Karnataka 1st PUC Question Bank for Maths

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

QUESTION BANK

CHAPTER-1
SETS
MCQ/ FB
1. Which of the following is not a sets
A) The collection of all the months of a year beginning with the letter J
B) The collection of all boys in your class.
C) The collection of ten most talented writers of India.
D) The collection of all natural numbers less than 100.
2. The set of intelligent students in a class is
A)A null set B) A singleton set C)A finite set D) Not a well defined collection
3. The set {x : x is a positive integer and 𝑥 2 < 40} in the roster form is
A) 1, 2,3, 4,5 B) 1, 4,9,16, 25,36 C) 49,64,81,100.... D) 1, 2,3, 4,5,6
4. The set A = {1, 4, 9, 16, 25, . . . } in set-builder form
A) {x : x = 2n-1, where n ∈ N} B) { x : x = 3n-2, where n ∈ N }
C) { x : x = 𝑛 , where n ∈ N}
2
D) { x : x = 2n, where n ∈ N }.
5. The set {x : x is a prime number which is divisor of 60} in the roster form is
A) 1, 2,3, 4,5,6,10,12,15, 20,30, 60 B) {1,2, 3, 5} C) {2,3,5} D) {2, 3}.
6. The set {x : x is a letter of the word MATHEMATICS},in the roster form is
A) {M,A,T,H,E,M,A,T,I,C,S } B) {M,A,T,H,E,I,C,S }
C) {H,E,M,A,T,I,S } D) { H,E,M,A,T,I,C }.
7. The set {x : x≠ 𝑥} represents
A) {0} B){ } C){1} D) { ∅ }
8. Which of the following sets is not disjoint set.
A) {1,2,3} 𝑎𝑛𝑑 {x│x ϵ N, x ≥ 4 & 𝑥 ≤ 6}
B) {a, e, i, o, u} 𝑎𝑛𝑑 {𝑐, 𝑑, 𝑔, 𝑓}
C) {x|x is an even integer } and {x|x is a odd integer}
D) {x|x is a positive prime ≤ 10} and {x|x is a positive even integer ≤ 10 }.
9. If R is the set of real number and Q is the set of rational number then, R − Q is
A) set of real number B) set of rational number
C) set of irrational number D) set of integer number.
10. If A⊂ 𝐵 Then the number of elements in A  B is equal to
A)n(A) B)n(B) C) n(A)+ n(B) D) n(A)- n(B).
11. Let U = {1,2,3,4,5,6,7,8,9} ,A = {1, 3, 5, 7,9} then the complement of set A is
A) {1, 3, 5, 7,9} B) {2,4,6,8,10} C) {2,4,8} D) {2,4,6,8}.
12. The number of non-empty subsets of the set {1, 2, 3, 4} is
A) 14 B)15 C)16 D) 17.
13. If A and B be any two sets, then ( A  B ) is equal to
1

A) A1  B1 B) A1  B1 C) A  B D) A  B
14. Let A and B be two sets such that A  B = A . Then, A  B is equal to
A)  B) B C) A D) none of these
15. Which of the following set is not empty set.
A) {𝑥 ∶ 1 < 𝑥 < 2, 𝑥 is a natural number}. B) {x : 𝑥 2 - 1 = 0 and x is a natural number}.
C) { x : 𝑥 2 = 4, x is odd } D) {x : x is an even prime number greater than 2}
16. The {𝑥|𝑥 ∈ 𝑅, −5 < 𝑥 ≤ 7} sets as intervals is
A) (−5,7) B) [−5,7] C) {−5,7} D) (−5,7].
17. The {𝑥|𝑥 ∈ 𝑅, 6 < 𝑥 < 10} sets as intervals is
A) (6,10) B) [6,10] C) {6,10} D) (6,10].

Question Bank: Department of School Education (Pre University ) 3

18. The {𝑥|𝑥 ∈ 𝑅, 𝑥 ≤ 1 } sets as intervals is
A) (−∞, 1) B) [−∞, 1] C) [ −∞, 1) D) (−∞, 1].
19. The intervals [−7,12) sets
A) {𝑥|𝑥 ∈ 𝑅, −7 < 𝑥 ≤ 12} B) {𝑥|𝑥 ∈ 𝑅, −7 ≤ 𝑥 ≤ 12}
C) {𝑥|𝑥 ∈ 𝑅, −7 < 𝑥 < 12} D) {𝑥|𝑥 ∈ 𝑅, −7 ≤ 𝑥 < 12}.
20. The intervals (3,8) sets
A) {𝑥|𝑥 ∈ 𝑅, 3 < 𝑥 ≤ 8} B) {𝑥|𝑥 ∈ 𝑅, 3 ≤ 𝑥 ≤ 8}
C) {𝑥|𝑥 ∈ 𝑅, 3 < 𝑥 < 8} D) {𝑥|𝑥 ∈ 𝑅, 3 ≤ 𝑥 < 8}.
21. Which of the following set is an infinite set.
A) {x : x ∈ N and (x – 1) (x –2) = 0} B){x : x ∈ N and x2 = 4}
C) {x : x ∈ N and 2x –1 = 0} D) {x : x ∈ N and x is prime}
22. Which of the following statement is incorrect
A) Every set is a subset of itself
B)Null set is a subset of all sets.
C)If B ⊂ A and B ≠ 𝐴, then B is called proper subset of A.
D)If the set A contains n elements then the number of proper subsets is given by 2n .
23. The number of proper subsets of the set {1, 2, 3} is
(A)5 B) 6 C) 7 D)8
24. Which is the following sets are not equal sets.
A) A = {a, b, c, d} and B = {c, a, b, d} B) A = {1,2,3,4} and B = {1,3,2,4}
C) A = {2, 4, 6, 8, 10}, B = { x : x is positive even integer and x ≤ 10}
D) {x|x is a positive prime ≤ 10} and {1,2,3,5,7}.
25. The length of the intervals (-5, 7 ]
(A) 10 (B) 12 (C) 11 (D) 2
26. Which of the following statement is correct statement.
i) 𝐴 ∪ 𝐴′ = 𝑈 ii) φ′ ∩ A =A iii) U′ ∩ A = A
A) only i and ii B) only ii and iii C) only i D) all i, ii and iii.
27. Which of the following statement is incorrect
A) { a, b } ⊄ { b, c, d } B){ a, e } ⊂ { x : x is a vowel in the English alphabet}
C) { a }∈ { a, b, c }. D){ a }⊂ { a, b, c }.
28. Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statement is correct
A){3, 4} ⊂ A B){3, 4} ∈ A C){1,2, 3} ⊂ A D) 4 ∈ A.
29. The number of sub set of a set A = φ is
A) 1 B) 2 C) 0 D) 3.
30. If X= { a, b, c, d } and Y = { f, b, d, g}, then X – Y is
A) {b, d } B) {f, g } C) {a, c } D) { a, c, f, g }.
31. Which of the following statement is correct
A){ 2, 3, 4, 5 } and { 3, 6} are disjoint sets.
B){ a, e, i, o, u } and { a, b, c, d }are disjoint sets.
C){ 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.
D){ 2, 6, 10 } and { 2,3, 7, 11} are disjoint.
32. If U = { a, b, c, d, e, f, g, h}, then the complements of complements of the sets {a, b, c}
A) { a, b, c } B) { d, e, f, g, h } C) { a, b, c, d, e, f, g, h } D) {…}
33. Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C, then
A) B = C B) A = C C) A ∪ B = A D) A ∩ B = A
34. Which of the following example of the null set
A) Set of odd natural numbers less than 3. B)Set of even prime numbers
C) { x : x is a natural numbers, x < 5 or x > 7 }
D) { y : y is a point common to any two parallel lines}.
Question Bank: Department of School Education (Pre University ) 4

35. Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be
considered as universal set (s) for all the three sets A, B and C
A) {0, 1, 2, 3, 4, 5, 6} B) φ
C){0,1,2,3,4,5,6,7,8,9,10} D) {1,2,3,4,5,6,7,8}
36. The solution set of the equation 𝑥 + x – 2 = 0 in roster form.
2

A) {1, 2 } B) {-1,2} C) {1,-2 } D) {-1,-2}.
1 2 3 4 5
37. Write the set E={2 , 3 , 4 , 5 , 6} in set builder form.
A) {x : x = n/n-1 , where n ∈ N} B) { x : x = n/n+1, where n ∈ N }
C) { x : x = n/n+1, where n ∈ N and n<7} D) { x : x = n/n+1, where n ∈ N and n<6}.
38. The set of all letters in the word TRIGONOMETRY in the roster form
A) {T,R,I,G,O,N,M,E,Y } B) {T,R,I,G,O,N,O,M,E,T,R,Y }
C) {T,R,I,G,O,N,M,E,,S,Y } D) {T,R,I,G,O,N,,E,Y }
39. The set {x : x is a two-digit natural number such that the sum of its digits is 8} in the roster
form
A) {17,26,35,44,53,62,71} B) {08,17,26,35,44,53,62,71,80 }
C) {08,17,26,35,44,53,62,71} D) {17,26,35,44,53,62,71,80}
40. Write {x : x is an integer and –3 ≤ x < 7} in the roster form
A) {-3,-2,-1,0,1,2,3,4,5,6,7} B) {-3,-2,-1,0,1,2,3,4,5,6,} C) (-3,7) D) [ -3,7)
41. Let A = { a, e, i, o, u } and B = { a, i, u }, then
A) A = B B) A ⊂ B C) A ∪ B = A D) A ∩ B = A
42. If R is the set of real numbers and Q is the set of rational numbers, then R ∩ 𝑸𝒄 ?
A) R - set of real numbers B) Q- set of rational numbers
C) T - set of irrational numbers D) Z- set of integers.
43. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find U∩ A𝒄 .
A) {2, 4, 6, 8, 10,12 } B) {1, 3, 5, 7, 9}
C) {2, 4, 6, 8, 10 } D) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
44. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one
angle different from 60°, then A𝒄 is
A) set of all triangles with at least two angles 60°
B) set of all triangles with all angles 60°
C) set of all triangles with at most one angle 60°
D) set of all triangles with exactly one angle 60°
45. How will you define a set of all real numbers?
A) {x: -1 < x < 1} B) {x: 0 < x < ∞} C) {x: -∞ < x < ∞} D) {x: -Z < x < +Z}
46. How will you define Union of two sets A and B?
A) {x: x € A or x € B} B) {x: x € A or x € B (or both)}
C) {x: x € A and B} D) {x: x € A – B}
47. Which of the following is not a set of letters of word PRINCIPAL?
A) {P,R,I,N,C,A,L} B) {C,A,P,I,N,R,L}
C) {P,R,I,N,C,I,P,A,L} D) {L,N,I,P,C,A,R}
48. Fill in the blanks to make each of the following a true statement :
(i)A ∪ A′ = ….. (ii) φ′ ∩ A = …. (iii) A ∩ A′ = …. (iv) U′ ∩ A = . …..
A) i) and ii) is U and iii) and iv) is φ B) i) and ii) is φ and iii) and iv) is U
C) i) and iii) is U and ii) and iv) is φ D) i) and iv) is φ and ii) and iii) is U
49. Statement I: The collection of all the months of a year beginning with the letter T is a set
Statement II: The collection of ten most talented writers of India is not a set
(A) Both statements are true
(B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true (D) Both statements are false.
Question Bank: Department of School Education (Pre University ) 5

50. Statement I: A set is defined as a well-defined collection of objects.
Statement II: A set can be a collection but a collection cannot be a set.
(A) Both statements are true
(B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true
(D) Both statements are false.
51. Statement I: The set builder notation has a certain rule or a statement that specifically
describes the common feature of all the elements of a set.
Statement II: A set of first five even natural numbers is {𝑥: 𝑥 ∈ 𝑁 , 𝑥 ≤ 10 𝑎𝑛𝑑 𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛}
(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. Assertion (A): The number of proper subsets of the set ∅ is 0
Reason (R): If A is a proper subsets of B, then at least one element of B is not in A
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of
the Assertion (A)
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct
explanation of the Assertion A)
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
53. Assertion (A): If A ∩ 𝐵 𝑐 = ∅ , then A = B.
Reason (R): set A and B are disjoint ,then A ∩ 𝐵 = ∅.
A) A is false but R is true B) A is false and R is false.
C)A is true but R is false D) A is false and R is false.
54. Representation of set using venn diagram

(A) 𝐴 − 𝐵 = { 5,7 } (B) 𝐵 − 𝐴 = { 1,2 } (C) 𝐴 ∪ 𝐵 = {1,2, 5,7} (D) 𝐴 ∩ 𝐵 = {3} .
55. For the figure given below, consider the following statements 1 and 2

Statement 1: (𝐴 − 𝐵) ∪ (𝐴 ∩ 𝐵) = {1,3,5,7,9,11,13,15,17,19}
Statement 2: 𝐴 − 𝐵 = 𝐴 − (𝐴 ∩ 𝐵)
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

Question Bank: Department of School Education (Pre University ) 6

56. Match List I to List II

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
57. Match List I to List II

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
58. Match List I to List II

Choose the correct answer from the options given below:
A) a-i, b-ii, c-iii, d-iv B) a-iii, b-iv, c-ii, d-i C) a-iii, b-iv, c-i , d-ii D) a-iii, b-iii, c-i ,d-ii
59. The number of elements in set {x : x is a letter of word TRIGONOMETRY} is _________
60. If U = { a, b, c, d, e, f, g, h}, and {a, b, c} then the number of elements in the complement of
set A is -------
61. The number of elements in set {𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎𝑛𝑑 𝑥 2 < 50} is _________
62. The number of elements in set {𝑥: 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎𝑛𝑑 𝑥 + 1 = 1} is _________
63. The number of elements in set of all letters of the word BETTER is _________
64. The number of subsets of the set {𝑎} is ---------
65. The number of improper subsets of the set ∅ is ---------

