑧 𝑥−2 𝑦−1 𝑧−1
34. Two lines 2 = = and = = are (Average)
−1 1 3 −5 2
(A) ||r (B) ⊥r (C) skew lines (D) lines
𝑥−1 𝑦−1 𝑧 𝑥−2 𝑦−1 1−𝑧
35. Two lines 3 = 4
= 5 and 3 = 4
= 5 are (Difficult)
(A) parallel (B) 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
(C) skew lines (D) intersecting in an acute angle.

SUBJECT: 35 – MATHEMATICS Page 99 of 164

36. Two lines with direction ratios a1, b1, c1 and a2, b2, c2 are (Easy)
𝑎1 𝑏1 𝑐1
(A) perpendicular, if = 𝑏 = 𝑐
𝑎2
(B) parallel, if a1a2 + b1b2 + c1c2 = 0
2 2
a1 b1 c1
(C) perpendicular if a12 + b12 + c12 = a22 + b22 + c22 (D) parallel , if = =
a2 b2 c2
𝑎
37. If the direction ratios of two parallel lines be a1, a2, a3 and b1, b2, b3, then 𝑎3 = (Easy)
1
𝑎1 𝑏1 𝑏 𝑏1
(A) 𝑎2
(B) 𝑏3
(C) 𝑏3 (D) 𝑏2
1
x +1 y − 2 z + 3 x −1 y + 2 z − 3
38. Statement 1: The angle between the straight lines = = and = =
2 5 4 1 2 −3
is 90o
Statement 2: Two lines with direction ratios a1, b1, c1 and a2, b2, c2 are perpendicular,
if a1a2 + b1b2 + c1c2 = 0 (Easy)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false.
39. Assertion(A) : The equation of a line parallel to y-axis and passing through the origin is r = k𝑗̂
Reason(R) : The direction ratios of y –axis is 0, k , 0. (Average)
A) A is false and R is true B) A is false and R is false
C) A is true and R is false D) A is true and R is true.
40. Statement 1: Skew lines are non - intersecting non - parallel lines
(𝑎 ⃗⃗⃗⃗1 ×𝑏
⃗⃗⃗⃗1 )∙(𝑏
⃗⃗⃗⃗2−𝑎 ⃗ 2)
Statement 2: The distance between two Skew lines is | ⃗⃗⃗⃗ ⃗
|. (Average)
|𝑏1 ×𝑏2 |
A) Statement 1 is true and Statement 2 is false
B) Statement 1 is false and Statement 2 is true
C) Statement 1 is true and Statement 2 is true
D) Statement 1 is false and Statement 2 is false
41. Statement 1: The angle between the lines whose d r’s are given by 2,-3,3 and 3, 3, 1 is 90o
𝑎 𝑏 𝑐
Statement 2: Two lines with dr’s a1, b1, c1 and a2, b2, c2 are perpendicular if 𝑎1 = 𝑏1 = 𝑐1 .
2 2 2
(Easy)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is true.
C) Statement 1 is true and Statement 2 is true.
D) Statement 1 is false and Statement 2 is false.
x −5 y + 4 z −6
42. Assertion: If the cartesian equation of a line is = = , then its vector form is
3 7 2
(
r = 5i − 4 j + 6k +  3i + 7 j + 2k )
Reason: The vector equation of the line which passes through the point (x 1, y1, z1) and parallel
to the 𝑏⃗ = x𝑖̂ + y𝑗̂ + z𝑘̂ is r = (x1 î + y1 ̂j + z1 k̂) + λ(x𝑖̂ + y𝑗̂ + 𝑧𝑘̂ ). (Easy)
A) A is false and R is true B) A is false and R is false
C) A is true and R is false D) A is true and R is true
43. Assertion (A): If a line makes angles α, β, γ with positive direction of the coordinate axes, then
sin2α + sin2β + sin2 γ = 2
Reason (R): The sum of squares of the direction cosines of a line is 1, (Easy)
A) A is false and R is true B) A is false and R is false
C) Both A and R are true and R is the correct explanation of A.
D) Both A and R are true and R is not the correct explanation of A.

SUBJECT: 35 – MATHEMATICS Page 100 of 164

44. Statement 1: The acute angle between the line r = i + j + 2k +  i − j ( ) and the x-axis is 4
Statement 2: The acute angle between the lines r = x1i + y1 j + z1k +  a1i + b1 j + c1k ( ) and
a1a2 + b1b2 + c1c2
( )
r = x2 i + y2 j + z2 k +  a2i + b2 j + c2 k is given by cos  =
a12 + b12 + c12 a2 2 + b2 2 + c2 2
(Difficult)

A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false.
45. Statement 1: The lines 𝑟 = 𝑖̂ + 2 𝑗̂ - 4 𝑘̂ + 𝜆 (2𝑖̂ + 3 𝑗̂ + 6𝑘̂) and 𝑟 = 3𝑖̂ + 3 𝑗̂ -5 𝑘̂+ 𝜇 (2𝑖̂ + 3 𝑗̂ + 6𝑘̂)
are parallel.
𝑎 𝑏 𝑐
Statement 2: Two lines with d. r. ’s a1, b1, c1 and a2, b2, c2 are parallel if 𝑎1 = 𝑏1 = 𝑐1 . (Easy)
2 2 2
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is true.
C) Statement 1 is true and Statement 2 is true.
D) Statement 1 is false and Statement 2 is false.
46. In the figure the equation of straight line l is (Easy)
A) 𝑟 = 𝑏⃗ , ϵR
⃗
B) 𝑟 = 𝑎 × 𝑏 , ϵR
C) 𝑟 = 𝑎 + 𝑏⃗ , ϵR
D) 𝑟 = 𝑏⃗ + 𝑎 , ϵR
47. If 𝐿1 𝑎𝑛𝑑 𝐿2 are in the direction of 𝑎 and 𝑏⃗ respectively and
 is angle between a & b then 𝑐𝑜𝑠𝜃 = (Easy)
⃗|
|𝑎⃗ + 𝑏 ⃗|
|𝑎⃗ × 𝑏
A) |𝑎||𝑏⃗ B) |𝑎||𝑏⃗
⃗ | ⃗ |
⃗
𝑎⃗ .𝑏
C) |𝑎||𝑏⃗ D) 0
⃗ |
48. In the figure the equation of straight line AB is (Difficult)
2(x−1) √2(y−2) z−3 x−1 y−2 z−3
(A) = = 0 (B) 3 = 2 = 0
√3 1 √ √
x+1 y+2 z+3 (x−1) √2(y−2) z−3
(C) √3 = = 1 (D) = =
1 √3 1 0
2

49. The direction ratios of z –axis is a, b, c , then a= -------- (Easy)
50. The direction cosine of negative x, -axis is a,0,0 then a=------- (Easy)
𝑥−1 2−𝑦 𝑧−3 𝑥−1 𝑦−1 𝑧−6
51. Lines 3 = 2 = 1 and 3𝑘 = 2 = −5 are perpendicular, then 𝑘 = − − − − − (Easy)

TWO MARKS QUESTIONS:
𝑥−1 2𝑧+3
1. Find the direction ratios of the line = 3y = . (Average)
2 4
𝑥−5 𝑦+4 𝑧−6
2. The Cartesian equation of a line is = = . Write its Vector form. (Easy)
3 7 2
3. Find the direction cosines of the line which makes equal angles with coordinate axis. (Easy)
4. Find the direction cosines of the line passing through two points (-2, 4, -5) and (1, 2, 3). (Easy)
5. Find the equation of the line which passes through the point (1,2,3) and parallel to the
vector3𝑖 + 2𝑗 − 2𝑘⃗ ,both are in vector and Cartesian form. (Easy)
𝑥+3 𝑦−1 𝑧+3 𝑥+1 𝑦−4 𝑧−5
6. Find the angle between the lines 3 = = 4 and 1 = 1 = 2 . (Easy)
5