Question Bank: Department of School Education (Pre University ) 7

TWO MARKS QUESTIONS:
1. Write down all the subsets of the set {1,2,3}. (U)
2. Write the solution set of the equation 𝑥 + 𝑥 − 20 = 0 in roster form.
2
(U)
3. Let A = {1,2,3,4,5,6}, B = {2,4,6,8}. Find A – B and B – A. (U)
4. Let A = {𝑎, 𝑏}, B = {𝑎, 𝑏, 𝑐}. Is A ⊂ B? What is AUB? (U)
5. If A = {3,5,7,9,11}, B = {7,9,11,13} and C = {11,13,15}, find 𝐴 ∩ (𝐵 ∪ 𝐶). (U)
6. If A = {3,5,7,9,11}, B = {7,9,11,13} and C = {15,17}, find 𝐴 ∩ (𝐵 ∪ 𝐶). (U)
7. If A = {3,6,9,12,15,18,21} and B = {4,8,12,16,20}, find A – B and B – A. (U)
8. If A = {3,6,9,12,15,18,21} and B = {2,4,6,8,10,12,14,16}, find A – B and B – A. (U)
9. If A = {3,6,9,12,15,18,21} and B = {5,10,15,20}, find A – B and B – A. (U)
10. If A = {2,4,6,8,10,12,14,16} and B = {4,8,12,16,20}, find A – B and B – A. (U)
11. If A = {4,8,12,16,20} and B = {5,10,15,20}, find A – B and B – A. (U)
12. If A = {2,4,6,8,10,12,14,16} and B = {5,10,15,20}, find A – B and B – A. (U)
13. If X = {𝑎, 𝑏, 𝑐, 𝑑} and Y = {𝑏, 𝑑, 𝑔, 𝑓} find X – Y and Y – X. (U)
14. If X = {𝑎, 𝑏, 𝑐, 𝑑} and Y = {𝑏, 𝑑, 𝑔, 𝑓} find X – Y and X∩Y. (U)
15. Let U = {1,2,3,4,5,6,7,8,9}, A = {1,2,3,4} and B = {2,4,6,8}, find (𝐴 ∪ 𝐵) . ′
(U)
16. Let U = {1,2,3,4,5,6,7,8,9}, A = {1,2,3,4} and B = {3,4,5,6}, find (𝐴 ∪ 𝐵)′ . (U)
17. Let U = {1,2,3,4,5,6,7,8,9}, A = {1,2,3,4} and B = {3,4,5,6}, find (𝐴 − 𝐵) . ′
(U)
18. If A = {3,5,7,9,11}, B= {7,9,11,13}, C= {11,13,15} and D = {15,17}, find (𝐴 ∩ 𝐵) ∪ (𝐶 ∪ 𝐷). (U)
19. If A = {3,5,7,9,11}, B = {7,9,11,13}, C = {11,13,15} and D = {15,17}, find (𝐴 ∪ 𝐵) ∩ (𝐶 ∪ 𝐷). (U)
20. Taking the set of natural numbers as the universal set, write the complements of the
following sets:
i) {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑜𝑑𝑑𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟}, ii) {𝑥: 𝑥𝑖𝑠𝑎𝑝𝑟𝑖𝑚𝑒𝑛𝑢𝑚𝑏𝑒𝑟}. (U)
21. Taking the set of natural numbers as the universal set, write the complement of the
following sets: i) {𝑥: 2𝑥 + 5 = 9}, ii) {𝑥: 𝑥 ≥ 7}. (U)
22. Draw the appropriate Venn diagram for 𝐴′ ∩ 𝐵 ′ . (U)
23. Draw the appropriate Venn diagram for (𝐴 ∩ 𝐵) . ′
(U)
24. Draw the appropriate Venn diagram for 𝐴′ ∪ 𝐵 ′ . (U)
25. Draw the appropriate Venn Diagram for (𝐴 ∪ 𝐵) . ′
(U)
26. List all the subsets of the set {−1,0,1}. (U)
27. Show that AUB = 𝐴 ∩ 𝐵 implies A = B. (S)
28. If A = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟} and B = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑒𝑣𝑒𝑛𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟}, find 𝐴 ∩ 𝐵. (U)
29. If A = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟} and B = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑜𝑑𝑑𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟}, find 𝐴 ∩ 𝐵. (U)
30. If A = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟} and B = {𝑥: 𝑥𝑖𝑠𝑝𝑟𝑖𝑚𝑒𝑛𝑢𝑚𝑏𝑒𝑟}, find 𝐴 ∩ 𝐵. (U)
31. Find the union of the sets A = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 ∧ 1 < 𝑥 ≤ 6} and
B = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 6 < 𝑥 < 10}. (U)
32. Find the intersection of the sets A = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑜𝑓 3} and
B = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟𝑙𝑒𝑠𝑠𝑡ℎ𝑎𝑛6}. (U)
33. Find the intersection of the sets A = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 1 < 𝑥 ≤ 6} and
B = {𝑥: 𝑥𝑖𝑠𝑎𝑛𝑎𝑡𝑢𝑟𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 6 < 𝑥 < 10}. (U)
34. Let A = {1,2,3,4,5,6,7,8,9,10} and B = {2,3,5}. Find A∩B. (U)
35. Find the union of the sets X = {1,3,5} and Y = {1,2,3}. (U)
36. Find the union of the sets A = {𝑎, 𝑒, 𝑖, 𝑜𝑢} and B = {𝑎, 𝑏, 𝑐}. (U)
37. Find the union of the sets A = {1,2,3} and B = ∅. (U)
38. If A and B are two sets such that A⊂B, then what is AUB? (U)
39. If A = {1,2,3,4}, B = {3,4,5,6} and C = {5,6,7,8}, find AUBUC. (U)
40. If A = {1,2,3,4}, B = {3,4,5,6} and C ={7,8,9,10} find AUBUC. (U)

Question Bank: Department of School Education (Pre University ) 8

THREE MARKS QUESTIONS:
1. Let U ={1,2,3,4,5,6}, 𝐴 = {2,3} and B = {3,4,5}. Show that (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵 ′ . (U)
2. Let U ={1,2,3,4,5,6,7,8,9}, 𝐴 = {2,4,6,8} and B = {2,3,5,7}. Show that (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵 ′ . (U)
3. Let U ={1,2,3,4,5,6,7,8,9}, 𝐴 = {2,4,6,8} and B = {2,3,5,7}. Show that (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵 ′ . (U)
4. If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17};
find (i) ( A ∩ B ) ∩ ( B ∪ C ) (ii) ( A ∪ D) ∩ ( B ∪ C)

Question Bank: Department of School Education (Pre University ) 9

CHAPTER-2
RELATIONS AND FUNCTIONS
MCQ/ FB
1. If (x + 1, y – 2) = (3,1), then the values of (x , y) is
A) (1, 2) B) (2, 3) C) (3 , 2 ) D) (2 , 1)
2. If P = {1, 2}, then the number of elements in P × P × P is
A) 2 B) 8 C) 4 D) 6.
𝑥 2 5 1
3. If ( 3 + 1, y – 3 ) = ( 3, 3 ), then the values of x and y are respectively
A)1, 2 B) 2,3 C) 3 , 2 D) 2 , 1.
4. If the set A has 3 elements and the set B = {3, 4 }, then the number of elements in A×B.
A) 2 B) 8 C) 4 D) 6.
5. If A = {1, 2} and B = {3, 4}, then the number of sub sets will A × B have
A) 4 B) 8 C) 32 D) 16.
6. If A × B = {(a, x),(a , y), (b, x), (b, y)},then the set A is
A) {a, x} B) {b, y} C) {a, b} D) {x, y}.
7. Cartesian product A × A has 9 elements among which are found (–1, 0) and (0,1), then the
set A is
A) {-1,0,1} B) {-1,0} C) {-1,1} D) {-1,1, 2}.
8. Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y) : y = x + 1 },then R is
A) { (2,3), (3,4), (4,5), (5,6)}. B) {(1,2), (2,3), (3,4), (4,5), (5,6)}.
C) {(1,2), (2,3), (3,4), (4,5)}. D) {(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)}.
9. If A = {1, 2} and B = {3, 4},then the number of relations from A to B is
A) 4 B) 8 C) 32 D) 16.
10. If A = {x, y, z} and B = {1, 2},then the number of relations from A to B is
A) 5 B) 6 C) 32 D) 64.
11. If R be the relation on Z defined by R = {(a,b): a, b ∈ Z, a – b is an integer},
then the domain e of R.
A) {0} B) ∅ C) Z D) R.
12. Which of the following is not true.
(A) A × B = {(a, b): a ∈ A, b ∈ B} (B) A × φ = φ
(C) If n(A) = p and n(B) = q, then n(A × B) = pq. (D) In general, A × B = B ×A
13. Let n(A) = n. Then the number of all relations on A is
n (n)! 2
A) 2 B) 2 C) 2 n D)None of these.
14. Which of the following relations given below is a function
(A) R = {(2,1),(3,1), (4,2)} (B) R = {(2,2),(2,4),(3,3), (4,4)}
(C) R = {(1,2),(2,3),(3,4), (4,5), (5,6), (5,7)} (D) R = {(2,1),(2,3),(3,3), (4,4)}.
15. The domain of the real functions f(x) = –|𝑥|
A) (0, ∞) B) (−∞, ∞) C) (−∞, 0) D) [−∞, ∞].
16. The range of the real functions f(x) = –|𝑥|
A) (0, ∞) B) (−∞, ∞) C) (−∞, 0) D) [−∞, ∞].
17. The domain of the real functions f(x) = √9 − x 2

A) (0,3) B) (−3,3) C) (−3,0) D) [−3,3].
18. The range of the real functions f(x) = √9 − x 2

A) (0,3) B) (−3,3) C) (−3,0) D) [0, 3].
x2 +3x+5
19. The domain of the real functions f(x) =x2−5x+4 is
A) 𝑅 − {1,4} B) 𝑅 − {1} C) 𝑅 D) 𝑅 − {4}.

Question Bank: Department of School Education (Pre University ) 10

20. The domain of the real functions f(x) = √x − 1
A) (1, ∞) B) (−∞, 1) C)[ −∞, 1] D) 1, ) .
21. The range of the real functions f(x) = √x − 1
A) [0, ∞) B) (−∞, ∞) C) (−∞, 0) D) [ 0, ∞].
22. The domain of the real functions f(x) = |x − 1|
A) (−∞, 0) B) (−∞, ∞) C) (0, ∞) D) [−∞, ∞].
23. The range of the real functions f(x) = |x − 1|
A) (0, ∞) B) (−∞, ∞) C) [ 0, ∞) D) [ 1, ∞].
24. The range of the real functions f(x) = 2 − 3x, x > 0 is
A) [−∞, 2] B) (−∞, 2) C) ( 2, ∞) D) [ 2, ∞).
25. The range of the real functions f(x) = 2 + x 2 , x ∈ R is
A) [−∞, 2] B) (−∞, ∞) C) ( 2, ∞) D) [ 2, ∞).
26. The domain and range of signum function
A) 𝑅 𝑎𝑛𝑑 {−1,0,1} B) 𝑅 𝑎𝑛𝑑 𝑍 C) 𝑅 𝑎𝑛𝑑 𝑍 + D) 𝑅 𝑎𝑛𝑑 𝑅
x2 +2x+1
27. The domain of the function f given by f(x) = x2− 8x+12 .
A)R – { 6, –2 } B)R – { 2,6 } C)R – [2, –6] D) R – (-2, 6)
28. Let A = {9,10,11,12,13} and let f : A→N be defined by f (n) = the highest prime factor of n.
Find the range of f.
(A) {(3,5,11,6,13} B) {2,3,11, 13} C) {3,5,11,6,13} D) {1, 13} .
29. Which of the following statement is false
(A) If P = {m, n} and Q = { n, m}, then P × Q = {(m, n),(n, m)}.
(B) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such
that x ∈ A and y ∈ B.
(C) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.
(D) If A and B are non-empty sets and either A or B is an infinite set, then so is A × B.
30. If the set A has 3 elements and the set B = {3,4,5}, then the number of elements in
(𝐴 × 𝐵) .......
31. Let A={1,2} and B={3,4}. The number of relations from A to B is ------------
32. If P = {1, 2}, then n( P × P × P )= ----------------
33. If the set A has 2 elements and the set B = {1,3,4,5}, then the number of elements in (𝐴 × 𝐵)
is -------
34. Let A = {1,2}. The number of relations on A is --------------
35. If A = {1,2, 3,4,5} and B = {1,2, 3,4,5,6,7,8}
Statement I: The relation R = {(1,2),(2,3),(3,4), (4,5), (5,6), (5,7)} is a function.
Statement II: let f : A→B is a function, if every elements of A has one and only one
elements of B
(A) Both statements are true
(B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true (D) Both statements are false.
36. From figure

Statement I: Figure 1 is a function, Statement II: Figure 2 is not a function
(A) Both statements are true
(B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true
(D) Both statements are false.
Question Bank: Department of School Education (Pre University ) 11

37. From figure, relation R from A to B

Statement I: Domain of the relation { 1,2,3 }
Statement II: Range of the relation { 2, 4 }
(A) Both statements are true (B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true
(D) Both statements are false.
38. Figure 1 Figure 2

Statement I:Figure 1 is only a relation, Statement II: Figure 2 is a function
(A) Both statements are true
(B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true
(D) Both statements are false.
39. Statement I: The domain of a relation R = {(4,5),(4,4),(3,5), (3,4)} is { 3 , 4 }.
Statement II: The domain of a relation is the set of all first coordinates in the relation
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.
40. Assertion (A): The relation R = {(1,2),(2,2),(3,2),(4,2),(5,2),(6,2)} is a function.
Reason (R): A function is a relation such that no two ordered pairs have the same
first coordinates and different second coordinates.
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of
the Assertion (A)
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct
explanation of the Assertion A)
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true
41. Let A={1,2,3,4,5} and R be a relation from A to A, defined as R = {(x, y): y = x + 1}. Then
1. The domain of the relation R is {1, 2, 3, 4,5}
2. The co domain of the relation R is {1,2,3,4,5}
3. The range of the relation R is { 2,3,4,5,6}
Which of the above statements are correct?
(A) 1 only (B) 2 only (C) 3 only (D) All 1, 2 and 3.
42. Let A={1,2} and B={3,4}.
1. R = {(1,1), (1,2), (1,3), (1,4)} is a relation from A to B
2. R ={(1,3), (1,4)} is a function from A to B
3. R = {(1,3), (2,3)} is a relation and also a function from A to B
Which of the above statements are correct?
(A) 1 only (B) 2 only (C) 3 only (D) All 1, 2 and 3.
Question Bank: Department of School Education (Pre University ) 12

43. Let f (x) = x2 and g (x) = √x, then
1. (f+g) (1) = 2 2. (g - f) (1) = 0 3. (gf) (1) = 1
Which of the above statements are correct?
(A) 1 only (B) 2 only (C) 3 only (D) All 1, 2 and 3.
44. Statement I: If f(x) = x and g(x) = x are two functions from R to R then (f÷g) (2) = 2.
2

Statement II: (f+g) (x) = f(x) + g(x).
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 domain of a relation R is the set of all first elements of the ordered pairs
in a relation R.
Statement 2: The range of a relation R is the set of all second elements of the ordered pairs
in a relation R.
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.
46. Statement 1: A real function has a set of real numbers or one of its subsets both as to its
domain and as its range.
Statement 2: Let A = {0,1,2,3,4,5} and let f : A→N be defined by f (x) = x+1 is a real function.
A) Statement 1 is true, and Statement 2 is false.
B) Statement 1 is true, and Statement 2 is true, Statement 1 is correct Explanation for
Statement 2
C) Statement 1 is true, and Statement 2 is true, Statement 1 is not a correct Explanation for
Statement 2
D) Statement 1 is false, and Statement 2 is false.
47. Figure 1 Figure 2

Statement I: Figure 1 is f(x)=|𝑥|, Statement II: Figure 2 is f(x)=𝑥 2
(A) Both statements are true
(B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true
(D) Both statements are false.
48. Figure 1 Figure 2

Statement I: Figure 1 is a constant function, Statement II: Figure 2 identity function.
(A) Both statements are true (B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true (D) Both statements are false.

Question Bank: Department of School Education (Pre University ) 13

49. From figure

Statement I: Domain of the function is R(Set of all real numbers)
Statement II: Range of the function is Z (Set of all positive integers)
(A) Both statements are true (B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true (D) Both statements are false.
50. From figure

Statement I: Domain of the function is R (Set of all real numbers)
Statement II: Range of the function is R-{0} (Set of all non zero real numbers)
(A) Both statements are true (B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true (D) Both statements are false.

TWO MARKS QUESTIONS:
1. A = {1,2,3,5} and B = {4,6,9}. Define a relation R from A to B by
R = {(𝑥, 𝑦): 𝑡ℎ𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑥 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝑜𝑑𝑑; 𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵}. Write R in roster form. (U)
2. The function ‘𝑡’ which maps temperature in degree Celsius into temperature in degree
9𝐶
Fahrenheit is defined by 𝑡(𝐶) = 5 + 32. Find the value of C, when 𝑡(𝐶) = 212. (A)
3. If (x + 1, y -2) = (3, 1), find the values of x and y. (U)
4. If A = {1,2,3}, B ={3,4} and C = {4,5,6}. Find 𝐴 × (𝐵 ∩ 𝐶). (U)
5. If A = {1,2,3}, B ={3,4} and C = {4,5,6}. Find (𝐴 × 𝐵) ∩ (𝐴 × 𝐶). (U)
6. If A = {1,2,3}, B ={3,4} and C = {4,5,6}. Find 𝐴 × (𝐵 ∪ 𝐶). (U)
7. If A = {1,2,3}, B ={3,4} and C = {4,5,6}. Find (𝐴 × 𝐵) ∪ (𝐴 × 𝐶). (U)
8. If P = {1,2}, form the set 𝑃 × 𝑃 × 𝑃. (U)
9. If 𝐴 × 𝐵 = {(𝑝, 𝑞), (𝑝, 𝑟), (𝑚, 𝑞), (𝑚, 𝑟)}, find A and B. (U)
𝑥 2 5 1
10. If (3 + 1, 𝑦 − 3) = (3 , 3), find the values of x and y. (U)
11. If 𝐺 = {7,8} and 𝐻 = {5,4,2}, find 𝐺 × 𝐻 and 𝐻 × 𝐺. (U)
12. If A = {−1,1}, find 𝐴 × 𝐴 × 𝐴. (U)
13. If A×B = {(𝑎, 𝑥), (𝑎, 𝑦), (𝑏, 𝑥), (𝑏, 𝑦)}. Find A and B. (U)
14. Let A = {1,2,3,4,5,6}. Define a relation R from A to A by R = {(𝑥, 𝑦): 𝑦 = 𝑥 + 1}
Write down the domain and range of R. (U)
15. Write the relation R = {(𝑥, 𝑥 3 ): 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10} ∈ 𝑟𝑜𝑠𝑡𝑒𝑟𝑓𝑜𝑟𝑚. (U)
16. Let A = {𝑥, 𝑦, 𝑧} and B = {1,2}. Find the number of relations from A to B. (S)
17. State whether the relation R = {(2,2), (2,4), (3,3), (4,4)} is a function or not. Justify your
answer. (U)
18. State whether the relation R = {(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)} is a function or not. Justify
your answer. (U)
19. State whether the relation R = {(1,3), (1,5), (2,5)} is a function or not. Justify your answer (U)
20. Find the domain and range of the function 𝑓(𝑥) = −|𝑥| (U)
21. Find the domain and range of the function 𝑓(𝑥) = √9−𝑥 2 (U)
Question Bank: Department of School Education (Pre University ) 14

𝑥2, 0 ≤ 𝑥 ≤ 3
22. The relation 𝑓 is defined by 𝑓(𝑥) = { The relation 𝑔 is defined by
3𝑥, 3 ≤ 𝑥 ≤ 10
𝑥2, 0 ≤ 𝑥 ≤ 2
𝑔(𝑥) = { , Show that 𝑓 is a function and g is not a function. (U)
3𝑥, 2 ≤ 𝑥 ≤ 10
𝑓(1.1)−𝑓(1)
23. If 𝑓(𝑥) = 𝑥 2 , find 1.1−1 (U)
24. Find the domain and range of the function 𝑓 defined by 𝑓(𝑥) = √𝑥 − 1 (U)
25. Find the domain and range of the function 𝑓 defined by 𝑓(𝑥) = − 1| |𝑥 (U)
26. Let 𝑓 = {(1,1), (2,3), (0, −1), (−1, −3)} be a function from 𝑍 to 𝑍 defined by 𝑓(𝑥) = 𝑎𝑥 + 𝑏,
for some integers 𝑎, 𝑏. Determine 𝑎, 𝑏. (U)
27. Let A = {9,10,11,12,13} and let 𝑓: 𝐴 → 𝑁 be defined by 𝑓(𝑛) = the highest prime factor of n.
Find the range of 𝑓. (U)
28. Let A = {1,2,3,4}, B = {1,5,9,11,15,16} and 𝑓 = {(1,5), (2,9), (3,1), (4,5), (2,11)}. Are the following
are true?
(i) 𝑓 is a relation from A to B
(ii) 𝑓 is a function from A to B. Justify your answer in each case. (U)
THREE MARKS QUESTIONS:
1. If A = {1,2},B = {1,2,3,4} and C = {5,6}. Verify that 𝐴 × (𝐵 ∩ 𝐶) = (𝐴 × 𝐵) ∩ (𝐴 × 𝐶). (U)
2. If P = {𝑎, 𝑏, 𝑐} and Q = {𝑟}, form the sets 𝑃 × 𝑄 and 𝑄 × 𝑃. Are these two products equal?
3. If A = {1,2},B = {1,2,3,4}, C = {5,6} and D = {5,6,7,8}. Verify that 𝐴 × 𝐶 is a subset of 𝐵 × 𝐷. (U)
4. Let A = {1,2} and B = {3,4}.Write𝐴 × 𝐵. How many subsets will 𝐴 × 𝐵 have? List them. (U)
5. The Cartesian product 𝐴 × 𝐴 has 9 elements among which are found (-1, 0) and (0, 1).
Find the set A and the remaining elements of 𝐴 × 𝐴. (S)
6. Let A = {1,2,∙∙∙ ,14}. Define a relation R from A to A by R = {(𝑥, 𝑦): 3𝑥 − 𝑦 = 0, 𝑤ℎ𝑒𝑟𝑒𝑥, 𝑦 ∈ 𝐴}.
Write down its domain, co domain and range. (U)
7. Define a relation R on the set 𝑁 of natural numbers by
R = {(𝑥, 𝑦): 𝑦 = 𝑥 + 5, 𝑥 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 4, 𝑥, 𝑦 ∈ 𝑁}. Depict this relationship
using roster form.
8. Write down the domain and range. (U)
Let A = {1,2,3,4,6}. Let R be the relation on A defined by
R = {(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝐴, 𝑏𝑖𝑠𝑒𝑥𝑎𝑐𝑡𝑙𝑦𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒𝑏𝑦𝑎}.
9. Write R in roster form. Find the domain and range of R. (U)
10. Determine the domain and range of the relation R defined by
R = {(𝑥, 𝑥 + 5): 𝑥 ∈ {0,1,2,3,4,5}} (U)
11. Let R be the relation on 𝑍 defined by R = {(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝑍, 𝑎 − 𝑏𝑖𝑠𝑎𝑛𝑖𝑛𝑡𝑒𝑔𝑒𝑟}.
Find the domain and range of R. (U)
12. Let 𝑁 be the set of natural numbers and the relation R be defined on𝑁 such that
R ={(𝑥, 𝑦): 𝑦 = 2𝑥, 𝑥, 𝑦 ∈ 𝑁}.
13. What is the domain and range of R? Is this relation a function? (U)
14. Let𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = 2𝑥 + 1 be two real functions.
𝑓
Find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓𝑔)(𝑥) and (𝑔) (𝑥). (U)
15. Let 𝑓(𝑥) = √𝑥 and 𝑔(𝑥) = 𝑥 be two real functions defined over the set of non-negative real
𝑓
numbers. Find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓𝑔)(𝑥) and (𝑔) (𝑥). (U)
16. Is the relation R = {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)} a function? Give reason. If it is a
function determine its domain and range. (U)
17. Is the relation R = a function? Give reason. If it is a function determine its domain and
range. (U)
18. A function 𝑓 is defined by 𝑓(𝑥) = 2𝑥 − 5.
Write down the values of (i) 𝑓(0), (ii) 𝑓(7), (iii) 𝑓(3). (U)
Question Bank: Department of School Education (Pre University ) 15

19. The function ‘𝑡’ which maps temperature in degree Celsius into temperature in degree
Fahrenheit is defined by
9𝐶
𝑡(𝐶) = 5 + 32. Find (i) 𝑡(0), (ii) 𝑡(28), (iii) 𝑡(−10). (A)
20. Let R be a relation from 𝑄 to 𝑄 defined by R = {(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝑄 ∧ 𝑎 − 𝑏 ∈ 𝑍}. Show that
(i) (𝑎, 𝑎) ∈ R for all 𝑎 ∈ 𝑄 (ii) (𝑎, 𝑏) ∈ R implies that (𝑏, 𝑎) ∈ R
(iii) (𝑎, 𝑏) ∈ R and (𝑏, 𝑐) ∈ R implies that (𝑎, 𝑐) ∈ R (U)
21. Let 𝑓 = {(1,1), (2,3), (0, −1), (−1, −3)} be a linear function from 𝑛𝑡𝑜𝑍 . Find 𝑓(𝑥). (A)
1 − 𝑥, 𝑥 < 0
22. The function 𝑓 defined by 𝑓(𝑥) = { 1, 𝑥 = 0 Draw the graph of 𝑓(𝑥). (S)
𝑥 + 1, 𝑥 > 0
23. Let 𝑓, 𝑔: 𝑅 → 𝑅 be defined, respectively by 𝑓(𝑥) = 𝑥 + 1, 𝑔(𝑥) = 2𝑥 − 3.
𝑓
Find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓𝑔)(𝑥) and (𝑔) (𝑥). (U)
24. Let R be a relation from 𝑁 to 𝑁 defined by R = {(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝑁 ∧ 𝑎 = 𝑏 2 }. Are the following
true?
(i) (𝑎, 𝑎) ∈ 𝑅, 𝑓𝑜𝑟𝑎𝑙𝑙𝑎 ∈ 𝑁 (ii) (𝑎, 𝑏) ∈ 𝑅, 𝑖𝑚𝑝𝑙𝑖𝑒𝑠(𝑏, 𝑎) ∈ 𝑅
(iii) (𝑎, 𝑏) ∈ 𝑅 ∧ (𝑏, 𝑐) ∈ 𝑅𝑖𝑚𝑝𝑙𝑖𝑒𝑠(𝑎, 𝑐) ∈ 𝑅. (S)
FIVE MARKS QUESTIONS:
1. Define Identity function. Draw the graph of it. Also write its domain and range. (S)
2. Define Signum function. Draw the graph of it. Also write its domain and range. (S)
3. Define Greatest integer function. Draw the graph of it. Also write its domain and range
4. Define Constant function. If the function f: 𝑅 → 𝑅 is defined by f(x) = 3 for each x ∈ 𝑅,
draw the graph of it. Also write its domain and range. (S)
1
5. Define Rational function. If the real valued function f: 𝑅 − {0} → 𝑅 defined by f(x) = 𝑥,
draw the graph of it. Also write its domain and range. (S)
6. Define polynomial function. . If the function f: 𝑅 → 𝑅 is defined by f(x) = 𝑥 2 , draw the
graph of it. Also write its domain and range. (S)
7. Define Modulus function. Draw the graph of it. Also write its domain and range.(S)
8. Draw the graph of the function𝑓(𝑥) = 𝑥 3 . Write its domain and range. (S)

Question Bank: Department of School Education (Pre University ) 16

CHAPTER-3
Trigonometric Functions
MCQ/ FB
1. Which of the following is incorrect
(A) If in a circle of radius r, an arc of length l subtends and angle of θ radians, then l = r θ
𝜋 𝜋 𝜋
(B) 𝑥 ° = 180 𝑥 in radian (C) 𝑥 𝐶 = 180 𝑥 in degree. (D) 30 minutes = 360 𝑟𝑎𝑑𝑖𝑎𝑛.
2. Which of the following is incorrect
(A )1 radian = 67° 16′ approximately (B) 1° = 0.01746 radian approximately.
l
(C) π radian = 180° (D) θc = r
3. 40° 20′ in radian measure
221𝜋 2𝜋 4𝜋 121𝜋
(A) (B) (C) (D)
540 9 9 540
4. 6 radians in degree measure approximately
(A) 343° (B)333° (C) 323° (D) 353°
5. 22° 30′ in radian measure
𝜋 𝜋 𝜋 𝜋
(A) 4 (B) 9 (C) 7 (D) 8
7𝜋
6. radians in degree measure
6
(A) 420° (B) 210° (C)−150° (D) 240°
7. −4 radians in degree measure approximately
(A) 228° (B)−228° (C) −118° (D) 118°
8. A wheel makes 60 revolutions in one minute. Then number of radians does
it turn in one second?
𝜋
(A) 𝜋 (B) 2𝜋 (C) 3𝜋 (D) 3
9. Which of the following is incorrect
(A) cos 𝜋 = − 1 (B) sin 𝜋 = 0 (C)sin 3𝜋/2 = −1 (D) cos 𝜋/2 = 1
10. Range of 𝑓(𝑥) =sinx is
(A) (−∞, ∞) (B) 𝑅 − [−1,1 ] (C)𝑅 − (−1,1) (D) [−1,1]
11. Domain of 𝑓(𝑥) = sin 𝑥 is
(2𝑛+1)𝜋
(A) (−∞, ∞) (B) 𝑅 − {𝑛𝜋: 𝑛 ∈ 𝑍} (C)𝑅 − { : 𝑛 ∈ 𝑍} (D) [−1,1]
2
12. Range of 𝑓(𝑥) =cosx is
(A) (−∞, ∞) (B) 𝑅 − [−1,1 ] (C)𝑅 − (−1,1) (D) [−1,1]
13. Domain of 𝑓(𝑥) =cosx is
(2𝑛+1)𝜋
(A) (−∞, ∞) (B) 𝑅 − {𝑛𝜋: 𝑛 ∈ 𝑍} (C)𝑅 − { : 𝑛 ∈ 𝑍} (D) [−1,1]
2
14. Domain of tangent of x is
(2𝑛+1)𝜋
(A) (−∞, ∞) (B) 𝑅 − {𝑛𝜋: 𝑛 ∈ 𝑍} (C)𝑅 − { : 𝑛 ∈ 𝑍} (D) [−1,1]
2
15. Range of 𝑓(𝑥) =tanx is
(A) (−∞, ∞) (B) 𝑅 − [−1,1 ] (C)𝑅 − (−1,1) (D) [−1,1]
16. Range of 𝑓(𝑥) =secx is
(A) (−∞, ∞) (B) 𝑅 − [−1,1 ] (C)𝑅 − (−1,1) (D) [−1,1]
17. Range of 𝑓(𝑥) =cosecx is
(A) (−∞, ∞) (B) 𝑅 − [−1,1 ] (C)𝑅 − (−1,1) (D) [−1,1]
18. Domain of 𝑓(𝑥) =secx of x is
(2𝑛+1)𝜋
(A) (−∞, ∞) (B) 𝑅 − {𝑛𝜋: 𝑛 ∈ 𝑍} (C)𝑅 − { : 𝑛 ∈ 𝑍} (D) [−1,1]
2
19. Domain of 𝑓(𝑥) =cosecx of x is
(2𝑛+1)𝜋
(A) (−∞, ∞) (B) 𝑅 − {𝑛𝜋: 𝑛 ∈ 𝑍} (C)𝑅 − { : 𝑛 ∈ 𝑍} (D) [−1,1]
2