SUBJECT: 35 – MATHEMATICS Page 101 of 164

7. Show that the line passes through the points (4,7,8) , (2,3,4) is parallel to the line passing
through the points (-1,2,1) and (1,2,5). (Easy)
8. Show that the line through the points (1,-1,2) ,(3,4,-2) is perpendicular to the line through the
points (0,3,2) and (3,5,6). (Easy)
9. Show that the points (2,3,4) ,(-1,-2,1) and (5,8,7) are collinear. (Easy)
10. Show that the points A (2, 3, – 4), B (1, – 2, 3) and C (3, 8, – 11) are collinear. (Easy)
11. Find the Cartesian equation of the line through the point (5, 2, -4) and which is parallel to the
vector 3𝑖̂ + 2 𝑗̂ - 8 𝑘̂. (Easy)
12. Find the equation of the line in vector form that passes through the point with position vector
2i − j + 4k and is in the direction i + 2 j − k . (Easy)
𝐱+𝟑 𝐲−𝟏 𝐳+ 𝟑 𝐱+𝟏 𝐲−𝟒 𝐳−𝟓
13. Find the angle between the pair of lines 𝟑 = = 𝟏 and 𝟏 = −𝟏 = 𝟐 (Easy)
𝟓
14. Find the angle between the pair of lines 𝑟 = 3𝑖 + 2𝑗 − 4𝑘⃗+λ(𝑖 + 2𝑗 + 2𝑘⃗ .) and
𝑟 = 5𝑖 − 2𝑗 + 𝜇(3𝑖 + 2𝑗 + 6𝑘⃗ ) (Easy)
(
15. Find the angle between the pair of lines r = 3i + j − 2k +  i − j − 2k )
(
and r = 2i − j − 56k +  3i − 5 j − 4k ) (Easy)
16. Find the angle between the pair of lines𝒓 ⃗ = 2𝑖̂ -5 𝑗̂ + 𝑘̂ + 𝝀 (𝟑𝑖̂ + 2 𝑗̂ + 6𝑘̂) and
̂
⃗𝒓 = 7𝒊̂ - 6 𝒋̂+ 𝝁 (𝒊̂ + 2 𝒋̂ +2 𝒌) (Easy)
𝟏−𝐱 𝟕𝐲 − 𝟏𝟒 𝐳− 𝟑 𝟕−𝟕𝐱 𝐲−𝟓 𝟔−𝐳
17. Find the value of p, so that the lines
𝟑
= 𝟐𝒑 = 𝟐 and 𝟑𝒑 = 𝟏 = 𝟓 are at right angles.
(Average)
𝐱−𝟏 𝐲−𝟐 𝐳− 𝟑 𝐱−𝟏 𝐲 −𝟏 𝐳−𝟔
18. Find the value of k, so that the lines −𝟑 = 𝟐𝒌 = 𝟐 and 𝟑𝒌 = 𝟏 = −𝟓 are at right angles.
(Easy)
THREE MARKS QUESTIONS:
1. Find the shortest between the lines l1 and l2 whose vector equations are
𝑟 = 𝑖 + 𝑗 + 𝜆(2𝑖 − 𝑗 + 𝑘) and 𝑟 = 2𝑖 + 𝑗 − 𝑘 + 𝜇(3𝑖 − 5𝑗 + 2𝑘). (Easy)
2. Find distance between the lines l1 and l2 given by 𝑟 = 𝑖 + 2𝑗 − 4𝑘 + 𝜆(2𝑖 + 3𝑗 + 6𝑘) and
𝑟 = 3𝑖 + 3𝑗 − 5𝑘 + 𝜇(2𝑖 + 3𝑗 + 6𝑘) (Easy)
𝑥+1 𝑦+1 𝑧+1 𝑥−3 𝑦−5 𝑧−7
3. Find shortest distance between the lines = −6 = 1 and 1 = −2 = 1 . (Average)
7
4. Derive the equation of the line in space passing through the point and parallel to the vector in
the vector form. (Easy)
5. Derive the equation of the line in space passing through the point and parallel to the vector
both in the Cartesian form. (Easy)
6. Derive the angle between two lines in vector and Cartesian form. (Average)
7. Derive the shortest distance between skew lines both in vector form. (Difficult)
8. Derive the distance between the parallel lines 𝑟 = ⃗⃗⃗⃗ 𝑎1 + 𝜆(𝑏⃗)and 𝑟 = ⃗⃗⃗⃗𝑎2 + 𝜇(𝑏⃗). (Difficult)
9. Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to
𝐱−𝟖 𝐲+𝟏𝟗 𝐳−𝟏𝟎 𝐱−𝟏𝟓 𝐲 − 𝟐𝟗 𝐳−𝟓
the two lines : = = and = = . (Difficult)
𝟑 −𝟏𝟔 𝟕 𝟑 𝟖 −𝟓
10. Find the distance between the lines l1& l2 whose vector equations are
⃗ = 𝑖̂ + 𝑗̂+ 𝝀 (2𝑖̂ - 𝑗̂ + 𝑘̂) and𝒓
𝒓 ⃗ = 2𝑖̂ + 𝑗̂ - 𝑘̂+ 𝝁 (3𝑖̂ -5 𝑗̂ + 2𝑘̂). (Average)
11. Find the shortest distance between the lines whose vector equations are
⃗𝒓 = (𝟏 − 𝒕)𝑖̂ + (t-2) 𝑗̂ +(𝟑 − 𝟐𝒕) 𝑘̂ and 𝒓 ⃗ = (𝒔 + 𝟏)𝑖̂ + (2s -1) 𝑗̂ - (𝟐𝒔 + 𝟏) 𝑘̂ (Difficult)
12. Find the shortest distance between the lines whose vector equations are
⃗𝒓 = (𝟏 − 𝒕)𝑖̂ + (t-2) 𝑗̂ +(𝟑 − 𝟐𝒕) 𝑘̂ and 𝒓 ⃗ = (𝒔 + 𝟏)𝑖̂ + (2s -1) 𝑗̂ - (𝟐𝒔 + 𝟏) 𝑘̂ (Difficult)
13. Find the distance between the lines 𝒓 ⃗ = 𝑖̂ + 2 𝑗̂ + 𝑘̂ + 𝝀 (𝑖̂ - 𝑗̂ + 𝑘̂)
and ⃗𝒓 = 2𝑖̂ - 𝑗̂ - 𝑘̂+ 𝝁 (2𝑖
̂ + 3 𝑗̂ + 𝑘̂) (Average)
14. Find the distance between the lines 𝒓 ⃗ = 𝑖̂ + 2 𝑗̂ - 4 𝑘̂ + 𝝀 (2𝑖̂ + 3 𝑗̂ + 6𝑘̂)
and ⃗𝒓 = 3𝑖̂ + 3 𝑗̂ -5 𝑘̂+ 𝝁 (2𝑖̂ + 3 𝑗̂ + 6𝑘̂) (Average)
15. Find the distance between the lines 𝒓 ⃗ = 𝑖̂ + 2 𝑗̂ + 3 𝑘̂ + 𝝀 (𝑖̂ - 3 𝑗̂ + 2𝑘̂)
And ⃗𝒓 = 4𝑖̂ + 5 𝑗̂ + 6 𝑘̂+ 𝝁 (2𝑖 ̂ + 3 𝑗̂ + 𝑘̂) (Average)

SUBJECT: 35 – MATHEMATICS Page 102 of 164

CHAPTER -12
LINEAR PROGRAMMING
MCQ /FB questions.(not included in the model papers of KSEB)
1. A general class of problems which seek to be maximize or, minimize is called. (Easy)
(A) The objective functions (B) Linear programming problem
(C) Optimization problems (D Feasible solution
2. Z = ax + by, where a, b are constants is a linear objective function. Variables x and y are called
(Easy)
(A) Decision variables (B) Dependent variables
(C) Independent variables (D) None of these
3. Every point of feasible region is called (Easy)
(A) Infeasible region (B) Optimal solution
(C) Feasible solution (D) None of these
4. Feasible region is the set of points which satisfy (Easy)
(A) The objective functions (B) Some the given constraints
(C) All of the given constraints (D) Non negative constraints.
5. Objective function of a linear programming problem is (Easy)
(A) a constraint (B) function to be optimized
(C) A relation between the variables (D) Corner Points.
6. A set of values of decision variables which satisfies the linear constraints and non-negativity
conditions of a L.P.P is called its (Easy)
(A) Unbounded solution (B) Optimum solution
(C) Feasible solution (D) Feasible region
7. The optimal value of the objective function is attained at the (Easy)
(A) points on X-axis (B) points on Y-axis
(C) corner points of the feasible region (D) none of these
8. In a LPP, the objective function is always (Easy)
(A) cubic function (B) quadratic function
(C) Linear function (D) constant.
9. The number of feasible solution of a L.P.P is (Easy)
(A) one (B) two (C) finite (D) infinite.
10. Let R be the feasible region for a linear programming problem, and let Z = ax + by
be the objective function. If R is bounded, then Z has (Easy)
(A) only a maximum value on R (B) only a minimum value on R
(C) both a maximum and a minimum value on R (D) no minimum value on R
11. Maximum or a minimum may not exist for a linear programming problem if (Easy)
(A) The feasible region is bounded (B) If the constraints are non linear
(C) if the objective function is continuous (D) The feasible region is unbounded
12. In a LPP, which of the following is correct (Average)
(A) A corner point of a feasible region is a point in the region which is the intersection of two
boundary lines.
(B) A feasible region is bounded if it can be enclosed within a circle
(C) A feasible region is unbounded that the feasible region does extend indefinitely in any
direction.
(D) If two corner points produce the same maximum (or minimum) value of the objective
function, then every point on the line segment joining these points will not give the same
maximum (or minimum) value.
13. The corner points of the feasible region determined by the system of linear constraints are
(0, 0), (0, 50), (30, 0), (20, 30). The objective function is Z = 4x + y , then maximum value of Z is
(Easy)
(A) 210 (B) 150 (C) 110 (D) 120.