Question Bank: Department of School Education (Pre University ) 17

20. Which of the following is not correct?
−1 1
(A) 𝑠𝑖𝑛𝜃 = 5 (B)𝑐𝑜𝑠𝜃 = 1 (C) 𝑠𝑒𝑐𝜃 = 2 (D) 𝑡𝑎𝑛𝜃 = 20
21. Which of the following is not correct?
(A) 4th quadrant sin function is increases from -1 to 0
(B) 3rd quadrant cot function is increases from 0 to ∞
(C) 2nd quadrant tan function is increases from –∞ to 0
(D) 1st quadrant cos function is decreases from 1 to 0.
3
22. If cos x = – 5 x lies in the third quadrant, then the values of sinx is
4 3 3 4
(A) – 5 (B)– 4 (C) 4 (D) 5.
3
23. If cos x = – 5 x lies in the third quadrant, then the values of tanx is
4 3 3 4
(A) – (B)– (C) (D)
3 4 4 3
3
24. If cos x = – 5 x lies in the third quadrant, then the values of cosecx is
5 5 5 5
(A) 4 (B) − 4 (C) 3 (D) − 3
3
25. If cot x = 4 x lies in the third quadrant, then the values of cosx is
4 4 3 3
(A) – 5 (B) 5 (C) 5 (D) − 5
5
26. If cot x = – 12 x lies in second quadrant, then the values of sec x
13 13 13 13
(A) –12 (B) (C) 5 (D) − 5
5
31𝜋
27. The value of sin 3
1 √3 1 √3
(A) 2 (B) (C) (D) − 2
2 √2
4
28. If 𝑡𝑎𝑛𝜃 = − 3 , then sin𝜃 is
4 4 4 4 4 4
(A)− 5 but not (B)− 5 or (C) 5 but not − 5 (D) None of these
5 5
15𝜋
29. The value of cot 4
1 1
(A) −1 (B) 1 (C) (D) −
√2 √2
29. sin (n + 1)xsin (n + 2)x + cos (n + 1)xcos (n + 2)x is equal to
(A)sinx (B)cosx (C)cos2x (D) sin2x
30. If sinx =0 ,then
𝜋
A)𝑥 = 𝑛𝜋 (B) 𝑥 = 2𝑛𝜋 (C) )𝑥 = (2𝑛 + 1) (D) 𝑥 = 0
2
31. If cosx =0 ,then
𝜋
A)𝑥 = 𝑛𝜋 (B) 𝑥 = 2𝑛𝜋 (C) )𝑥 = (2𝑛 + 1) 2 (D) 𝑥 = 0
𝜋
32. The value of 𝑡𝑎𝑛 8 is equal to
1 1
(A) √2 + 1 (B) 1 − √2 (C) (D)
2 √2+1
𝜋
33. The value of 𝑐𝑜𝑠 12 is
√3+1 √3−1
(A) (B) 2√2 (C)√2 + 1 (D) √2 − 1
2√2
34. If tan A = 1/2 and tan B = 1/3, then the value of A + B is
(A) π/6 (B) π (C) 0 (D) π/4
𝜋
35. The value of 𝑠𝑖𝑛 12 is
√3+1 √3−1
(A) (B) 2√2 (C)√2 + 1 (D) √2 − 1
2√2

Question Bank: Department of School Education (Pre University ) 18

36. 1-sin245° = ___________
37. The value of cos 1° cos 2° cos 3° … cos 179° is---------
38. If y = cos x, then what is the maximum value of y IS ---------
39. If cos(-t) = 0.34, what is cos(t) IS -------
40. If tan x = 5, then tan (2 x) =------
41. If cos t = 3/4, and sin t < 0, then sin (3 t) = -------
42. Statement I: 𝑠𝑖𝑛 (𝑛 + 1)𝑥𝑠𝑖𝑛 (𝑛 + 2)𝑥 + 𝑐𝑜𝑠 (𝑛 + 1)𝑥𝑐𝑜𝑠 (𝑛 + 2)𝑥 is equal to 𝑐𝑜𝑠𝑥.
Statement II: 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑦 + 𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑦 = sin (𝑥 + 𝑦)
(A) Both statements are true (B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true (D) Both statements are false.
43. Match List I with List II
List I List II
a) Domain of 𝑠𝑖𝑛𝑥 i) (−∞ , ∞) − {𝑛𝜋: 𝑛 ∈ 𝑍}
b) Domain of 𝑐𝑜𝑡𝑥 ii) [−1 , 1]
c)Range of 𝑐𝑜𝑠𝑥 iii) (−∞ , ∞)

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
44. Match List I with List II
List I List II
a) 𝑠𝑖𝑛𝑥 is positive i) 𝐼𝐼 𝑎𝑛𝑑 𝐼𝑉 𝑞𝑢𝑎𝑑𝑟𝑒𝑛𝑡
b) 𝑐𝑜𝑡𝑥 is negative ii) 𝐼 𝑎𝑛𝑑 𝐼𝐼 𝑞𝑢𝑎𝑑𝑟𝑒𝑛𝑡
c)𝑠𝑒𝑐𝑥 is positive iii) 𝐼 𝑎𝑛𝑑 𝐼𝑉 𝑞𝑢𝑎𝑑𝑟𝑒𝑛𝑡

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
List I List II
𝜋
a) 𝑠𝑖𝑛 (𝜋 − 4 ) i) 2

b) 𝑐𝑜𝑠𝑒𝑐 (𝜋 − 6 )
𝜋 ii) −2
𝜋 1
c)𝑠𝑒𝑐 (𝜋 − 3 ) iii)
√2

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-iii, b-ii, c-I C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.
46. Assertion (A): 𝐼𝑓 𝑐𝑜𝑠(−𝑥) = 0.34, 𝑡ℎ𝑒𝑛 𝑐𝑜𝑠(𝑥) = 0.34.
Reason (R): 𝑐𝑜𝑠 (−𝑥) = −𝑐𝑜𝑠𝑥
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of
the Assertion (A)
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation
of the Assertion A)
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is false.

Question Bank: Department of School Education (Pre University ) 19

𝜋 𝜋 𝜋 𝜋 1
47. Statement I: sin 3 sin 12 + cos 3 cos 12 = .
√2
Statement II: 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑦 + 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑦 = cos (𝑥 − 𝑦)
(A) Both statements I and II are true and Statement II is the correct explanation of the
Statement I.
(B) Statement I is true and statement II is true and Statement II is not the correct
explanation of the Statement I.
(C) Statement I is false and statement II is true (D) Both statements I and II are false.
𝜋 𝜋
48. Statement I: 3𝑠𝑖𝑛 6 + 4𝑠𝑖𝑛3 6 =1.
Statement II: 3𝑠𝑖𝑛 𝑥 + 4𝑠𝑖𝑛3 𝑥 =sin3x.
(A) Both statements I and II are true and Statement II is not the correct explanation of the
Statement I.
(B) Statement I is true and statement II is true and Statement II is the correct explanation
of the Statement I.
(C) Statement I is false and statement II is true (D) Both statements I and II are false.
𝜋 1
49. Statement 1. 𝑠𝑖𝑛 (2𝜋 + 6 ) = 2
𝜋 1
Statement 2. 𝑐𝑜𝑠 (2𝜋 − 3 ) = 2
(A) Both statements are true
(B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true (D) Both statements are false.
−4
50. A point is in III- Quadrant and on the Unit Circle. If its x-coordinate is 5 , then the y-

coordinate of the point is
3 3 4 3
(A) 5 (B) − 5 (C) 5 (D) – 4.

51. Find the point on the Unit Circle associated with the rotation -π/2.
(A) (0,1) (B) [0, −1 ] (C) (−1, 0) (D) [1,0 ].
52. If point (a , b) is on the Unit Circle associated with the rotation x.
1. 𝑆𝑖𝑛𝑥 = 𝑏 2. 𝐶𝑜𝑠𝑥 = 𝑎 3. 𝑐𝑜𝑠(−𝑥) = −𝑎
𝑊ℎ𝑖𝑐ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡𝑠 𝑎𝑟𝑒 𝑐𝑜𝑟𝑟𝑒𝑐𝑡?
(𝐴)1 𝑎𝑛𝑑 2 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝐵) 1 𝑎𝑛𝑑 3 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝐶) 3 𝑜𝑛𝑙𝑦 (𝐷) 𝐴𝑙𝑙 1, 2 𝑎𝑛𝑑 3.
53. If cos(-x) = 0.34, then the value of cos(x) is
(A) -0.34 (B) 0.66 (C) 0.34 (𝐷) - 0.66
4
54. If cos x = 5 , then the value of cos (2 x) =
8 7 7 24
A) 25 B) 25 C) − 25 D) 25
3
55. If sin x = 5 , then the value of cos (2 x) =
18 7 7 24
A) 25 B) 25 C) − 25 D) 25

Question Bank: Department of School Education (Pre University ) 20

56. The terminal side of an angle 𝜃 in standard position passes
through the point (3, −4)as in the following figure, then sinθ =
3 3 4 3
A) − 4 (B) 5 C) − 5 D) − 5
57. The graph drawn below depicts

(A)sine 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (B) 𝑐𝑜𝑠𝑖𝑛𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (C) tan 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (D) 𝑐𝑜𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 .
58. The graph drawn below depicts

(A)sine 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (B) 𝑐𝑜𝑠𝑖𝑛𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (C) tan 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (D) 𝑐𝑜𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛0
59. The graph drawn below depicts

(A)sine 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (B) 𝑐𝑜𝑠𝑖𝑛𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (C) sec 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (D) 𝑐𝑜𝑠𝑒𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
60.

Statement I :Figure1 graph of y = cos x Statement II :Figure2 graph of y = sin x
(A) Both statements are true (B) Statement I is true and statement II is false
(C) Statement I is false and statement II is true (D) Both statements are false.
Two marks questions
1. A wheel makes 360 revolutions in one minute. Through how many radians does it turn
in one second. (U)
2. Find the degree measure of the angle subtended at the centre of a circle of radius 100
22
cm by an arc of length 22cm. (use= 7 ) (U)
3. If in two circles, the arcs of the same length subtend angles 600 and 750 at the centre,
find the ratio of their radii. (U)
Question Bank: Department of School Education (Pre University ) 21

4. A minute hand of watch is 1.5cm long. How for does its tip moves in 40 minutes.(A)
5. Find the angle in radian through which a pendulum swings if its length 75cm and the
tip describe an arc of length. (U)
6. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor
arc of the chord.
7. Find the angle in radian through which a pendulum swings if its length 75cm and the
tip describe an arc of length
(i) 10 cm (ii) 15 cm (iii)21 cm
11
8. Convert radians into degree measure. (i)-4 radians (ii) 6 radians (iii) 16 radians
9. Find the radian measure corresponding to the degree measure 400 20’
10. Find the value of (i) 𝑠𝑖𝑛7650 (ii) cosec(-14100) (iii) 𝑐𝑜𝑠(−17100 )
31𝜋 19𝜋 −11𝜋 −15𝜋 𝜋
(iv) 𝑠𝑖𝑛 ( ) (v) 𝑡𝑎𝑛 ( ) (vi) 𝑠𝑖𝑛 ( ) (𝑣𝑖𝑖) 𝑐𝑜𝑡 ( ) (viii) Find the value of tan
3 3 3 4 8
11. Prove that cos  − sin  = cos2 .
4 4
(A)
12. Prove that 2 + 2 + 2 cos  = 2 cos  . (S)
13. Prove the following:
2𝑡𝑎𝑛𝑥 2𝑡𝑎𝑛𝑥 1−𝑡𝑎𝑛2 𝑥
i)𝑠𝑖𝑛2𝑥 = 1+ 𝑡𝑎𝑛2𝑥 (𝑈) (𝒊𝒊 )𝑡𝑎𝑛2𝑥 = 1−𝑡𝑎𝑛2 𝑥 (𝑈) iii) 𝑐𝑜𝑠2𝑥 = 1+𝑡𝑎𝑛2 𝑥 (𝑈)
iv)𝑠𝑖𝑛3𝑥 = 3𝑠𝑖𝑛𝑥 − 4𝑠𝑖𝑛 𝑥(𝑈)
3
𝒗 ) 𝑐𝑜𝑠3𝑥 = 4𝑐𝑜𝑠 3 𝑥 − 3𝑐𝑜𝑠𝑥 (𝑈)
2𝑡𝑎𝑛𝑥
𝒗𝒊 )tan3x=3tanx−tan3x1−3tan2x (𝑈) vii) 𝑠𝑖𝑛2𝑥 = 1+𝑡𝑎𝑛2 𝑥 (𝑈)
𝑐𝑜𝑡𝑥𝑐𝑜𝑡𝑦−1 𝑐𝑜𝑡𝑥𝑐𝑜𝑡𝑦+1
viii) 𝑐𝑜𝑡(𝑥 + 𝑦) = 𝑐𝑜𝑡𝑥+𝑐𝑜𝑡𝑦 (𝑈) ix) 𝑐𝑜𝑡(𝑥 + 𝑦) = 𝑐𝑜𝑡𝑦−𝑐𝑜𝑡𝑥 (𝑈)
𝜋 𝜋 5𝜋 𝜋 𝜋 𝜋 𝜋 −1
14. Prove that (i) 3𝑠𝑖𝑛 𝑠𝑒𝑐 − 4𝑠𝑖𝑛 𝑐𝑜𝑡 = 1 (ii) 𝑠𝑖𝑛2 + 𝑐𝑜𝑠 2 − 𝑡𝑎𝑛2 =
6 3 6 4 6 3 4 2
𝜋 2 7𝜋 2𝜋 3 𝜋 5𝜋 𝜋
(iii) 2𝑠𝑖𝑛2 6 + 𝑐𝑜𝑠𝑒𝑐 𝑐𝑜𝑠 =2 (iv) 𝑐𝑜𝑡 2 6 + 𝑐𝑜𝑠𝑒𝑐 6 + 3𝑡𝑎𝑛2 6 = 6
6 3
3𝜋 𝜋 𝜋
15. 2𝑠𝑖𝑛2 4 + 2𝑐𝑜𝑠 2 4 + 2𝑠𝑒𝑐 2 3 = 10
16. Find the value of (i) 𝑠𝑖𝑛150 (ii) 𝑡𝑎𝑛150 (iii) 𝑠𝑖𝑛750
𝜋 𝜋 𝜋 𝜋
17. Prove that 𝑐𝑜𝑠 (4 − 𝑥) 𝑐𝑜𝑠 ( 4 − 𝑦) − 𝑠𝑖𝑛 ( 4 − 𝑥) 𝑠𝑖𝑛 ( 4 − 𝑦) = 𝑠𝑖𝑛(𝑥 + 𝑦)
𝑐𝑜𝑠(𝜋+𝑥) 𝑐𝑜𝑠(−𝑥)
18. Prove that 𝜋 = 𝑐𝑜𝑡 2 𝑥
𝑠𝑖𝑛(𝜋−𝑥) 𝑐𝑜𝑠( +𝑥)
2
3𝜋 3𝜋
19. Prove that 𝑐𝑜𝑠 ( 2 + 𝑥) 𝑐𝑜𝑠(2𝜋 + 𝑥) [𝑐𝑜𝑡 ( 2 − 𝑥) + 𝑐𝑜𝑡(2𝜋 + 𝑥)] = 1
20. Prove that cos 4x = 1 – 8si𝑛2 x co𝑠 2 x
Three marks questions.
1. Prove the following:
(i) (sin 3𝑥 + 𝑠𝑖𝑛𝑥)𝑠𝑖𝑛𝑥 + (𝑐𝑜𝑠3𝑥 − 𝑐𝑜𝑠𝑥)𝑐𝑜𝑠𝑥 = 0
(ii) Prove that 𝑠𝑖𝑛3𝑥 = 3𝑠𝑖𝑛𝑥 − 4𝑠𝑖𝑛3 𝑥
3𝑡𝑎𝑛𝑥−𝑡𝑎𝑛3 𝑥
(iii)Prove that 𝑡𝑎𝑛3𝑥 = 1−3𝑡𝑎𝑛2 𝑥
(iv) Prove that 𝑐𝑜𝑠3𝑥 = 4𝑐𝑜𝑠 3 𝑥 − 3𝑐𝑜𝑠𝑥
𝜋
𝑡𝑎𝑛( +𝑥) 1+𝑡𝑎𝑛𝑥 2
(v) Prove that 4
𝜋 = (1−𝑡𝑎𝑛𝑥)
𝑡𝑎𝑛( −𝑥)
4
3𝜋 3𝜋
(vi) Prove that 𝑐𝑜𝑠 ( 4 + 𝑥) − 𝑐𝑜𝑠 ( 4 − 𝑥) = −√2𝑠𝑖𝑛𝑥
𝜋 𝜋
(vii) Prove that 𝑐𝑜𝑠 (4 + 𝑥) + 𝑐𝑜𝑠 (4 − 𝑥) = √2 𝑐𝑜𝑠𝑥
(viii) Show that tan 3 x tan 2 x tan x = tan 3x – tan 2 x – tan x
 x+ y
(ix) Prove that (cos x + cos y ) 2 + (sin x − sin y ) 2 = 4cos 2  
 2 