SUBJECT: 35 – MATHEMATICS Page 103 of 164

14. The corner points of the feasible region determined by the system of linear constraints are
(0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. Condition on p and q so that the
maximum of Z occurs at both the points (15, 15) and (0, 20) , then (Average)
(A) p = q (B) p = 2q (C) q = 2p (D) q = 3p.
15. Corner points of the feasible region determined by the system of linear constraints are (0, 3),
(1, 1) and (3, 0). Let Z = px + qy, where p, q > 0. Condition on p and q so that the minimum
of Z occurs at (3, 0) and (1, 1) is (Average)
(A) p = 2q (B) p = q/2 (C) p = 3q (D) p = q
16. Corner points of the feasible region for an LPP are (0, 5), (4, 3), (0, 6) .
Let Z = 200 x + 500y be the objective function. The Minimum value of Z occurs at (Easy)
(A) (0, 5) (B) (4,3) (C) (0, 6)
(D) line segment joining the points (0, 5) and (4, 3).
17. Corner points of the feasible region for an LPP are (0,10), (5,5), (0, 20), (15,15).
Let Z = 3x + 9y be the objective function. The maximum value of Z occurs at (Easy)
(A) (0, 20) (B) (15,15)
(C) line segment joining the points (15, 15) and (0, 20)
(D) line segment joining the points (5, 5) and (15, 15).
18. The corner points of the feasible region determined by the system of linear constraints are
(2, 72), (15, 20) and (40, 15). The objective function is Z = 6 x + 3 y,
then maximum value of Z is (Easy)
(A) 228 (B) 150 (C) 285 (D) 320.
19. Assertion (A): The maximum value of Z = 5x + 3y, satisfying the conditions
x ≥ 0, y ≥ 0 and 5x + 2y ≤ 10, is 15.
Reason (R): The optimal value of the objective function is attained at the corner points of the
feasible region. (Average)
A) A is false but R is true B) A is false and R is false
C) Both A and R are true and R is the correct explanation of A.
D) Both A and R are true and R is not the correct explanation of A.
20. Assertion (A): The minimum value of the objective Z = x + 3y. satisfying the conditions
2x + y ≤ 20, x ≥ 0, y ≥ 0 is 0.
Reason (R): In a LPP, the minimum value of the objective function Z =ax + by is always 0, if
origin is one of the corner point of the feasible region. (Average)
A) A is false and R is true B) A is false and R is false
C) Both A and R are true and R is the correct explanation of A.
D) Both A and R are true and R is not the correct explanation of A.
Six mark questions
Solve the following linear programming problems graphically
1. Maximize Z=4x+y subject to constraints x + y  50,3x + y  90,x  0, y  0 (Easy)
2. Maximize Z= 3x+4y subject to x + y ≤ 4, 𝑥 ≥ 0, 𝑦 ≥ 0 (Easy)
3. Minimize Z = 200x + 500y, subject to the constraints x + 2y ≥10, 3x + 4y ≤ 24, x ≥ 0, y ≥ 0.
(Easy)
4. Maximize 𝑍 = 3𝑥 + 2𝑦 subject to constraints 𝑥 + 2𝑦 ≤ 10,3𝑥 + 𝑦 ≤ 15, 𝑥, 𝑦 ≥ 0 (Average)
5. Minimize Z = – 3x + 4y, subject to constraints x + 2y ≤ 8, 3x + 2y ≥ 12, x ≥ 0, y ≥ 0. (Average)
6. Maximize and minimize : z = 5 x + 10 y subject to constraints
𝑥 + 2𝑦 ≤ 120, 𝑥 + 𝑦 ≥ 60, 𝑥 − 2𝑦 ≥ 0, 𝑥, 𝑦 ≥ 0. (Difficult)
7. Maximize and minimize : 𝑧 = 𝑥 + 2𝑦 subject to constraints
𝑥 + 2𝑦 ≥ 100, 2𝑥 − 𝑦 ≤ 0, 2𝑥 + 𝑦 ≤ 200, 𝑥, 𝑦 ≥ 0. (Difficult)
8. Minimize and Maximize Z = 3x + 9y
subject to the constraints: x + 3y ≤60, x + y ≥ 10 and x ≤ y and x ≥ 0, y ≥ 0. (Difficult)
9. Maximize Z = 5x + 3y, subject to constraints
3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥0. (Difficult)

SUBJECT: 35 – MATHEMATICS Page 104 of 164

Additional Questions
10. Maximize z= x+y subject to constraints x + 2 y  40, 2 x + y  50 , 𝑥 ≥ 0, 𝑦 ≥ 0 (Easy)
11. Maximize z= x+y subject to constraints x + 2 y  28,3x + y  24 , 𝑥 ≥ 0, 𝑦 ≥ 0 (Easy)
12. Maximize z = 17.5 x + 7 y subject to constraints x + 3 y  12,3x + y  12 , 𝑥 ≥ 0, 𝑦 ≥ 0 (Average)
13. Maximize z = 7 x + 10 y subject to constraints 2 x + 3 y  120 , 2 x + y  80 , 𝑥 ≥ 0, 𝑦 ≥ 0 (Easy)
14. Maximize z = 5 x + 6 y subject to constraints 25 + 8 y  200,10 x + 8 y  240 , 𝑥 ≥ 0, 𝑦 ≥ 0 (Easy)
15. Minimize and Maximize Z = 4x + y, subject to constraints
x + y ≥50, 3x + y ≤90, x, y ≥ 0. (Easy)
16. Maximize Z = 250x + 75y, subject to the constraints 5x + y ≥100, x + y ≤ 60, x ≥ 0, y ≥ 0.
(Average)

SUBJECT: 35 – MATHEMATICS Page 105 of 164

CHAPTER-13
PROBABILITY
MCQ /FB questions.
1) If 𝐸 & 𝐹 are any events then witch of the following is incorrect (Easy)
A) 𝑃(𝐹|𝐹) = 1 B) 𝑃(𝑆|𝐹) < 1
C) 𝑃(𝐸 ′ |𝐹) = 1 − 𝑃(𝐸|𝐹) D) 𝑃((𝐸 ∪ 𝐹)|𝐺) = 𝑃(𝐸|𝐺) + 𝑃(𝐹|𝐺) − 𝑃((𝐸 ∩ 𝐹)|𝐺)
7 9 4
2) If 𝑃(𝐴) = , 𝑃(𝐵) = 𝑎𝑛𝑑 𝑃(𝐴 ∩ 𝐵) = then 𝑃(𝐴|𝐵) is (Easy)
13 13 13
4 4 9 4
A) 13
B)9 C)13 D)7.
1 3 1
3) If 𝑃(𝐸) = 4 , 𝑃 (𝐹) = 4 𝑎𝑛𝑑 𝑃(𝐸 ∩ 𝐹) = 4 then 𝑃(𝐸|𝐹) is (Easy)
1 1 2
A) 4 B) 1 C) 3 D) 3
.
4) If 𝑃(𝐵) = 0.5 𝑎𝑛𝑑 𝑃(𝐴 ∩ 𝐵) = 0.32 then 𝑃(𝐴|𝐵) (Easy)
8 1 4 16
A) 25 B)2 C)25 D)25.
1
5) If 𝑃(𝐴) = 2 𝑎𝑛𝑑 𝑃(𝐵) = 0 then find 𝑃(𝐴|𝐵) is (Easy)
1
A) 2 B)1 C)0 D)𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡𝑠.
1
6) If 𝑃(𝐴) = 2 𝑎𝑛𝑑 𝑃(𝐵) = 0 then find 𝑃(𝐵|𝐴) is (Easy)
1
A) 2 B)1 C)0 D)𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡𝑠.
7) If 𝑃(𝐸) = 0.6, 𝑃(𝐹) = 0.3 𝑎𝑛𝑑 𝑃(𝐸 ∩ 𝐹) = 0.2 then 𝑃(𝐸|𝐹) 𝑖𝑠 (Easy)
1 2 3
A) 3 B) 3 C)0 D) 2.
8) If 𝑃(𝐸) = 0.6, 𝑃(𝐹) = 0.3 𝑎𝑛𝑑 𝑃(𝐸 ∩ 𝐹) = 0.2 then 𝑃(𝐹|𝐸) 𝑖𝑠 (Easy)
1 2 3
A) 3 B) 3 C)0 D) 2
.
9) If A is a subset of B and P(A) ≠ 0, then (𝐵|𝐴) is (Easy)
1
A) 1 B)0 C)2 D)𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡𝑠.
10) If 𝐴 ∩ 𝐵 = 𝜙 𝑎𝑛𝑑 P(A) ≠ 0, 𝑡ℎ𝑒𝑛 𝑃(𝐵|𝐴) is (Easy)
1
A) 2 B)1 C)0 D)𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡𝑠.
5 2
11) If 2𝑃(𝐴) = 𝑃(𝐵) = 𝑎𝑛𝑑 𝑃(𝐴|𝐵) = 5, then 𝑃(𝐴 ∩ 𝐵) is (Average)
13
5 15 19 2
A) B)26 C)26 D) 13.
26
5 2
12) If 2𝑃(𝐴) = 𝑃(𝐵) = 𝑎𝑛𝑑 𝑃(𝐴|𝐵) = 5 then 𝑃(𝐴 ∪ 𝐵) is (Average
13
5 15 11 2
A) 26 B)26 C)26 D) 13.
6 5 7
13) If 𝑃(𝐴) = 11 , 𝑃(𝐵) = 11 𝑎𝑛𝑑 𝑃 (𝐴 ∪ 𝐵) = 11 then 𝑃(𝐴/𝐵) is (Average)
5 6 4 4
A) 7 B) 7 C)5 D) 6.
6 5 7
14) If 𝑃(𝐴) = 11 , 𝑃(𝐵) = 11 𝑎𝑛𝑑 𝑃 (𝐴 ∪ 𝐵) = 11 then 𝑃(𝐵/𝐴) is (Average)
5 6 4 4
A) B) C) D) .
7 7 5 6
6 5 7
15) If 𝑃(𝐴) = 11 , 𝑃(𝐵) = 11 𝑎𝑛𝑑 𝑃 (𝐴 ∪ 𝐵) = 11 then 𝑃(𝐴 ∩ 𝐵) is (Average)
2 4 4 4
A) 11 B) C) D) .
11 5 6
16) If 𝑃(𝐴) = 0.8, 𝑃(𝐵) = 0.5, 𝑃(𝐵|𝐴) = 0.4 then 𝑃(𝐴 ∩ 𝐵) 𝑖𝑠 (Average)
A) 0.32 B)0.2 C)0.4 D)0.3
17) If 𝑃(𝐴) = 0.8, 𝑃(𝐵) = 0.5, 𝑃(𝐵|𝐴) = 0.4 then 𝑃(𝐴|𝐵) 𝑖𝑠 (Average)
16 4 1 4
A) 25 B) C) D)
5 2 25
18) If 𝑃(𝐴) = 0.8, 𝑃(𝐵) = 0.5, 𝑃(𝐵|𝐴) = 0.4 then 𝑃 (𝐴 ∪ 𝐵) 𝑖𝑠 (Average)
A) 0.32 B) 0.98 C) 0.9 D) 0.72