Question Bank: Department of School Education (Pre University ) 22

𝑥−𝑦
(x) Prove that (𝑐𝑜𝑠𝑥 − 𝑐𝑜𝑠𝑦)2 + (𝑠𝑖𝑛𝑥 − 𝑠𝑖𝑛𝑦)2 = 4 𝑠𝑖𝑛2 ( 2 )
(xi) Prove that 𝑐𝑜𝑡𝑥𝑐𝑜𝑡2𝑥 − 𝑐𝑜𝑡2𝑥𝑐𝑜𝑡3𝑥 − 𝑐𝑜𝑡3𝑥𝑐𝑜𝑡𝑥 = 1
(xii) Prove that 𝑠𝑖𝑛2𝑥 + 2 𝑠𝑖𝑛4𝑥 + 𝑠𝑖𝑛6𝑥 = 4 𝑐𝑜𝑠 2 𝑥𝑠𝑖𝑛4𝑥
2. Prove the following:
sinx + sin3x sinx − siny  x−y 
( i) = tan2x (ii) = tan   (U)
cosx + cos3x cos x + cos y  2 
sin5x + sin3x cos7x + cos5x
(iii) = tan4x (iv ) = cotx (U)
cos5x + cos3x sin7x − sin5x
cos9x − cos5x sin2x
(v) =− (U)
sin17x − sin3x cos10x
3. Prove the following:
sinA + sin2A + sin3A
( i) = tan2A (A)
cos A + cos2A + cos3A .
(ii) sin( A + B) sin( A − B) = sin2 A − sin2 B . (A)
(iii) cos ( A + B) cos ( A − B) = cos2 A − sin2 B . (A)
(iv) Prove that cos 6𝑥 = 32 𝑐𝑜𝑠 6 𝑥 − 48𝑐𝑜𝑠 4 𝑥 + 18𝑐𝑜𝑠 2 𝑥 − 1
4 𝑡𝑎𝑛𝑥−4𝑡𝑎𝑛3 𝑥
(v) Prove that tan 4𝑥 = 1−6 𝑡𝑎𝑛2𝑥+𝑡𝑎𝑛4 𝑥.
(vi) Prove that 𝑐𝑜𝑠 2 2𝑥 − 𝑐𝑜𝑠 2 6𝑥 = 𝑠𝑖𝑛4𝑥 sin 8𝑥.
3 12 𝜋
(v) If 𝑠𝑖𝑛𝑥 = 5 and 𝑐𝑜𝑠𝑦 = − 13 , 𝑤ℎ𝑒𝑟𝑒 2 < 𝑥 , 𝑦 < 𝜋 then find the value of 𝑠𝑖𝑛(x+y)
(vi) Prove that 𝑠𝑖𝑛2 6𝑥 − 𝑠𝑖𝑛2 4𝑥 = 𝑠𝑖𝑛2𝑥 sin 10𝑥.
Five marks questions
1) Prove geometrically that 𝑐𝑜𝑠(𝑥 + 𝑦) = 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑦 − 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑦.
4 𝑥 𝑥 𝑥
2) If 𝑡𝑎𝑛𝑥 = − 3 𝑤ℎ𝑒𝑟𝑒 𝑥 in quadrant II , then find the value of sin 2 , cos 2 , tan 2 . .
1 𝑥 𝑥 𝑥
3) If 𝑠𝑖𝑛𝑥 = 4 , 𝑥 in quadrant II , then find sin 2 , cos 2 , tan 2 .
1 𝑥 𝑥 𝑥
4) If 𝑐𝑜𝑠𝑥 = − 3 , 𝑥 in quadrant III , then find sin 2 , cos 2 , tan 2 .
3 3𝜋 𝑥 𝑥 𝑥
5) If 𝑡𝑎𝑛𝑥 = 4 𝑤ℎ𝑒𝑟𝑒 𝜋 < 𝑥 < 2 then find the value of sin 2 , cos 2 . , tan 2 . .
6) Prove that 𝑐𝑜𝑡 4𝑥(𝑠𝑖𝑛5𝑥 + 𝑠𝑖𝑛3𝑥) = 𝑐𝑜𝑡𝑥(𝑠𝑖𝑛5𝑥 − 𝑠𝑖𝑛3𝑥)
𝑠𝑖𝑛5𝑥−2 𝑠𝑖𝑛3𝑥+𝑠𝑖𝑛𝑥
7) Prove that = 𝑡𝑎𝑛𝑥
cos 5𝑥−cos 𝑥
(𝑠𝑖𝑛7𝑥+𝑠𝑖𝑛5𝑥)+(𝑠𝑖𝑛9𝑥+𝑠𝑖𝑛3𝑥)
8) Prove that (𝑐𝑜𝑠7𝑥+𝑐𝑜𝑠5𝑥)+(𝑐𝑜𝑠9𝑥+𝑐𝑜𝑠3𝑥) = 𝑡𝑎𝑛6𝑥
𝑥 3𝑥
9) Prove that 𝑠𝑖𝑛3𝑥 + 𝑠𝑖𝑛2𝑥 − 𝑠𝑖𝑛𝑥 = 4 𝑠𝑖𝑛𝑥𝑐𝑜𝑠 2 𝑐𝑜𝑠 2
𝑥 9𝑥 5𝑥
10) Prove that 𝑐𝑜𝑠2𝑥𝑐𝑜𝑠 2 − 𝑐𝑜𝑠3𝑥𝑜𝑠 2 = 𝑠𝑖𝑛5𝑥𝑠𝑖𝑛 2
𝜋 𝜋 3
11) Prove that 𝑐𝑜𝑠 2 𝑥 + 𝑐𝑜𝑠 2 (𝑥 + 3 ) + 𝑐𝑜𝑠 2 (𝑥 − 3 ) = 2.
−3
12) Find If 𝑐𝑜𝑠𝑥 = 5 , x lies in the third quadrant, find the values of other five trigonometric
functions.
−5
13) If 𝑐𝑜𝑡𝑥 = 12 , x lies in second quadrant,, find the values of other five trigonometric functions.
3
14) If 𝑐𝑜𝑡𝑥 = 4 , x lies in third quadrant,, find the values of other five trigonometric functions.
−1
15) If 𝑐𝑜𝑠𝑥 = , x lies in third quadrant,, find the values of other five trigonometric functions.
2
3
16) If 𝑠𝑖𝑛𝑥 = 5 , x lies in second quadrant,, find the values of other five trigonometric functions.
13
17) If 𝑠𝑒𝑐𝑥 = 5 , x lies in fourth quadrant, find the values of other five trigonometric functions.

Question Bank: Department of School Education (Pre University ) 23

CHAPTER-4
Complex numbers and Quadratic Equations
MCQ / FB

1. If 4x+i(3x–y) = 3 +i(– 6), where x and y are real numbers, then the values of x.
3 4 3 4
(A) 4 (B) 3 (C) − 4 (D) − 3
2. If 4x+i(3x–y) = 3 +i(– 6), where x and y are real numbers, then the values of y.
39 33 39 33
(A) − 4 (B) 4 (C) 4 (D) − 4
3. The additive inverse of -2+3i
(A) 2-3i (B) -2-3i (C) 2+3i (D) -2+3i
4. The multiplicative identity of Z= a+ib is
(A) a-ib (B) 0+i0 (C) 1+1i (D) 1+0i
5. The multiplicative inverse of 2 – 3i.
2 – 3𝑖 2 + 3𝑖 −2 – 3𝑖 2+ 3𝑖
(A) 13 (B) (C) (D) 13 .
√13 √13
6. The multiplicative inverse of -i
1+ 𝑖 𝑖
(A) 𝑖 (B) (C) (D) −𝑖
√2 √2
7. The multiplicative inverse of 4 – 3i.
4+ 3𝑖 4+ 3𝑖 −4 – 3𝑖 −4+ 3𝑖
(A) . (B) (C) (D) .
25 √25 25 25
8. The multiplicative inverse of √5 + 3𝑖 is
√5+3𝑖 √5−3𝑖 −√5+3𝑖 −√5−3𝑖
(A) . (B) 14 (C) 14 (D)
14 14
9. The conjugate of a complex number -5-3i
(A) 5-3i (B) -5+3i (C) -5-3i (D) 5+3i.
10. The real numbers x ,if (x – 3i) (3 + 5i) is the conjugate of –6 – 24i
(A) 6 (B) -7 (C) -6 (D) 7.
1+𝑖 𝑛
11. If ( ) = 1, then the smallest value of n
1−𝑖
(A) 2 (B) 3 (C) 4 (D) 6.
12. The square roots of – 1
(A) 1 ,-1 (B) – 𝑖 (C) 𝑖 (D) 𝑖, −𝑖
13. The square roots of -3
(A) 3,-3 (B) √3 ,- √3 (C) √3 𝑖, - √3 𝑖 (D) √3 𝑖.
14. If (5 – 3𝑖) = a + ib, then a =
3

(A) 125 (B) 10 (C) −10 (D) 25
15. The modulus of the complex number -1+i√3.
(A) 2 (B) √2 (C) 4 (D) -2.
16. Which of the following is incorrect for any two complex numbers z1 and z2
𝑍 |𝑍 |
(A) |z1z2| =|z1||z2| (B) ̅̅̅̅̅̅
𝑍1 𝑍2 = ̅̅̅
𝑍1 ̅̅̅
𝑍2 (C) |𝑍1 | = |𝑍1| (D) √−1√−1 = 1.
2 2
1+𝑖
17. The modulus of the complex number
1−𝑖
(A) 2 (B) √2 (C) 1 (D) 3.
1
18. The modulus of the complex number
1+𝑖
1 1
(A) (B) √2 (C) 1 (D) 2.
√2
19. Simplest form (1+i)6
(A) 8𝑖 (B) 64𝑖 (C) −8𝑖 (D) −64𝑖

Question Bank: Department of School Education (Pre University ) 24

20. Which of the following is incorrect
(A) √𝑎 √𝑏 = √𝑎𝑏 for all real numbers
(B) if any of a and b is zero, then √𝑎 √𝑏 = √𝑎𝑏 =0
(C) √𝑎 √𝑏 ≠ √𝑎𝑏 , if both a and b are negative real numbers
(D) √𝑎 √𝑏 = √𝑎𝑏 for all positive real number a and b
21. If z (5+i√2 )= 1+ i√2, then z is
1 1 7 + 4 2i
(A) 3- 6√2 𝑖 (B) (3- 6√2 𝑖) 27 (C) (3- 6√2 𝑖) 23 (D) .
27
22. 𝑖 1 +i2 + i3 + i4 =
(A) 2 (B) 1 (C) 0 (D) -1.
𝑎+𝑖𝑏
23. If = x+iy, then the value of 𝑥 2 + 𝑦 2
𝑎−𝑖𝑏
(A) 𝑎2 + 𝑏 2 (B) 1 (C) 0 (D) -1.
24. Imaginary part of 𝑖 −35 is
(A) 𝑖 (B) -i (C) 1 (D) -1.
25. Simplest form of 𝑖 −39
is
(A) 𝑖 (B) -i (C) 1 (D) -1.
𝑎−𝑖𝑏
26. If 𝑥 − 𝑖𝑦 = √ then (𝑥 2 + 𝑦 2 )2 is
𝑐−𝑖𝑑

𝑎2 +𝑏2 𝑎2 +𝑏2
(A) 𝑐 2 +𝑑2 (B) √𝑐 2 +𝑑2 (C) 1 (D) -1.
27. The graph drawn below depicts

(A) 𝑍 = 3 + 4𝑖 (B) 𝑍 = 3 − 4𝑖 (C) 𝑍 = 4 + 3𝑖 (D) 𝑍 = 4 − 3𝑖
28. For the figure given below,

1
(A) 𝑍 ∗ = 𝑍 (B) 𝑍 ∗ = −𝑍 (C) 𝑍 ∗ = 𝑍̅ (D) 𝑍 ∗ = 𝑍
29. For the figure given below,|𝑧| 𝑖𝑠

(A) 4 (B) 1 (C) √17 (D) 17.

Question Bank: Department of School Education (Pre University ) 25

30. Match List I with List II

List I List II
a) Additive inverse of 1+ i 1
i) 1+ i
b) conjugate of 1+ i ii) 1 − i
c) multiplicative inverse of 1+ i iii) −1 − i

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
2 + 2𝑖
31. Assertion (A): The multiplicative inverse of 2 + 2𝑖 is
√8
𝑍̅
Reason (R): The multiplicative inverse of Z is |𝑍|2
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation
of the Assertion (A)
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct
explanation of the Assertion A)
(C) Assertion (A) is true and Reason (R) is false
(D) Assertion (A) is false and Reason (R) is true
32. STATEMENT 1: The modulus of 3 − 4𝑖 is 5
STATEMENT 2: The modulus of 𝑎 + 𝑏𝑖 is √a2 + b 2
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct
explanation of the STATEMENT 1
(C) STATEMENT 1 is true and STATEMENT 2 is false
(D) STATEMENT 1 is false and STATEMENT 1 is true
33. STATEMENT 1: 𝑇ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖 (𝑖𝑜𝑡𝑎) 𝑖𝑠 (−1)1/2 STATEMENT 2: 𝒊 . 𝒊 = √−1√−1= 1.
(A) Both STATEMENT 1 and 2 are true (B)Both STATEMENT 1 and 2 are false.
(C) STATEMENT 1 is true and STATEMENT 2 is false.
(D) STATEMENT 1 is false and STATEMENT 1 is true.
1 3
34. STATEMENT 1: If 𝑧 = 3 + 𝑖,then the real part of 𝑖𝑠
𝑧 10
1 𝑥−𝑖𝑦
STATEMENT 2: If 𝑧 = 𝑥 + 𝑖𝑦, then 𝑧 = 𝑥 2 +𝑦 2.
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct
explanation of the STATEMENT 1
(C)STATEMENT 1 is true and STATEMENT 2 is false.
(D) STATEMENT 1 is false and STATEMENT 1 is true.
35. Assertion (A):The value of 1 + i2 + i4 + i6 + … + i20 is 0
Reason (R): i2𝑛 = ±1. If n is multiple of 4, then i𝑛 = −1 and n is not a multiple of 4 then i𝑛 = 1
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation
of the Assertion (A)
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct
explanation of the Assertion A)
(C) Assertion (A) is false and Reason (R) is false
(D) Assertion (A) is false and Reason (R) is true

Question Bank: Department of School Education (Pre University ) 26

𝟏+𝒊 2𝒏
36. If ( ) = 1, then the smallest positive value of n is = ------------
𝟏−𝒊
37. If the complex number z = x + iy satisfies the condition |z | = 4, represents a circle with
centre (0, 0) and radius r is = ------------
38. The value of 𝑖 2025 is 𝑘𝑖 ,then k is ……..
|𝒛|
39. If z = x + iy , then = ……….
|𝐳̅ |
𝟐+𝒊
40. | | =………
𝟏−𝟐𝒊
41. The positive smallest integer n, for witch (1 + 𝑖)2𝑛 = (𝑖 − 𝑖)2𝑛 is = ------------
3
42. If 5i (− 𝑖) = a +ib ,then a =……
5
43. If z(2-i) = 3+i, then z2 is equal to a+ib then a =….
44. If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then 𝑥 2 + 𝑦 2 =…….
45. 1.If √𝑎 + 𝑖𝑏 = 𝑥 − 𝑖𝑦, then √𝑎 − 𝑖𝑏 = 𝑥 + 𝑖𝑦.
2. |𝑥 + 𝑖𝑦| = |𝑥 − 𝑖𝑦|. 3. If |𝑧| = 0, 𝑡ℎ𝑒𝑛 𝑧 = 0 + 𝑖0.
Which of the above statements are correct?
(A) 1 only (B) 2 only (C) 3 only (D) All 1, 2 and 3.
46. If z = x+iy, then match List I with List II

List I List II
a) z 𝑧̅ i) 𝑥 2 + 𝑦 2
b) 𝑧 − 𝑧̅ ii) 2x
c) 𝑧 + 𝑧̅ iii) 𝑖2𝑦
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-i, b-iii, c-ii.
47. If 𝑧𝑖 = 𝑥𝑖 +i𝑦𝑖 ,then match List I with List II

List I List II
a) 𝑧1 𝑧2 i) (𝑥1 − 𝑥2 ) + 𝑖(𝑦1 − 𝑦2 )
b) 𝑧1+ 𝑧2 ii) (𝑥1 𝑥2 − 𝑦1 𝑦2 ) + 𝑖(𝑥1 𝑦2 + 𝑦1 𝑥2 )
c) 𝑧1− 𝑧2 iii) (𝑥1 + 𝑥2 ) + 𝑖(𝑦1 + 𝑦2 )

Choose the correct answer from the options given below:
A) a-ii , b-iii, c-i B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii
Two Marks Questions
1. Express the following in the form of a + ib
5 + 2i
1) (U ) 2) (5 − 3i)3 (U ) 3) (1 − i) 4 (U )
1 − 2i
3
1 
3
 1 
3
 18  1 25 
4)  + 3i  (U ) 5)  −2 − i  (U ) 6) i +    ( A)
3   3    i  
(3 + i 5)(3 − i 5)
7) ( A) 8) ( − 3 + −2)(2 3 − i) (U )
( 3 + 2i) − ( 3 − i 2)

Question Bank: Department of School Education (Pre University ) 27

2. Simplify
 −1 
3
1 
1)(−5i)  i  (U ) 2)( −i)(2i)  i  (U )
8   8 
 −3 
3)(5i)  i  (U ) 4) i 9 + i19 (U )
 5 
−39
5) i (U ) 6)3(7 + i7) + i(7 + i7) (U )
7)(1 − i) − (−1 + i6) (K ) 8) i −35 (K )
3. Find the multiplicative inverse of each of the following complex numbers
1). 2 − 3i (U ) 2). 4 − 3i (U ) 1+ i 2 + 3i
5). ( A) 6). ( A)
3). 5 + 3i (U ) 4). − i (U ) 1− i 3 + 4i
1+𝑖 𝑚
4. If (1−𝑖) = 1, then find the least positive integral value of m
Three Marks Questions
1. If 4x + i(3x – y) = 3 + i(-6), where x and y are real numbers, then find the values of x and y (U)
2. Find the real numbers x and y if (x – iy)(3 + 5i) is the conjugate of -6 – 24i. (U)
𝑎−𝑖𝑏 𝑎2 + 𝑏 2
3. If x – iy = √ prove that (𝑥 2 + 𝑦 2 )2 = 𝑐 2+ 𝑑2 (U)
𝑐−𝑖𝑑
𝑎+𝑖𝑏
4. If x + iy = , prove that x2 + y2 = 1 (U)
𝑎−𝑖𝑏
2
(𝑥+𝑖)2 (𝑥 2 +1)
5. If a+ib = , prove that a2 + b2 = (2𝑥 2 +1)2 (S)
2𝑥 2 + 1
𝑢 𝑣
6. If (x + iy)3 = u + iv, then show that + 𝑦 = 4(𝑥 2 − 𝑦 2 ) (S)
𝑥
3−2𝑖 sin 𝜃
7. Find real θ such that is purely real (S)
1−2𝑖 sin 𝜃
(3 − 2i )(2 + 3i )
8. Find the conjugate of , (S)
(1 + 2i )(2 − i )
𝑍 +𝑍2 +1
9. If Z1 = 2 – i, Z2 = 1 + i then find | 1 | (S)
𝑍1 −𝑍2 +1
𝑧1 𝑧2
10. Let z1 = 2 – 𝑖, z2 = -2 + 𝑖 find Re( ) (S)
̅̅̅
𝑧1
11. If (a + i b) (c + i d) (e + i f) (g + i h) = A + i B
then how that (a2 + b2) (c2 + d2) (e2 + f 2 ) (g2 + h2) = A2 + B2 (U)
(3+𝑖√5)(3−𝑖√5)
12. Express in a+ib form
(√3+𝑖√2)−(√3−𝑖√2)
3
1 25
13. Evaluate [𝑖 + ( 𝑖 ) ]
18

1 2 3−4𝑖
14. Reduce ( − 1+𝑖) ( 5+𝑖 ) to the standard form.
1−4𝑖
15. Find the real numbers x and y if (x – iy)(3 + 5i) is the conjugate of -6 – 24i
16. Find the number of non-zero integral solutions of the equation |1-i|x = 2x

Question Bank: Department of School Education (Pre University ) 28

CHAPTER-5
LINEAR INEQUALITIES
MCQ / FB
1. Which of the following is incorrect statement
A) 7 > 5 is examples of numerical inequalities.
B) y ≤ 4 is examples of literal inequalities
C) 3 < x < 5 is examples of double inequalities.
D) ax + b < 0 is example of slack inequalities.
2. Which of the following is incorrect statement
A) ax + by ≤ c is slack inequalities. B) ax + by > c is slack inequalities .
C) a𝑥 2 + bx + c ≤ 0 is slack inequalities. D) ax + by ≥ c is slack inequalities.
3. Which of the following is incorrect statement, if a ≤ b
A) a + k ≤ b + k ,if k ∈ 𝑅 B) a - k ≤ b – k, if k ∈ 𝑅
C) ak ≤ bk if k > 0 D) a /k ≤ b /k, if k ≠ 0.
4. Which of the following is incorrect statement, if a ≤ b
A) a k ≤ b k ,if k > 0 B) a k ≥ b k ,if k < 0
C) ak ≤ bk if k < 0 D) a/k ≥ b/k, if k < 0
5. The set of solution of the inequality 24x < 100, when x is a natural number
A) {0,1,2,3,4,5}. B) {1,2,3,4,5} C) {1,2,3,4 } D) {5,6,7 … … . }.
6. The number of solutions of the inequality 24x < 100, when x is a natural number
A) 6 B) 5 C) 4. D) 3.
7. The set of solution of the inequality 30 x < 200 when x is an integer.
A) {0,1,2,3,4,5. 6 } B) {1,2,3,4,5,6 }
C) {……-2,-1,01,2,3,4,5,6}. D) {……-2,-1,01,2,3,4,5,6…….}.
8. The solution of the inequality x < 0 for real x.
A) (−∞, 0]. B) [ 0, ∞) C) (−∞, 0 ) D) (0, ∞).
9. The solution of the inequality x ≤ 0 for real x.
A) (−∞, 0]. B) [ 0, ∞) C) (−∞, 0 ) D) (0, ∞).
10. The solution of the inequality x ≥ 0 for real x.
A) (−∞, 0]. B) [ 0, ∞) C) (−∞, 0 ) D) (0, ∞).
11. The solution of the inequality x > 0 for real x.
A) (−∞, 0]. B) [ 0, ∞) C) (−∞, 0 ) D) (0, ∞).
12. The solution of the inequality -1 < x < 1 for real x.
A) [ -1 , 1]. B) (−∞, −1] ∪ [ 1, ∞) C) (−1,1 ) D) (−∞, −1) ∪ (1, ∞).
13. The solution of the inequality -1 ≥ x and x ≥ 1 for real x.
A) [ -1 , 1]. B) (−∞, −1] ∪ [ 1, ∞) C) (−1,1 ) D) (−∞, −1) ∪ (1, ∞).
14. The set of solution of the inequality 4x + 3 < 6x +7, when x is real number
A) {−2, −1,01,2,3,4,5,6 … . . }. B) {… … … − 4, −3, −2} C) (−∞, −2 ) D) (−2, ∞).
15. The set of solution of the inequality 4x + 3 < 6x +7, when x is real number
A) [−∞, −2]. B) {… … … − 4, −3, −2} C) (−∞, −2 ) D) (−2, ∞).
16. The solution of the inequality 3(x – 1) ≤ 2 (x – 3) for real x.
A) (−∞, −3]. B) [ −3, ∞) C) (−∞, −3 ) D) (−3, ∞).
17. The solution of the inequality 3(2 – x) ≥ 2 (1 – x) for real x.
A) (−∞, 4]. B) [ −4, ∞) C) (−∞, −4 ) D) (−4, ∞).
x x
18. The solution of the inequality x + + < 11 for real x.
2 3
A) (-∞,6]. B) [ 6,∞) C) (-∞,6 ) D) (6,∞).

Question Bank: Department of School Education (Pre University ) 29

19. The solution of

A) (−∞, 0]. B) [ −∞, 3) C) (−4, 3 ) D) (−∞, 3).
20. For the figure given below, consider the following statements 1,2 and 3

Statement 1: The solution of the inequality 2 ≤ 𝑥 < 6 for real x.
Statement 2: solution of the inequality 2 ≤ 𝑥 < ∞ for real x.
Statement 3: solution of the inequality - ∞ < 𝑥 ≤ 6 for real x.
A) Statement 1 is true and Statement 2 and 3 are false
B) Statement 1 and 3 are true but Statement 2 is false
C) Statement 2 and 3 are true but Statement 1 is false
D) All the Statements 1 ,2 and 3 are true.
21. If 7x + 3 < 5x + 9. then the graph of the solutions on number line is
A)

B)

C)

D)
3
22. For the graph given below solution set for the real line

A) (−5, ∞). B) [ −5, ∞) C) (−∞, −5 ) D) (−5,1).
23. For the graph given below solution set for the real line

A) (−∞, 5) B) [ −∞, 5) C) (−5, 5 ) D) (−∞, −5).
24. The solution of −8 ≤ 5𝑥 − 3 < 7 𝑖𝑠
A) -1 ≤ 𝒙 < 𝟐. B) -1 < 𝑥 < 2 C) -1 ≤ 𝑥 ≤ 2 D) -1 < 𝑥 ≤ 2
25. The number of pairs of consecutive odd natural numbers, both of which are Larger than
10, such that their sum is less than 40.
A) 1. B) 2 C) 3 D) 4.