SUBJECT: 35 – MATHEMATICS Page 106 of 164

19) If A and B are any two events such that P(A) + P(B) – P(A and B) =P(A), then
P(B/A) = (Average)
𝑃(𝐵)
A) 1 B)0 C)𝑃(𝐴) D)𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡𝑠.
20) If A and B are two event such that P(A)≠ 0 and P(B | A) = 1, then (Easy)
(A) A ⊂B (B) B ⊂A (C) B = ∅ (D) A = ∅.
21) If A and B are events such that P(A|B) = P(B|A), then (Average)
(A) A ⊂B but A ≠B (B) A = B (C) A ∩B = ∅ (D) P(A) = P(B).
22) If P(A|B) > P(A), then which of the following is correct : (Average)
(A) P(B|A) < P(B) (B) P(A ∩ B) < P(A).P(B) (C) P(B|A) > P(B) (D) P(B|A) = P(B)
23) If A and B are two events such that A ⊂B and P(B) ≠0, then which of the following
is correct? (Average)
𝑃(𝐵)
(A) 𝑃(𝐵/𝐴) = (B) P(A∩B) < P(A) (C) P(A|B) = P(A) (D) 𝑃(𝐵/𝐴) = 1
𝑃(𝐴)
24) A die is rolled, consider an events 𝐸 = {1,3,5}, 𝐹 = {2,3}, then P(𝐸|𝐹)is (Average)
1 1 1 2
A) 3 B)2 C)5 D)3
25) A die is rolled, consider an events 𝐸 = {1,3,5}, 𝐹 = {2,3}, then𝑃(𝐹|𝐸)is (Average)
1 1 1 2
A) 3 B)2 C)5 D)3
26) A die is rolled, consider an events 𝐸 = {1,3,5}, 𝐺 = {2,3,4,5}, then𝑃(𝐺|𝐸)is (Average)
1 1 1 2
A) 3 B)2 C)5 D)3
27) A is rolled, consider an events 𝐸 = {1,3,5}, 𝐺 = {2,3,4,5}, then𝑃(𝐸|𝐺)is (Average)
1 1 1 2
A) 3 B)2 C)5 D)3
28) A die is rolled, consider an events 𝐸 = {1,3,5}, 𝐹 = {2,3}and G = {2,3,4,5} then 𝑃(𝐸 ∪ 𝐹|𝐺)is
(Average)
1 1 3
A) 1 B)2 C)4 D)4
′ |𝐵)
29) If 𝑃(𝐴) = 0.4 𝑃(𝐵) = 0.5 𝑎𝑛𝑑 𝑃(𝐴 ∩ 𝐵) = 0.25 then 𝑃(𝐴 𝑖𝑠 (Average)
5 1 1 3
A) B) C) D) .
8 2 4 4
30) In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and
20% read both Hindi and English newspapers. A student is selected at random. If she
reads Hindi newspaper, find the probability that she reads English newspaper. (Difficult)
1 1 2 1
A) 3 B)2 C) 3 D) 4.
31) In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and
20% read both Hindi and English newspapers. A student is selected at random. If she
reads English newspaper, find the probability that she reads Hindi newspaper. (Difficult)
1 1 2 1
A) 3 B)2 C) 3 D) 4.
32) Two cards are drawn random without replacement from a pack of 52 playing cards.
Find the probability that both are black cards. (Average)
1 1 25 25
A) 26 B)4 C)102 D)104.
33) A Urn contains 10 black and 5 white balls, 2 balls are drawn one after the other without
replacement. What is the probability that both drawn balls are black. (Average)
3 4 1 2
A) 7 B)9 C)9 D)21
34) Three cards drawn successively without replacement from a pack of 52 well shuffled cards.
What is the probability that 1st two cards are king and 3rd card drawn is ace. (Difficult)
2 1 2 1
A) 13×13×13 B) C) D) .
13×13×13 13×17×25 13×17×25
35) If A and B are independent events then (Easy)
A) A and B’ are dependent B) A’ and B are dependent
C) A’ and B’ are dependent D) 𝑃 (𝐴 ∪ 𝐵) = 1 − 𝑃(𝐴′ )𝑃(𝐵′ ).
36) Two events A and B are said to be independent, if, (Easy)
(A) A and B are mutually exclusive (B) P(A′∩B′) = [1 – P(A)] [1 – P(B)]
(C) P(A/B) = P(B) (D) P(B/A) = P(A).

SUBJECT: 35 – MATHEMATICS Page 107 of 164

37) If A and B are two independent events then the probability of occurrence of
at least one of A and B is (Average)
′ )𝑃(𝐵′ ) ′ )𝑃(𝐵′ )
A) 1 + 𝑃(𝐴 B) 1 − 𝑃(𝐴) − 𝑃(𝐵) C) 1 − 𝑃(𝐴 D) 1 − (𝑃(𝐴) + 𝑃(𝐵))
38) Two events E and F are independent events, which of the following is not true (Easy)
A) P(F|E) = P (F) B) P (E|F) = P (E)
C) P(E ∪ F ) = P (E) + P (F) D)P(E ∩ F) = P(E) . P (F).
3 1
39) If A & 𝐵 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑒𝑣𝑒𝑛𝑡𝑠 and 𝑃(𝐴) = , 𝑃(𝐵) = ,then 𝑃(𝐴 ∩ 𝐵) is (Easy)
5 5
1 3 22 2
A) 3 B)25 C)25 D)3
3 3 1
40) Let E and F be two events such that 𝑃(𝐸) = 5 , 𝑃(𝐹) = 10 and 𝑃(𝐸 ∩ 𝐹) = 5.
then E and F are (Easy)
A) dependent events B) independent events
C) mutually exclusive events D) 𝑒𝑥ℎ𝑎𝑢𝑠𝑡𝑖𝑣𝑒 𝑒𝑣𝑒𝑛𝑡𝑠.
1 1
41) If A and B are two independent events such that 𝑃(𝐴) = 4 and 𝑃(𝐵) = 2 then
P(not A and not B) (Average)
1 3 7 1
A) 3 B)8 C)8 D)2
42) If A and B are independent events with 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.4 then 𝑃(𝐵|𝐴) is (Easy)
3 2 3 7
A) 10 B)5 C)25 D)10
43) If A and B are independent events with 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.4 then 𝑃(𝐴|𝐵) (Easy)
3 2 3 7
A) 10 B)5 C)25 D)10
44) If A and B are independent events with 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.4 then 𝑃(𝐴 𝑎𝑛𝑑 𝐵) (Easy)
A) 0.3 B) 0.4 C)0.12 D)0.7
45) If A and B are independent events with 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.4 then 𝑃(𝐴 𝑜𝑟 𝐵) (Easy)
A) 0.3 B) 0.42 C)0.12 D) 0.58.
46) If A and B are independent events with 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.4 then 𝑃(neither A nor B) (Easy)
A) 0.3 B) 0.42 C)0.12 D) 0.58
1 1 1
47) If A and B are two events such that P (A) = 2, P (B) = 4 and P(𝐴 ∩ 𝐵) = 8 ,
then P (not A and not B) (Average)
1 3 7 1
A) 3 B)8 C)8 D)2
1 7 1
48) If 𝑃(𝐴) = 2 , 𝑃(𝐵) = 12 𝑎𝑛𝑑 𝑃(𝑛𝑜𝑡 𝐴 𝑜𝑟 𝑛𝑜𝑡 𝐵) = 4 then A and B are. (Average)
A) dependent events B) independent events
C) mutually exclusive events D) 𝑒𝑥ℎ𝑎𝑢𝑠𝑡𝑖𝑣𝑒 𝑒𝑣𝑒𝑛𝑡𝑠.
49) If A and B are independent events such that 𝑃(𝐴) = 0.3 𝑎𝑛𝑑 𝑃(𝐵) = 0.6,
then P(A and not B) is (Average)
A) 0.12 B)0.18 C)0.28 D)0.42
50) If A and B are independent events such that 𝑃(𝐴) = 0.3 𝑎𝑛𝑑 𝑃(𝐵) = 0.6,
then P (neither A nor B) is (Average)
A) 0.12 B)0.18 C)0.28 D)0.42
1 3
51) A and B are events such that 𝑃(𝐴) = , 𝑃(𝐴 ∪ 𝐵) = , 𝑃(𝐵) = 𝑞, then the value of q if A and B
2 5
are mutually exclusive (Average)
3 1 1 7
A) 10 B)10 C)5 D)10
1 3
52) A and B are events such that 𝑃(𝐴) = , 𝑃(𝐴 ∪ 𝐵) = , 𝑃(𝐵) = 𝑞, then the value of q if A and B
2 5
are independent (Average)
3 1 1 7
A) 10 B)10 C)5 D)10
53) An electronic assembly consists of two subsystems say A and B. From previous
testing procedures, the following probabilities are assumed to be known, P(A fails) = 0.2,
P(B fails alone) = 0.15 and P(A and B fail) = 0.15, then P(A fails alone) (Difficult)
A) 0.15 B) 0.5 C) 0.05 D) 0.75