Question Bank: Department of School Education (Pre University ) 30

26. The number of pairs of consecutive even positive integers, both of which are larger than
5 such that their sum is less than 23.
A) 1. B) 2 C) 3 D) 4.
27. Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he
should get in the third test to have an average of at least 60 marks.
A) 30 B) 35 C) 40 D) 45.
28. The solution of |x| < 1 for real x.
A) [ -1 , 1]. B) (−∞, −1] ∪ [ 1, ∞) C) (−1,1 ) D) (−∞, −1) ∪ (1, ∞).
29. The solution of |x| > 1 for real x.
A) [ -1 , 1]. B) (−∞, −1] ∪ [ 1, ∞) C) (−1,1 ) D) (−∞, −1) ∪ (1, ∞).
30. The solution of |x − 1| ≤ 1 for real x.
A) [ 0,1]. B) [ 0 , 2] C) (0, 2 ) D) (−2,0).
𝑥−1
31. Statement 1: The solution of ≥ 3, for real x is [ 13, ∞)
4
Statement 2: If 𝑎 ≤ 𝑏, then 𝑎𝑘 ≤ 𝑏𝑘 𝑖𝑓 𝑘 > 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): Solution of the system of inequalities 4x - 12 ≥ 0 and 2x - 7 ≤ 5 is [3,6].
Reason (R): If 4x - 12 ≥ 0 and 2x - 7 ≤ 5, then x ≥ 3 and x ≤ 6.
A) A is false but R is true B) A is false and R is false
C) A is true but R is false D) A is false and R is false
33. Statement 1 : Solution of linear inequality – 3x + 15 < – 12 is x ∈ (−∞, 9).
Statement 2 : If 𝑎 ≤ 𝑏, then 𝑎𝑘 ≥ 𝑏𝑘 𝑖𝑓 𝑘 < 0
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
34. Number of solutions of the inequality 5x+3 < 18, when x is a natural number is ………
35. Number of solutions of the inequality − 2 ≤ 5𝑥 + 3 < 7 𝑖𝑠, when x is a natural number is
………
36. If 4x + 4 < 6x +5, then x belongs to the interval is (a, ∞),then a =……….
37. If -a < -b, then positive a and b carries the relation
A) 𝑎 < 𝑏. B) −𝑎 > −𝑏 C) 𝑎 > 𝑏 D) - 𝑎 > 𝑏
38. If a < b and k is a positive integer then 𝑎 − 𝑘 … … 𝑏 − 𝑘
A) < B) > C) ≥ D) ≤.
1 1
39. If x ≥ y ,then ……𝑦
𝑥
A) < B) > C) ≥ D) ≤.
40. If A ={ x: −∞ < 𝑥 ≤ 1, 𝑥 ∈ 𝑅 } and B ={ x: −2 ≤ 𝑥 < ∞, 𝑥 ∈ 𝑅 },then A∩B is
A) [-2, 1]. B) [−∞, 1] C) (−2, 1 ) D) (−∞, ∞)
Two marks Questions:
1. 1) Solve 30x < 200when
i) x is a natural number, ii) x is an integer. (k)
2. Solve 24x < 100,when
i) x is a natural number. ii) x is an integer. (k)
3. Solve−12x > 30, when
i) x is a natural number. ii) x is an integer. (k)

Question Bank: Department of School Education (Pre University ) 31

4. Solve 5x − 3 < 3x + 1 when
i) x is an integer, ii) x is a real number. (k)
5. Solve 5x − 3 < 7, when
i) x is an integer. ii) x is a real number. (k)
6. Solve 3x + 8 > 2,when
i) x is an integer. ii) x is a real number. (k)
II. Solve following inequalities for real number 𝐱
1) 4x + 3 < 6x + 7 (k) 2)4x + 3 < 5x + 7 (k) 3) 3x − 7 > 5x − 1 (k)
5−2x x
4) 3(x − 1) ≤ 2(x − 3) (k) 5) 3(2 − x) ≥ 2(1 − x) (k) 6) 3 ≤ 6 − 5 (U)
x x x x 3(x−2) 5(2−x)
7) x + 2 + 3 < 11 (U) 8) 3 > 2 + 1 (U) 9) ≤ (U)
5 3
1 3x 1
10) 2 [ 5 + 4] ≥ 3 (x − 6) (U) 11) 2(2x + 3) − 10 < 6(x − 2) (U)
x (5x−2) (7x−3)
12) 37 − (3x + 5) ≥ 9x − 8(x − 3) (U) 13) 4 < − (U)
3 5
(2x−1) (3x−2) (2−x) 5−3x
14) ≥ − (U) 5) −8 ≤ 5x = 3 < 7 (A) 16) −5 ≤ ≤8 (A)
3 4 5 2
7x
17) 2 ≤ 3x − 4 ≤ 5 (A) 18) 6 ≤ 3(2x − 4) < 12 (A) 19) −3 ≤ 4 − 2 ≤ 18 (A)
3(x−2) 3x (3x+11)
20) −15 < 5 ≤ 0 (A) 21) −12 < 4 − −5 ≤ 2 (A) 22) 7 ≤ 2 ≤ 11 (A)
III. Solve the following inequalities and show the graph of the solution in each case on
number line
3𝑥−4 𝑥+1
1) 7𝑥 + 3 < 5𝑥 + 9 (k) 2) 2 ≥ 4 − 1 (U) 3) 3𝑥 − 2 < 2𝑥 + 1 (k)
𝑥 (5𝑥−2) (7𝑥−3)
4) 5𝑥 − 3 ≥ 3𝑥 − 5 (k) 5) 3(1 − 𝑥) < 2(𝑥 + 4) (k) 6)2 ≥ 3 − 5 (U)
IV
1. Find the minimum marks he should get in the annual examination to have an average of
at least 60 marks. (A)
2. The marks obtained by a student of Class XI in first and second terminal examination
are 62 and 48, respectively
3. Find all pairs of consecutive odd natural numbers, both of which are larger than 10,
such that their sum is less than 40. (A)
4. Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he
should get in the third test to have an average of at least 60 marks. (A)
5. To receive Grade ‘A’ in a course, one must obtain an average of 90 marks of more in five
examinations (each of 100 marks). If Sunita’s marks in first four examinations are
87,92,94 and 95, find minimum marks that Sunita must obtain in fifth examination to
get grade ‘A’ in the course. (A)
6. Find all pairs of consecutive odd positive integers both of which are smaller than 10
such that their sum is more than 11. (A)
7. Find all pairs of consecutive even positive integers, both of which are larger than 5 such
that their sum is less than 23. (A)
V. Solve the following system of inequalities and represent the solution graphically on
the number line.
8. 3𝑥 − 7 < 5 + 𝑥, 1 − 5𝑥 ≤ 1 (A)
9. 5𝑥 + 1 > −24, 5𝑥 − 1 < 24 (A)
10. 2(𝑥 − 1) < 𝑥 + 5, 3(𝑥 + 2) > 2 − 𝑥 (A)
11. 3𝑥 − 7 > 2(𝑥 − 6), 6 − 𝑥 > 11 − 2𝑥 (A)
12. 5(2𝑥 − 7) − 3(2𝑥 + 3) ≤ 0,2𝑥 + 19 ≤ 6𝑥 + 47 (A)

Question Bank: Department of School Education (Pre University ) 32

VI. Statement Problems:
13. In an experiment, a solution of hydrochloric acid is to be kept between 300 and 350
Celsius. What is the range of temperature in degree Fahrenheit if conversion formula is
5
given by C=9 (𝐹 − 32), where C and F represent temperature in degree Celsius and
degree Fahrenhiet , respectively. (S)
14. A manufacturer has 600 liters of a 12% solution of acid. How many liters of a 30% acid
solution must be added to it so that acid content in the resulting mixture will be more
than 15% but less than 18%? (S)
15. A solution is to be kept between 680 F and 770 F. What is the range in temperature in
degree Celsius (C) if the Celsius/ Fahrenheit (F) conversion formula is given by
9
𝐹 = 5 𝐶 + 32? (S)
16. A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The
resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640
liters of the 8% solution, how many liters of the 2% solution will have to be added? (S)
17. How many liters of water will have to be added to 1125 liters of the 45% solution of acid
so that the resulting mixture will contain more than 25% but less than 30% acid
content? (S)

Question Bank: Department of School Education (Pre University ) 33

CHAPTER-6
PERMUTATIONS AND COMBINATIONS
MCQ / FB
1. The number of 4 letter words, with or without meaning, which can be formed out of the letters
of the word ROSE, where the repetition of the letters is not allowed.
(A)6 (B)12 (C)24 (D) 256 .
2. The number of 4 letter words, with or without meaning, which can be formed out of the letters
of the word ROSE, where the repetition of the letters was allowed.
(A)6 (B)64 (C)24 (D) 256
3. Given 4 flags of different colors, then the number of different signals can be generated. If
a signal requires the use of 2 flags one below the other
(A)6 (B)12 (C)24 (D) 4.
4. The number of 2 digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if the
digits can be repeated
(A)10 (B)8 (C)120 (D) 6.
5. The number of 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming
that repetition of the digits is allowed
(A)25 (B)120 (C)60 (D) 125.
6. The number of 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming
that repetition of the digits is not allowed
(A)25 (B)120 (C)60 (D) 125
7. The number of 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits
can be repeated
(A)108 (B)216 (C)36 (D) 60.
8. A coin is tossed 3 times and the outcomes are recorded. The number of possible outcomes
(A)2 (B)4 (C)6 (D) 8.
9. The value of 7!-5! is
(A) 120 (B)5040 (C)4920 (D) 5160.
1 1 𝑥
10. If 6! + 7! = 8! , then x =
(A) 64 (B)49 (C)56 (D) 81.
1 1 𝑥
11. If 8! + 9! = 10! , then x =
(A) 64 (B)81 (C)100 (D) 72.
12. Which of the following is correct
8! 7!
(A) 3! + 4!= 4! (B) 6!2! =8 (C) 5! = 7!-5! (D) nPr.=r! nCr.
13. The value of 6P3 -5P3 is
(A) 120 (B)100 (C)220 (D) 20.
14. The number of ways in which a Chairman and a Vice-Chairman can be chosen
from amongst a group of 12 persons assuming that one person can not hold more than one
position is
(A) 144 (B)120 (C)132 (D) 12.
15. The number of permutations of the letters of the word ALLAHABAD is
9! 9! 9! 7!
(A) 4!2!2! (B) 4!2! (C) 4!! (D) 4!2!.
16. The number of permutations of the letters of the word INSTITUTE is
9! 9! 9! 4!
(A) 2!2! (B) 3!2! (C) 3! (D) 2!2!.
17. The number of ways of rearranging the letters of the word ROOT is
(A) 24 (B)12 (C)6 (D) 4.

Question Bank: Department of School Education (Pre University ) 34

18. The number of even numbers can be formed using all the digits 2, 3, 4, 5, 6 when repetition
is not allowed
(A) 5𝑃5 (B) 3 . 5𝑃5 (C) 3 . 4𝑃4 (D) 3. 3𝑃3 .
19. 1. The number of permutations of n objects, where p objects are of the same kind and rest
are all different= n!/p!
2. The number of permutations of n objects, where P1objects are of one kind, P2 are of
𝑛!
second kind, ..., Pk are of kth kind and the rest, if any, are of different is 𝑃 !𝑃 !….𝑃 !
1 2 𝑘
3. The number of permutations of n different objects taken r at a time, where repetition is
not allowed, is nPr.
Which of the above statements are correct?
(A) 1 and 2 only (B) 2 and 3 only (C) 1 and 3 only (D) All 1, 2 and 3
20. The number of ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs
of the same color are indistinguishable
9! 5! 6! 7!
(A) 4!3!2! (B) 4!3!2! (C) 4!3!2! (D) 4!3!2!.
21. The number of 4-digit numbers can be formed by using the digits 1 to 9 if repetition of
digits is not allowed
(A) 6561 (B)3024 (C)126 (D) 2024.
22. The number of 3-digit numbers can be formed by using the digits 1 to 9 if repetition of
digits is not allowed is
(A) 9P3 (B) 9C3 (C)3! (D) 93 .
23. The number of 4-digit numbers are there with no digit repeated is
(A) 10P4 (B) 9P4 (C)3 9P3 (D) 9 9P3
24. The number lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3, 4, 5, if
the repetition of the digits is not allowed is
(A) 900 (B) 100 (C)120 (D) 800
25. The number of words, with or without meaning, can be formed using all the letters of the
word AGAIN, using each letter exactly once is
(A) 120 (B)60 (C)30 (D)100.
26. The number of words, with or without meaning, can be formed using all the letters of the
word EQUATION, using each letter exactly once is
8!
(A) 8! (B)9! (C)2! (D) 2024.
27. The number of permutations of the letters of the word DAUGHTER is
8! 8!
(A) 2!2! (B) 2! (C) 8! (D) 7!
28. The number of permutations of the letters of the word EXAMINATION is
11! 11! 11!
(A) 2!2!2! (B) 2!2! (C) 11! (D) 6!
29. The number of words, with or without meaning can be made from the letters of the word
MONDAY, assuming that no letter is repeated, if 4 letters are used at a time
(A) 720 (B)60 (C)360 (D) 720.
30. Which of the following is incorrect
nPr.
(A) 𝑟! = nCr (B) n Cr = n C n-r (C) n Cr + nCr-1= n+1Cr (D) n C n = nPn
31. If n C a= n C b, then n =
(A) a (B)b (C)𝑎 − 𝑏 (D) a + b.
32. If C8 = C9, then C17 is
n n n

(A) 17! (B)1 (C)17 (D) 18.
33. The number of ways of choosing 4 cards from a pack of 52 playing cards .
4
(A) (13𝐶1 ) (B)4(13𝐶1 ) (C)52C4 (D) 13C4.

Question Bank: Department of School Education (Pre University ) 35

34. The number of ways of choosing 4 cards from a pack of 52 playing cards ,
if four cards are of the same suit is
4
(A) (13𝐶1 ) (B)4(13𝐶1 ) (C)52C4 (D) 4(13C4)
35. The number of ways of choosing 4 cards from a pack of 52 playing cards ,if four cards
are belong to four different suits is
4
(A) (13𝐶1 ) (B)4(13𝐶1 ) (C)52C4 (D) 13C4.
36. The number of ways of choosing 4 cards from a pack of 52 playing cards , if four cards
are face cards t is
(A) 12C4 (B) 52C12 (C)52C4 (D) 4.12C4.
37. The number of ways of choosing 4 cards from a pack of 52 playing cards , two are red
cards and two are black cards is
2
(A) (26𝐶2 ) (B)4(13𝐶1 ) (C) 226C4 (D) 26C4.
38. The number of ways of choosing 4 cards from a pack of 52 playing cards , if cards are of
the same color is
2
(A) (26𝐶2 ) (B)4(13𝐶1 ) (C) 2( 26C4 ) (D) 26C4.
39. If nC8 = nC2, then nC2 is
(A) 90 (B)10 (C)45 (D) 36.
40. The number of chords can be drawn through 21 points on a circle
(A) 110 (B) 210 (C)208 (D) 410.
41. The number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if
each selection consists of 3 balls of each color.
(A) 1000 (B) 3000 (C)2000 (D) 4000.
42. The number of diagonals which can be drawn in a polygon of n sides
n (n + 3) n (n − 1) n ( n − 2) n (n − 3)
(A) (B) (C) (D)
2 2 2 2
43. If 15C3r = 15C3+r then r =
3
(A) 2 (B) 3 (C)6 (D) 2.
44. There are n points in a plane of which p points are collinear. How many lines can be formed
from these points?
(A) nC2 – pC2- 1 (B) nC2 – pC2 (C) nC2 – pC2 + 1 (D) nC2+pC2- 1
45. The number of 4 letter words, with or without meaning, which can be formed out of the
letters of the word BANK, where the repetition of the letters is not allowed is --------( U )
46. The number of squares can be created on a chessboard if there are 4 horizontal lines and 4
vertical lines is ----
47. The value of 1! – 0! is -------
100 1 𝑥
48. If 10! = 8! + 9! ,then x is -------
12!
49. Compute (10!)(2!) is -------
𝑛!
50. Evaluate , when n = 5, r = 2. is -------
𝑟!(𝑛−𝑟)!
51. Match List I with List II
List I List II
a) 5𝐶0 i) 20
b) 𝑃2
5 ii) 10
c) 5𝐶2 iii) 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
Question Bank: Department of School Education (Pre University ) 36

52. Match List I with List II
List I List II
a) 15𝐶5 i) 15𝐶10
b) 15𝑃5 ii) 15𝐶5 + 15𝐶4
c) 16𝐶5 iii) 5! 15𝐶5
Choose the correct answer from the options given below:
A) a-i , b-iii, c-ii B) a-iii, b-ii, c-I C) a-ii, b-i, c-iii D) a-iii, b-i, c-ii

53. STATEMENT 1: If nC8 = nC2, then nC1 =10
STATEMENT 2: If n Ca = n Cb, then a + b =n.
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct explanation
of the STATEMENT 1
(C) STATEMENT 1 is true and STATEMENT 2 is false.
(D) STATEMENT 1 is false and STATEMENT 1 is true
54. STATEMENT 1: The number of permutations of n different objects taken r at a time,
where repetition is not allowed, is nPr..
STATEMENT 2: The number of 4-digit numbers can be formed by using the digits 1 to 9 if
repetition of digits is allowed is 94 .
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct explanation
of the STATEMENT 1
(C) STATEMENT 1 is true and STATEMENT 2 is false
(D) STATEMENT 1 is false and STATEMENT 1 is false.
55. Assertion (A): Ravi travels from Mumbai to Jammu in 7 different ways. But he is allowed
to return to Mumbai by any way except the one he used earlier. The number of ways can he
complete his journey is 42
Reason(R): There are 30 people in a group. If all shake hands with one another ,then number of
handshakes are 30𝐶2 .
(A) Both Assertion (A) and Reason (R) are true .
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
56. If nPr = 3024 and nCr = 126 then r =…….
57. There are 20 points in a plane, how many triangles can be formed by these points if 5 are
colinear?
A) 1130 B) 550 C) 1129 D) 1140.