SUBJECT: 35 – MATHEMATICS Page 108 of 164

54) An electronic assembly consists of two subsystems, say, A and B. From previous
testing procedures, the following probabilities are assumed to be known, P(A fails) = 0.2,
P(B fails alone) = 0.15 and P(A and B fail) = 0.15, then P(A fails |B has failed) is (Difficult)
A) 0.15 B) 0.5 C) 0.05 D) 0.75.
55) The probability of obtaining an even prime number on each die, when a pair of
dice is rolled is (Average)
1 1 1 1
A) 36 B)6 C) 18 D)4.
56) A die is tossed thrice. Find the probability of getting an odd number at least once. (Average
1 1 3 7
A)
8
B)4 C) 4 D)8.
1 1
57) Probability of solving a specific problem independently by A and B are 2 and 3 respectively.
If both try to solve the problem then the probability that the problem is solved (Average)
1 2 1 5
A) 6 B)3 C)3 D)6
1 1
58) Probability of solving a specific problem independently by A and B are and respectively.
2 3
If both try to solve the problem then the probability that exactly one of them solves
the problem is (Average)
1 2 1 1
A) 6 B)3 C)3 D) 2
59) Two balls are drawn at random with replacement from a box containing 10 black and 8
red balls, then the probability that both are red is (Average)
25 16 20 28
A) 81 B)81 C)81 D) 153
60) Two balls are drawn at random with replacement from a box containing 10 black and
8 red balls find the probability that 1st ball is black and second is red (Average)
25 16 20 28
A) 81 B)81 C)81 D) 153
61) Two balls are drawn at random with replacement from a box containing 10 black and
8 red balls find the probability that One of them is black and other is red (Average)
40 20 16 56
A) B) C) D)
81 81 81 153
62) Statement 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 (Average)
and P(E ∩ F) = 0.2, then P(E|F) = 2/3
𝑃(E ∩ F)
Statement 2: Let E and F be two events with a random experiment, then P(E|F) = P(E)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is false.
63) Statement 1: If P(A) = 0 and P(B) ≠ 0, then find P(A /𝐵) = 0 (Easy)
Statement 2: If P(A) = 0, then P(A∩B) = 0
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for
Statement 1
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
D) Statement 1 is false and Statement 2 is true.
64) For any two independent events A and B. P(A)= x and P(B)= y (Difficult)
Assertion (A): The probability that at least one of the events A and B occurs is x + y -2 xy.
Reason (R):𝑃(E ∩ F / ) = 𝑃(E)𝑃(F / ).
A) A is false and R is true B) A is false and R is false
C) Both A and R are true and R is the correct explanation of A.
D) Both A and R are true and R is not the correct explanation of A.

SUBJECT: 35 – MATHEMATICS Page 109 of 164

65) Assertion (A): In rolling a die, event A= {1, 3, 5} and event B = {3, 6} are mutually (Easy)
independent events.
Reason (R):If A and B are two independent events then 𝑃(E ∩ F) = 𝑃(𝐸)𝑃(𝐹).
A) A is false and R is true B) A is false and R is false
C) Both A and R are true D) A is true and R is false
1 1
66) Assertion (A) : Let A and B be two events such that P (A) = , P (B) = 2 and
5
(Easy)
1
P(A/B)= 5, then A and B are independent events.
Reason (R) : If A and B are two independent events then P ( B/A) = P(B) .
A) A is false and R is true
B) Both A and R are true and R is the correct explanation of A.
C) Both A and R are true and R is not the correct explanation of A.
D) A is false and R is false
5 4 𝑘
67) If 𝑃(𝐵) = 12 , 𝑃(𝐴 ∩ 𝐵) = 12 𝑎𝑛𝑑 𝑃(𝐴|𝐵) = 5 ,then k= ----------- (Average)
′ 𝑘
68) If 𝑃(𝐹) = 0.6 𝑎𝑛𝑑 𝑃(𝐸 ∩ 𝐹) = 0.2 then find 𝑃(𝐸 |𝐹) = 3 ,then k= ------------- (Average)
1
69) If 𝑃(𝐴) = 3 𝑎𝑛𝑑 𝑃(𝐵|𝐴) = 0 then 𝑃(𝐵) = − − − − (Easy)
1
70) If P(A) = 0 and P(B) = 4, then P(A /𝐵) =--------- ` (Easy)
71) If A and B are two events such that A is a sub set of B and P(A) ≠ 0, then P(B/A) =- (Easy)
72) If A≠ ∅ is a subset of B then 𝑃(𝐵|𝐴) = − − − − − (Easy)
73) If P (𝐴) ≠ 0 , then 𝑃(𝐴|𝐴) = − − − − − (Easy)
3 1 3
74) If 𝑃(𝐴) = , 𝑃(𝐵) = 3 and A and B are independent events then 𝑃(𝐴 ∩ 𝐵) = 𝑘 ,then k= (Easy)
5
75) For the figure given below the P(Second Pick R) is (Average)

1 1 3 1
A) 2 B)4 C)10 D) 5.
76) For the figure given below P (B/A )= (Easy)

4 4 4 5
A) B) C) D) .
7 12 9 12
77) For the figure given below the P(Walk) is (Average)

11 8 1 7
A) B)15 C)15 D) 18.
15

SUBJECT: 35 – MATHEMATICS Page 110 of 164

78) For the figure given below the P(B / /A) is (Average)

1 4 2 3
A) 5 B)5 C)3 D) 4.

79. A coin is flipped and a dice is rolled. (Average)
1
Statement 1: The probability of getting a ‘tail’ and a 6’ is 12
5
Statement 2: The probability of getting a ‘tail’ and a not 6’ is 12.
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is true and Statement 2 is true.
C) Statement 1 is false and Statement 2 is true.
D) Statement 1 is false and Statement 2 is false.
80. For the figure given below (Average)

20
Statement 1: P(both are white)=56
30
Statement 2: P(at least one white) =56
6
Statement 3: P(both are black) =56
Which of the above statements are correct?
(A) 1 and 3 only (B) 2 and 3 only (C) 1 and 3 only (D) All 1, 2 and 3.

Two mark questions.
1. If A and B are independent events with 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.4 then
find 𝑃(not A and not B). (Average)
2. If A and B are independent events such that 𝑃(𝐴) = 0.3 𝑎𝑛𝑑𝑃(𝐵) = 0.6, find P(A and B). (Average)
3. If A and B are independent events such that 𝑃(𝐴) = 0.3 𝑎𝑛𝑑𝑃(𝐵) = 0.6,
find P(A and not B). (Average)
5 2
4. If 2𝑃(𝐴) = 𝑃(𝐵) = 𝑎𝑛𝑑 𝑃(𝐴|𝐵) = then find 𝑃(𝐴 ∪ 𝐵). (Average)
13 5
5. Prove that 𝑃(𝐴𝑙 |𝐵) = 1 − 𝑃(𝐴|𝐵). (Average)
6. If A and B are independent events with 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.4 then find 𝑃(𝐴 ∪ 𝐵). (Average)
6 5 7
7. If 𝑃(𝐴) = 11 , 𝑃(𝐵) = 𝑎𝑛𝑑𝑃(𝐴 ∪ 𝐵) = then find 𝑃(𝐵|𝐴). (Average)
11 11
1 3
8. A and B are an events such that 𝑃(𝐴) = 2 𝑃(𝐴 ∪ 𝐵) = 5 , 𝑃(𝐵) = 𝑞,then find q if A and B are
Independent. (Average)
1 1 2
9. Let A and B are two events such that 𝑃(𝐴) = , 𝑃(𝐴|𝐵) = 2 𝑎𝑛𝑑 𝑃(𝐵|𝐴) = 3
4
then find𝑃(𝐵). (Average)
10. A fair die is rolled consider an events 𝐸 = {2,4,6} 𝑎𝑛𝑑 𝐹 = {1,2} then find 𝑃(𝐸|𝐹). (Easy)
11. A fair die is rolled consider an events E = {1,3,5} and F = {2,3,5}, then find P(F/E). (Easy)

SUBJECT: 35 – MATHEMATICS Page 111 of 164

12. A couple has two children. Find the probability that both children are males if it is known that
at least one of the children is male. (Average)
13. Mother, Father and son line up at random for a family picture , Find P(E/F).
If E: son on one end, F: father in middle. (Average)
14. Consider an experiment of tossing two fair coins simultaneously. Find the probability that both
are heads. Given that at least one of them is head. (Average)
15. A couple has 2 children find the probability that both are female if it is known that elder child
is female. (Average)
16. Given that the 2 number appear in on throwing two dices are different.
Find the probability of an event the sum of the number is 4. (Average)
17. Find the conditional probability of obtaining the sum 8 given that the red die resulted
is a number less than 4. (Average)
18. In a hostel 60% of students read Hindi newspaper, 40% of students read English newspaper
and 20% read both Hindi and English newspapers. A student is selected at random then,
If she reads Hindi newspaper find the probability that she also reads English
newspapers. (Difficult)
19. A coin is tossed 3 times then find 𝑃(𝐸|𝐹), where E : Head on third toss and
F : Head on first two tosses . (Average)
20. A coin is tossed 3 times then find 𝑃(𝐸|𝐹),where E :at least two heads and
F : at most two heads. (Average)
21. A black and red dice are rolled. Find the conditional probability of obtaining the sum greater
than 9. Given that black die resulted as 5. (Average)
22. If A and B are independent events, then prove that A and B’ are also independent. (Average)
23. If A and B are independent events, then prove that A’ and B are also independent. (Average)
24. If A and B are independent events, then prove that A’ and B’ are also independent. (Average)
25. If A and B are two independent events then prove that the probability of occurrence of
at least one of A and B is given by 1 − 𝑃(𝐴′ ). 𝑃(𝐵′ ). (Average)
1 7 1
26. If 𝑃(𝐴) = , 𝑃(𝐵) = 12 𝑎𝑛𝑑 𝑃(𝑛𝑜𝑡𝐴 𝑜𝑟 𝑛𝑜𝑡𝐵) = 4 then state whereas A or B are independent (Average)
2
1 1 1
27. If A and B are two events such that 𝑃(𝐴) = 𝑎𝑛𝑑 𝑃(𝐵) = 2. Then 𝑃(𝐴 ∩ 𝐵) = 8 then
4
Find P(not A and not B) . (Average)
28. Find the probability of getting even prime number on each die, when a pair of dice is rolled.
(Average)
29. Two cards are drawn random without replacement from a pack of 52 playing cards.
Find the probability that both are black cards. (Average)
30. Two cards are drawn successfully with replacement from a pack 52 cards
Find the probability distribution of number of ace cards. (Average)
31. An Urn contains 10 black and 5 white balls, 2 balls are drawn one after the other without
replacement. What is the probability that both drawn balls are black. (Average)
32. Three cards drawn successively without replacement from a pack of 52 well shuffled cards.
What is the probability that 1st two cards are king and 3rd card drawn is ace. (Average)
33. Two balls are drawn at random with replacement from a box containing 10 black and
8 red balls find the probability that both are red. (Average)
34. Two balls are drawn at random with replacement from a box containing 10 black and
8 red balls find the probability that 1st ball is black and second is red. (Average)
35. A die is tossed thrice. Find the probability of getting an odd number at least once. (Average)
1 1
36. Probability of solving a specific problem independently by A and B are 2 and respectively.
3
If both try to solve the problem independently. Find the probability that the problem is solved.
(Average)
37. If P(𝐸1 ) = ½ , P(𝐸2 )= ½ and P(𝐴/𝐸1 )= ½, P(𝐴/𝐸2 )= ¼ . Find P(𝐸1 /𝐴). (Average)

SUBJECT: 35 – MATHEMATICS Page 112 of 164

Three mark questions.
1. A die is thrown twice and sum of the numbers appeared is observed to be six.
What is the conditional probability that the number 4 has appeared at least once. (Average)
2. A die is thrown 3 times. Events A and B are defined as follows.
Event A :4 on first throw and Event B : 6 and 5 on second and third throw.
Find the probability of ‘A’ given that ‘B’ has already occurred. (Average)
3. A pair of die are thrown, an event A and B are as follows, A : the sum of 2 numbers on the die
is 8 and B : there is an even number on the first die.
Find the conditional probability 𝑃(𝐵|𝐴). (Average)
4. 10 cards numbered from 1 to 10 are placed in a box mix up thoroughly and 1 card is drawn
random, if it is known that the number on the drawn card is more than 3.
What is the probability that it is an even number. (Average)
5. An instructor has question bank consisting of 300 easy true/false questions, 200 difficult
true/false questions, 500 easy multiple choice questions and use 400 difficult MCQ’s. If a
question is selected at random from the question bank. What is the probability that it will be a
easy question given that it’s a MCQ. (Difficult)
6. One card is drawn at random from a well shuffled deck of 52 cards. Find where events E and F
are independent. E : the card drawn is a spade and F: the card drawn is an ace. (Difficult)
7. A die is marked 1,2,3 in red and 4,5,6 in green is tossed. Let ‘A’ be an event that ‘the number
is even’ and B be an event that ‘the number is red. Are A and B independent. (Difficult)
8. An unbiased dies is thrown twice, let A be an event ‘odd number on the first thrown’ let ‘B’ be
an event odd number on the 2nd thrown check the independence of the events A and B.(Difficult)
9. A die is thrown, if E be an event, the number appearing is a multiple of 3 and F be an
event the number appearing is even, then find whether E and F are independent. (Difficult)
10. A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls.
One of the two bags is selected at random and a ball is drawn from the bag.
What is the probability that the ball is red. (Difficult)
11. An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and
is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and
then a ball is drawn at random. What is the probability that the second ball is red ? (Difficult)
12. Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls.
One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II . The ball so
drawn is found to be red in colour. Find the probability that the transferred ball is black.(Difficult)
13. A bag contains 3 red and 4 black balls, another bag contains 5 red and 6 black balls.
One of the two bags is selected at random and a ball is drawn from the bag which is found to
be red. Find the probability that the ball is drawn from the first bag. (Difficult)
14. There are three coins, one is a two headed coin, another is a biased coin that comes up
head 75% 𝑜𝑓𝑡ℎ𝑒𝑡𝑖𝑚𝑒𝑎𝑛𝑑𝑡ℎ𝑖𝑟𝑑𝑖𝑠an unbiased coin. One of the three coins is chosen at random
and tossed it shows head. What is the probability that it was the two headed coin. (Difficult)
15. Given three identical boxes I, II and III, each containing two coins. In box I, both coins are gold
coins, in box II, both are silver coins and in the box III, there is one gold and one silver coin. A
person chooses a box at random and takes out a coin. If the coin is of gold, what is the
probability that the other coin in the box is also of gold? (Difficult)
16. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers.
The probabilities of an accident are 0.01, 0.03 and 0.15 respectively. One of the insured
person meets with an accident. What is the probability that he is a scooter driver? (Difficult)
17. A manufacturer has three machine operators A, B and C. The first operator A produces 1%
defective items, where as the other two operators B and C produce 5% and 7% defective
items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and
C is on the job for 20% of the time. A defective item is produced, what is the probability that it
was produced by A? (Difficult)

SUBJECT: 35 – MATHEMATICS Page 113 of 164

18. A doctor is to visit a patient from the past experience it is known that the probabilities that
3 1 1 2
he will come by train, bus scooter or by other means of transportation are , 5 , 10 𝑎𝑛𝑑 5
10
1 1 1
respectively. The probability that he will be late are 4 , 3 & 12. If he comes by train,
bus scooter respectively. But he comes by the means of transport he will not be late.
When he arrive, is late. What is the probability be will come by trains. (Difficult)
19. Of the students in a college it is known that 60% reside in hotel and 40% are day scholar
(not residing in hostel) previous year results report that 30% of the student who reside in
hostel attain ‘A’ grade and 20% of day scholars attain ‘A’ grade in their annual examination.
At the end of the year one student is chosen at random from college and he has ‘A’ grade.
What is the probability that the student is a hosteller? (Difficult)
20. In a factory which manufactures bolts, machines A, B and C manufacture respectively
25%, 35% and 40% of the bolts. Of their outputs, 5, 4 & 2 percent are respectively defective
bolts. A bolt is drawn at random from the product and is found to be defective.
What is the probability that it is manufactured by the machine B? (Difficult)
21. In answering a question on a multiple choice test a student either knows the answer or guesses.
Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses.
Assuming that a student who guesses the answer will be correct with probability 1/4 .
What is the probability that a student knows the answer given that he answered it correctly.
(Difficult)
22. A man is know to speak truth 3 out of 4 times. He throws a die and reports that it is six.
Find the probability that it is actually six. (Difficult)
4
23. Probability that a person speaks truth is 5, A coin is tossed a person reports that head
appears. Find the probability that it is actually head. (Difficult)
24. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are
drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.
(Difficult)
25. A factory has two machines A and B. Past record shows that machine A produced 60% of the
items of output and machine B produced 40% of the items. Further, 2% of the items produced
by machine A and 1% produced by machine B were defective. All the items are put into
one stockpile and then one item is chosen at random from this and is found to be defective.
What is the probability
that it was produced by machine B? (Difficult)
26. Suppose that the reliability of a HIV test is specified as follows: Of people having HIV, 90% of
the test detect the disease but 10% go undetected. Of people free of HIV, 99% of the test are
judged HIV–ve but 1% are diagnosed as showing HIV+ve. From a large population of which
only 0.1% have HIV, one person is selected at random, given the HIV test,
and the pathologist reports him/her as HIV+ve. What is the probability that the person
actually has HIV? (Difficult)
***************

SUBJECT: 35 – MATHEMATICS Page 114 of 164

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EXAMINATION AND ASSESSMENT BOARD
II PUC MATHEMATICS (35) EXAMINATION – 1, 2025
Duration: 3hrs Max Marks: 80
Instructions:
1) The question paper has five Parts namely 𝐀, 𝐁, 𝐂, 𝐃 and 𝐄. Answer all the Parts.
2) Part - A has 𝟏𝟓 multiple choice questions, 5 fill in the blank questions.
3) For Part-A questions, only the first written answers will be considered for evaluation.
4) Use the graph sheet for question on Linear Programming Problem in Part-E.
PART-A
I. Answer all the Multiple-choice questions: (𝟏𝟓 × 𝟏 = 𝟏𝟓)
1. A relation 𝑅 in a set 𝐴 is called reflexive relation if
a) (𝑎, 𝑎) ∈ 𝑅 For all 𝑎 ∈ 𝐴 b) (𝑎, 𝑎) ∈ 𝑅 For at least one 𝑎 ∈ 𝐴
c) (𝑎, 𝑏) ∈ 𝑅 Implies (𝑏, 𝑎) ∈ 𝑅 d) (𝑎, 𝑏) ∈ 𝑅 and (𝑏, 𝑐) ∈ 𝑅 Implies (𝑎, 𝑐) ∈ 𝑅
−1 1
2. The principal value of sin ( 2) is
√
𝜋 𝜋 𝜋 𝜋
a) 2 b) 3 c) 4 d) 6
3. Match list -I With list -II.
List -I List -II
−𝜋 𝜋
A) Domain of sin−1 𝑋 i) ( 2 , 2 )

B) Range of tan−1 𝑋 ii) [0, 𝜋]

C) Range of cos−1 𝑋 iii) [−1,1]
Choose the correct answer from the options given below:
a) 𝐴 − 𝑖, 𝐵 − 𝑖𝑖, 𝐶 − 𝑖𝑖𝑖 b) 𝐴 − 𝑖𝑖𝑖, 𝐵 − 𝑖𝑖, 𝐶 − 𝑖
c) 𝐴 − 𝑖𝑖, 𝐵 − 𝑖, 𝐶 − 𝑖𝑖𝑖 d) 𝐴 − 𝑖𝑖𝑖, 𝐵 − 𝑖, 𝐶 − 𝑖𝑖
4. For a 2 × 2 matrix 𝐴 = [𝑎𝑖𝑗 ] whose elements are given by 𝑎𝑖𝑗 = 2𝑖 − 𝑗 is equal to
2 3 1 0 1 1 1 2
a) [ ] b) [ ] c) [ ] d) [ ]
1 2 3 2 2 2 2 1
5. Let 𝐴 be a non-singular matrix of order 3 × 3, then |𝑎𝑑𝑗𝐴| is equal to
a) |𝐴| b) 3|𝐴| c) |𝐴|3 d) |𝐴|2
′ 𝜋
6. If 𝑓(𝑥) = cos 2x, then 𝑓 (4 ) is
a) 2 b) -2 c) √2 d) −√2
7. For the given figure consider the following statements 1 and 2 :

Statement 1: Left hand derivative of 𝑦 = 𝑓(𝑥) at 𝑥 = 1 is -1 .
Statement 2: The function 𝑦 = 𝑓(𝑥) is differentiable at 𝑥 = 1.
Then which of the following are true?
a) Statement 1 is true, statement 2 is false
b) Statement 1 is false, statement 2 is true
c)Both statements 1 and 2 are true
d) both statements 1 and 2 are false
8. The absolute maximum value of the function 𝑓 given by 𝑓(𝑥) = 𝑥 3 , 𝑥 ∈ [−2,2] is
a) 2 b) 0 c) -2 d) 8

SUBJECT: 35 – MATHEMATICS Page 115 of 164

9. ∫ 𝑒 𝑥 (sin 𝑥 − cos 𝑥)𝑑𝑥 Is
a) −𝑒 𝑥 cos 𝑥 b) 𝑒 𝑥 cos 𝑥 c) 𝑒 𝑥 sin 𝑥 d) 𝑒 𝑥 sin2 𝑥
𝑑𝑦
𝑑3 𝑦 𝑑2 𝑦
10. The degree of differential equation 𝑑𝑥 3 + 𝑑𝑥 2 + 𝑒 = 0 is 𝑑𝑥

a) 1 b) 3 c) 2 d) not defined
11. The direction cosines of the vector 𝑎 = 𝑖ˆ − 𝑗ˆ + 2𝑘ˆ are
1 −1 2 1 −1 2 1 −1 2 −1 1 2
a) 5 , 5 , 5 b) 6 , 6 , 6 c) 6 , 6 , 6 d) , ,
√ √ √ √ √ √ √6 √6 √6
12. The angle between two vectors 𝑎 and 𝑏⃗ with |𝑎| = √3, |𝑏⃗| = 2 and 𝑎 ⋅ 𝑏⃗ = √6 is
𝜋 𝜋 𝜋 𝜋
a) 6 b) 3 c) 4 d) 2
13. The equation of 𝑦-axis in space is
a) 𝑥 = 0, 𝑦 = 𝑜 b) 𝑥 = 0, 𝑧 = 0 c) 𝑦 = 0, 𝑧 = 0 d) 𝑦 = 0
1 2
14. If 𝑃(𝐴) = 2 𝑃(𝐵 ∣ 𝐴) = 3 then 𝑃(𝐴 ∩ 𝐵) is
1 1 3
a) 3 b) 2 c) 1 d) 5
1 1 1
15. Assertion [A]: For two events 𝐸 and 𝐹 if 𝑃(𝐸) = 5 , 𝑃(𝐹) = 2 and 𝑃(𝐸 | 𝐹) = 5 . Then 𝐸 and 𝐹 are
independent events
Reason [R]: If 𝐸 and 𝐹 are two independent events then 𝑃(𝐹 ∣ 𝐸) = 𝑃(𝐹)
Then which of the following are true?
a) [𝐴] is true but [𝑅] is false b) Both [𝐴] and [𝑅] are false
c) Both [𝐴] and [𝑅] are true d) [A] is false but [R] is true
II. Fill in the blanks by choosing the appropriate answer from those given in the bracket
5
[0,2,1, 9 , −1,6] (𝟓 × 𝟏 = 𝟓)
3
16. The value of cos (sec −1 (2) − sin−1 (√2 )) is
𝑑𝑦
17. If 𝑦 = sin−1 (cos 𝑥) then 𝑑𝑥 = ____
13
18. The value of ∫7 1𝑑𝑥 = ____
19. The projection of vector 𝑖ˆ + 𝑗ˆ along the vector 𝑖ˆ − 𝑗ˆ is ____
4 9
20. If 𝑃(𝐴 ∩ 𝐵) = 13 and 𝑃(𝐵) = 13 then 𝑃(𝐴′ ∣ 𝐵) = ____
PART -B
III. Answer any six of the following questions: (𝟔 × 𝟐 = 𝟏𝟐)
21. Find the equation of the line through the points (1,2) and (3,6) using determinants.
𝑑𝑦 𝑦
22. If √𝑥 + √𝑦 = √10 then show that 𝑑𝑥 + √𝑥 = 0
23. A balloon which is always remains spherical has a variable radius find the rate at which its
volume is increasing with radius when the radius is 10 cm .
24. Find the interval in which the function given by 𝑓(𝑥) = 4𝑥 3 − 6𝑥 2 − 72𝑥 + 30 is decreasing
25. Find ∫ cot 𝑥 ⋅ log (sin 𝑥)𝑑𝑥.
𝑑2 𝑦
26. Verify that the function 𝑦 = 𝑎sin 𝑥 + 𝑏cos 𝑥 is a solution of differential equation 𝑑𝑥 2 + 𝑦 = 0.
27. If 𝑎 = 𝑖ˆ + 𝑗ˆ + 𝑘̂ , 𝑏ˆ = 2𝑖ˆ − 𝑗ˆ + 3𝑘ˆ, and 𝑐 = 𝑖ˆ − 2𝑗ˆ + 𝑘ˆ then find unit vector parallel to the vector
2𝑎 − 𝑏⃗ + 3𝑐
𝑥−1 𝑦−2 𝑧−3 𝑥−1 𝑦−1 𝑧−6
28. If the lines −3 = 2𝑘 = 2 and 3𝑘 = 1 = −2 are perpendicular to each other, then find the
value of 𝑘
29. An urn contains 10 black and 5 white balls, two balls are drawn from the urn one after the
other without replacement. What is the probability that both drawn balls black?

SUBJECT: 35 – MATHEMATICS Page 116 of 164

PART -C
IV. Answer any six of the following questions: (𝟔 × 𝟑 = 𝟏𝟖)
3
30. Check whether the relation 𝑅 in 𝑅 defined by 𝑅 = {(𝑎, 𝑏): 𝑎 ≤ 𝑏 } is reflexive, symmetric and
transitive.
63 5 3
31. Prove than tan−1 (16) = sin−1 (13 ) + cos −1 (5)
32. Express [ 1 5] as the sum of symmetric and a skew - symmetric matrix.
−1 2
𝑑𝑦 𝑡
33. Find 𝑑𝑥 if 𝑥 = 𝑎 (cos 𝑡 + log (tan 2)) and 𝑦 = 𝑎sin 𝑡.
34. Find the two positive numbers 𝑥 and 𝑦 such that 𝑥 + 𝑦 = 60 and 𝑥𝑦 3 is maximum
2𝑥
35. Evaluate ∫ 𝑥 2+3𝑥+2 𝑑𝑥.
36. Find the area of triangle ABC where positions vectors A, B, C are 𝑖ˆ − 𝑗ˆ + 2𝑘̂ , 2𝑗ˆ + 𝑘ˆ , 𝑗ˆ + 3𝑘ˆ
respectively
37. Derive the equation of a line in space through given point and parallel to a given vector 𝑏⃗ in
the vector form
38. In two identical boxes, box I contains 2 gold coins, while box II contains one gold and one
silver coin, A person chooses a box at random and takes out a coin, if the coin is of gold,
what is the probability that the other coin in the box is also a gold?
PART-D
V. Answer any four of the following questions: (𝟒 × 𝟓 = 𝟐𝟎)
𝑥−2
39. IF 𝐴 = 𝑅 − (3) and 𝐵 = 𝑅 − {1} and 𝑓: 𝐴 → 𝐵 is a function defined by 𝑓(𝑥) = (𝑥−3) is 𝑓 one-one
and onto? Justify your answer
1
40. If 𝐴 = [−4] and 𝐵 = [−1 2 1,] verify that (𝐴𝐵)′ = 𝐵′ 𝐴′ .
3
41. Solve the following system of linear equations by matrix method.
4𝑥 + 3𝑦 + 2𝑧 = 60,2𝑥 + 4𝑦 + 6𝑧 = 90,6𝑥 + 2𝑦 + 3𝑧 = 70.
42. If 𝑦 = (tan−1 𝑥)2 then show that (𝑥 2 + 1)2 𝑦2 + 2𝑥(𝑥 2 + 1)𝑦1 = 2
1 1
43. Find the integral of 𝑥 2+𝑎2 with respect to 𝑥 and hence find ∫ 𝑥 2−6𝑥+13 𝑑𝑥.
44. Find the area of circle 𝑥 2 + 𝑦 2 = 𝑎2 by method of integration
𝑑𝑦 𝜋
45. Solve the differential equation cos2 𝑥 𝑑𝑥 + 𝑦 = tan 𝑥 (0 ≤ 𝑥 < 2 )
PART -E
VI. Answer the following questions:
𝜋
𝑎 𝑎
46. Prove that ∫0 𝑓(𝑥)𝑑𝑥 = ∫0 (𝑎 − 𝑥)𝑑𝑥 and hence evaluate ∫04 log (1 + tan 𝑥)𝑑𝑥. 6
OR
Solve the following linear programming problem graphically:
Minimize and maximize 𝑍 = 5𝑥 + 10𝑦, Subject to 𝑥 + 2𝑦 ≤ 120, 𝑥 + 𝑦 ≥ 60, 𝑥 − 2𝑦 ≥ 0 and
𝑥 ≥ 0. 𝑦 ≥ 0 6

3 1
47. If 𝐴 = [ ] show that 𝐴2 − 5𝐴 + 7𝑙 = 0 and hence find 𝐴−1 4
−1 2
OR
𝑘cos 𝑥 𝜋
𝜋−2𝑥
, 𝑥≠2 𝜋
Determine the value of 𝑘 if 𝑓(𝑥) = { 𝜋 is continues at 𝑥 = 2 4
3, 𝑥=2

SUBJECT: 35 – MATHEMATICS Page 117 of 164

GOVERNMENT OF KARNATAKA
KARNATAKA SCHOOL EXAMINATION AND ASSESSMENT BOARD
II PUC MATHEMATICS (35) EXAMINATION – 2, 2025
Duration: 3hrs Max Marks: 80
Instructions:
1) The question paper has five Parts namely 𝐀, 𝐁, 𝐂, 𝐃 and 𝐄. Answer all the Parts.
2) Part - A has 𝟏𝟓 multiple choice questions, 5 fill in the blank questions.
3) For Part-A questions, only the first written answers will be considered for evaluation.
4) Use the graph sheet for question on Linear Programming Problem in Part-E.

PART-A
I. Answer all the Multiple-choice questions: (𝟏𝟓 × 𝟏 = 𝟏𝟓)
1. The relation 𝑅 in the set [1,2,3] given by 𝑅 = {(2,3)} is
a) Reflexive b) Symmetric c) Transitive d) Equivalence
2. The principal value branch of cot −1 𝑥 is
−𝜋 𝜋 −𝜋 𝜋
a) [0, 𝜋] b) (0, 𝜋) c) [ 2 , 2 ] d) ( 2 , 2 )
−𝜋 𝜋
3. Statement 1: If 2 ≤ 𝑥 ≤ 2 then sin−1 (sin 𝑥) = 𝑥
Statement 2: If 0 ≤ 𝑥 ≤ 𝜋 then cos (cos −1 𝑥) = 𝑥
a) Statement 1 and statement 2 are true
b) Statement 1 and statement 2 are false
c) Statement 1 is true but statement 2 is false
d) Statement 2 is true but statement 1 is false
4. If a matrix has 8 elements, then the total number of possible matrices of different order
here
a) 4 b) 6 c) 2 d) 8
′
5. If A is a square Matrix with |𝐴| = 8, then the value of |𝐴𝐴 | =
1
a) 8 b) 64 c) 16 d)
8
𝑑𝑦
6. If 𝑦 = log (log 𝑥), then 𝑑𝑥 =
log 𝑥 1 1 1
a) 𝑥
b) 𝑥 c) log 𝑥 d) 𝑥log 𝑥
−1 𝑑𝑦
7. If 𝑦 = sin 𝑥 + sin−1 √1 − 𝑥 2 , then 𝑑𝑥 =
1
a) b) -1 c) 0 d) 1
√1−𝑥 2
8. The total revenue in rupees received from the sale 𝑥 units of a product is given
𝑅(𝑥) = 3𝑥 2 + 36𝑥 + 5 the marginal revenue when 𝑥 = 15 is
a) 90 b) 96 c) 116 d) 126
𝑥 1 1
9. ∫ 𝑒 (𝑥 − 2 ) 𝑑𝑥 =
𝑥
𝑒𝑥 𝑒𝑥 −𝑒 𝑥
a) 𝑥 + 𝐶 b) 𝑥 2 + 𝐶 c) 𝑒 𝑥 + 𝐶 d) 𝑥 2 + 𝐶
√3 𝑑𝑥
10. ∫1
1+𝑥 2
𝜋 −𝜋 𝜋 −𝜋
a) 12 b) 12 c) 3 d) 4
11. The position vector of a point which divides the join of points with position vectors 3𝑎 − 2𝑏⃗
and 𝑎 + 𝑏⃗ externally in the ratio 2: 1 is
5𝑎⃗
a) 4𝑎 − 𝑏⃗ b) 4𝑏⃗ − 𝑎 c) 2𝑎 + 𝑏⃗ d)
3
12. The value of 𝑥 for which 𝑥(𝑖ˆ + 𝑗ˆ + 𝜅ˆ) is a unit vector is
1
a) ±√3 b) 3 c) ± 3 d) 1
√
13. The direction cosines of negative 𝑍 - axis are
a) 0,0,1 b) 0,0, −1 c) −1, −1,0 d) 0,1,0
14. If A and B are two events such that 𝑃(𝐴 ∣ 𝐵) = 𝑃(𝐵 ∣ 𝐴), then
a) 𝑃(𝐴) = 𝑃(𝐵) b) 𝐴 = 𝐵 c) 𝐴 ∩ 𝐵 = ∅ d) 𝐴 ⊏ 𝐵 but 𝐴 ≠ 𝐵
SUBJECT: 35 – MATHEMATICS Page 118 of 164