Question Bank: Department of School Education (Pre University ) 37

58. In how many ways can we select 6 people out of 10, of which a particular person is not
included?
A) 10𝐶4 B) 9𝐶6 C) 9𝐶5 D) 10𝐶6
59. In how many different ways can five friends sit for a photograph of five chairs in a row?
A) 120 ways B) 24 ways C) 240 ways D)720 ways.
60. There are 20 chair patterns and ten table layouts available to a party planner. How many
different ways can she create a set of tables and chairs for the party.
A) 230 B) 200 C) 300 D) 400
61. The total number of ways of answering 5 objective questions, each questions having 5
choices is
A) 54 B) 45 C) 10 D) 9
62. The total number of ways 4 INDIENS and 3 ENGLISHMEN and 2 AMERICAN can be seated
in a row so that all Persons of the same nationality sit together.
A) 4! 3! 2! B) 9! C) 3(4! 3! 2! ) D) 3! 4! 3! 2!
63. The value of 6C4 + 5C3 + 5C2 is
A) 35 B) 210 C) 15 D) 20.
64. The total number of ways of answering 10 true, false questions, is
A) 102 B) 210 C) 10 D) 20
65. The number of ways of choosing 2 cards from a pack of 52 playing cards , if cards are
Ace is
A) 52𝐶2 B) 4𝐶2 C) 12𝐶2 D) 48𝐶2 .
Two marks questions
1. Given 4 flags of different colors, how many different signals can be generated, if a signal
requires the use of 2 flags one below the other? (U)
2. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
repetition of the digits is allowed? (U)
3. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
repetition of the digits is not allowed? (U)
4. How many 4-letter codes can be formed using the first 10 letters of the English alphabet,
if no letter can be repeated? (U)
5. How many 5-digit telephone numbers can be formed using the digits 0 to 9 if each
number starts with 67 and no digit appears more than once? (U)
6. A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are
there? (U)
7. Find the number of permutations of the letters of the word ALLAHABAD. (U)
8. How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits
is not allowed? (U)
9. In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs
of the same color are indistinguishable? (A)
10. How many 4-digit numbers are there with no digit repeated? (S)
11. How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is
repeated? (U)
12. From a committee of 8 persons, in how many ways can we choose a chairman and a vice
chairman assuming one person cannot hold more than one position? (U)
13. How many words, with or without meaning, can be formed using all the letters of the word
EQUATION, using each letter exactly once? (U)
14. If 𝑛𝐶9 = 𝑛𝐶8 , 𝑓𝑖𝑛𝑑 𝑛𝐶17 (K)
Question Bank: Department of School Education (Pre University ) 38

15. A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In
how many ways can this be done? (U)
16. If 𝐶8 = 𝐶2 , 𝑓𝑖𝑛𝑑 𝐶2 .
𝑛 𝑛 𝑛
(K)
17. How many chords can be drawn through 21 points on a circle? (A)
18. In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
19. In how many ways can one select a cricket team of eleven from 17 players in which only 5
players can bowl if each cricket team of 11 must include exactly 4 bowlers? (A)
20. A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black
and 3 red balls can be selected. (A)
21. In how many ways can a student choose a programme of 5 courses if 9 courses are
available and 2 specific courses are compulsory for every student? (S)
22. Find the number of different signals that can be generated by arranging at least 2 flags in
order (one below the other) on a vertical staff, if five different flags are available. (A)
1 1 𝑥
23. If 8! + 9! = 10!, find x. (K)
1 1 𝑥
24. If 6! + 7! = 8!, find x. (K)
25. How many numbers lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3,
4, 5 if the repetition of the digits is not allowed? (U)
Three marks questions
1. Find the value of n such that 𝑛𝑝5 = 42. 𝑛𝑝3, n4 (K)
2. Find r, if5. 𝑝𝑟 = 6. 𝑝𝑟−1.
4 5
(K)
𝑛𝑝 5
3. Find the value of n such that 𝑛−1𝑝4 = 3, (K)
4
4. Find the number of different 8-letter arrangements that can be made from the letters of the
word DAUGHTER so that (i) all vowels occur together (ii) all vowels do not occur together.
5. Find n, If 𝑛−1𝑝3 : 𝑛𝑝4 = 1: 9. (K)
6. Find r if 𝑝𝑟 = 2. 𝑝𝑟−1.
5 6
(K)
7. In how many distinct permutations of the letters of the word MISSISSIPPI the four I’s do
not come together? (U )
8. Determine the number of 5 card combinations out of a deck of 52 cards if each selection
of 5 cards has exactly one king.
9. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
(i) Repetition of the digits is allowed? (ii) Repetition of the digits is not allowed?
10. Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no
digit is repeated. How many of these will be even?
11. In how many ways can the letters of the word ASSASSINATION be arranged so that all the
S’s are together ?
12. How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are
divisible by 10 and no digit is repeated ?
13. If the different permutations of all the letter of the word EXAMINATION are listed as in a
dictionary, how many words are there in this list before the first word starting with E ?
14. How many words, with or without meaning, can be formed using all the letters of the word
EQUATION at a time so that the vowels and consonants occur together.
15. How many words, with or without meaning, each of 2 vowels and 3 consonants can be
formed from the letters of the word DAUGHTER ?
16. How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4,
2, 4? (A)
17. How many words, with or without meaning, each of 3 vowels and 2 consonants can be
formed from the letters of the word INVOLUTE? (A)

Question Bank: Department of School Education (Pre University ) 39

Five marks questions:
1. What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how
many of these
(i) four cards are of the same suit, (ii) four cards belong to four different suits. ( A )
2. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be
selected if the team
has (i) no girl ? (ii) at least one boy and one girl ? (iii) at least 3 girls ? (A)
3. A committee of 7 is to be formed from 9 boys and 4 girls. In how many ways can this be
done when the committee consists of: (i) exactly 3 girls? (ii) at least 3 girls ?
(iii) at most 3 girls ? (A)
4. In an examination, a question paper consists of 12 questions divided into two parts i.e.,
Part I and Part II, containing 5 and 7 questions, respectively. A student is required to
attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a
student select the questions? (A)
5. 4 cards are chosen from a pack of 52 playing cards In how many of these
i) are face cards ii) two are red cards and two are black cards iii) cards are of the same
color? (A)
6. How many words, with or without meaning can be made from the letters of the word
MONDAY, assuming that no letter is repeated, if. (i) 4 letters are used at a time, (ii) all
letters are used at a time, (iii) all letters are used but first letter is a vowel? (U)
7. Find the number of arrangements of the letters of the word INDEPENDENCE. In how
many of these arrangements,(i) do the words start with P (ii) do all the vowels always
occur together
(iii) do the vowels never occur together (iv) do the words begin with I and end in P?
8. A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how
many ways can this be done? How many of these committees would consist of 1 man and
2 women?
9. In how many ways can the letters of the word PERMUTATIONS be arranged if the
i) words start with P and end with S, (ii) vowels are all together,
iii) there are always 4 letters between P and S?
******

Question Bank: Department of School Education (Pre University ) 40

CHAPTER-7
BINOMIAL THEOREM
MCQ / FB
1. The number of non-zero terms in the expansion of (𝑎 + 𝑏)𝑛 is
(a)n-1 (b) n+1 (c) n (d) n+2.
2. The number of non-zero terms in the expansion of (𝑎 + 𝑏) is
6

(a)5 (b) 6 (c) 7 (d) 8
3. The coefficients Cr occurring in the binomial theorem are known as
n

(a) coefficients (b) binomial coefficients (c) Combination (d) binomial term .
4. In the expansion of (a+b)n, the sum of the indices of a and b is
(a)n-1 (b) n+1 (c) n (d) n+2.
5. In the expansion of (a - b) stated as
n,
𝑛−𝑘 𝑛−𝑘
(a) ∑𝑛𝑘=0 𝑎𝑛−𝑘 𝑏 𝑘 (b)- ∑𝑛𝑘=0 𝑎𝑛−𝑘 𝑏 𝑘 (c) ∑𝑛𝑘=0(−𝑎)𝑎 𝑏 𝑘 (d) ∑𝑛𝑘=0 𝑎 𝑎 (−𝑏)𝑘
6. The coefficient of 𝑥 𝑘 𝑖𝑛 𝑡ℎ𝑒 𝑒𝑥pansion of (𝑥 + 1)𝑛
(a)𝑛𝐶𝑘 (b) 𝑛 − 1𝐶𝑘 (c) 𝑛𝐶𝑘−1 (d) 𝑛 − 1𝐶𝑘−1
7. The sum of all the coefficients in the binomial expansion of (𝑥 + 1)𝑛
(a) 2𝑛 (b) 2𝑛−1 (c) n (d) 2𝑛+1
8. The sum of all the coefficients in the binomial expansion of (𝑥 − 1)𝑛
(a) 2𝑛 (b) 2𝑛−1 (c) n (d) 0.
9. The sum of all the binomial coefficients in expansion of (𝑎 + 𝑏)𝑛
(a) 2𝑛 (b) 2𝑛−1 (c) n (d) 2𝑛+1
10. In the expansion of ( 1 + x ) the sum of coefficients of odd powers of x is
n

(a)0 (b) 2n − 1 (c) 2 n (d) 2n−1
11. In the expansion of ( 1 + x )n the sum of coefficients of even powers of x is
(a)0 (b) 2n − 1 (c) 2 n (d) 2n−1
12. The value of 10𝐶1 + 10𝐶2 + 10𝐶3 + ⋯ … … … . . +10𝐶10 is
(a) 1024 (b) 512 (c) 1023 (d) 256.
13. The value of 10𝐶0 − 10𝐶1 + 10𝐶2 − 10𝐶3 … … … … . +10𝐶10 is
(a) 1024 (b) 512 (c) 0 (d) 256
14. The value of 10𝐶1 + 10𝐶3 + 10𝐶5 … … … … . +10𝐶9 is
(a) 1024 (b) 512 (c) 1023 (d) 256
𝑛 𝑟
15. 𝑘=0 3 𝑛𝐶𝑟 =
∑
(a) 2𝑛 (b) 3𝑛 (c) 4𝑛 (d) n
16. The value of 𝑛𝐶0 + 2. 𝑛𝐶1 + 4. 𝑛𝐶2 + 8. 𝑛𝐶3 … … … … . +2𝑛 . 𝑛𝐶𝑛 is
(a) 2𝑛 (b) 3𝑛 (c) 4𝑛 (d) n
17. The value of 1 + 3. 20𝐶1 + 9. 20𝐶2 + 27. 20𝐶3 … … … … . +320 is
(a) 220 (b) 320 (c) 420 (d) 202
18. The number of terms in the expansion of (2𝑥 − 3𝑦)6 is = ⋯ … … … ..
19. The sum of the binomial coefficients of the first and the last term of (a + b)n = ……
20. The real part of (1 + i)4 is =…….
21. The number of terms in (1 + 2x +x2)60 is ........
22. The number of terms in (1 + x +x2)5 is ........
23. 6n– 5n always leaves remainder when divided by 25 is ......

Question Bank: Department of School Education (Pre University ) 41

24. STATEMENT 1: The largest coefficient in the expansion of (1 + 𝑥)8 is 8𝐶5
STATEMENT 2: The greatest coefficient will always occur in the middle term.
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct explanation
of the STATEMENT 1
(C) STATEMENT 1 is true and STATEMENT 2 is false.
(D) STATEMENT 1 is false and STATEMENT 1 is true
25. Assertion (A): If the coefficient of 3rd term in the expansion of (1 + 𝑥)𝑛 is 10,then n = 5
Reason(R): The coefficient of rth term in the expansion of (1 + 𝑥)𝑛 is 𝒏𝑪𝒓−𝟏 .
(A) Both Assertion (A) and Reason (R) are true .
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
26.1. The total number of terms in the expansion
(1 + 𝑥)10 + (1 − 𝑥)10 after simplification will be 6.
2. The total number of terms in the expansion (1 + 𝑥)10 is 11.
3. (r+1)th term in the expansion (1 + 𝑥)𝑛 is x 𝑟 𝑛𝐶𝑟 .
𝑊ℎ𝑖𝑐ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡𝑠 𝑎𝑟𝑒 𝑐𝑜𝑟𝑟𝑒𝑐𝑡?
(𝐴)1 𝑎𝑛𝑑 2 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝐵) 1 𝑎𝑛𝑑 3 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝐶) 3 𝑜𝑛𝑙𝑦 (𝐷) 𝐴𝑙𝑙 1, 2 𝑎𝑛𝑑 3.
27. STATEMENT 1: (1.1) 10000
is greater than 1000
STATEMENT 2: (1.1)10000 = (1 + 0.1)10000
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of the
STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct explanation
of the STATEMENT 1
(C) STATEMENT 1 is true and STATEMENT 2 is false.
(D) STATEMENT 1 is false and STATEMENT 1 is true
28. In the expansion of (1 + 𝑎𝑥)𝑛 , 𝑛 ∈ 𝑁 the coefficient of x and 𝑥 2 are 8 and 24 respectively,
then
(a) a = 2 , n = 4 (b) a = 4 , n = 2 (c) a = 2 , n = 3 (d) a = 3, n = 4.
29. Match List I with List II

List I List II
a) coefficient of 𝑥 2 in (𝟏 + 𝒙)𝟔 i) −15
b) coefficient of 𝑥 2 in (𝟏 − 𝒙)𝟔 ii)15
c)The number of terms in (1 + 𝑥)6 – (1 − 𝑥)6 iii) 3

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-iii, b-ii, c-I C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.
30.If (𝑎 + 𝑏) = 𝑎 + 5𝑎 𝑏 + 10𝑎 𝑏 + 10𝑎 𝑏 + 5𝑎𝑏 + 𝑏 5
5 5 4 3 2 2 3 4

𝑎𝑛𝑑 (𝑎 + 𝑏)6 = 𝑎6 + 6𝑎5 𝑏 + 𝑙𝑎4 𝑏 2 + 𝑚𝑎3 𝑏 3 + 𝑛𝑎2 𝑏 4 + 6𝑎𝑏 5 + 𝑏 6 , l ,m and n are respectively
(a) 15,15,15 (b) 20,15,15 (c) 15,20,15 (d) 20,15,20.

Question Bank: Department of School Education (Pre University ) 42

TWO /THREE MARKS QUESTIONS:
1. Using Binomial Theorem evaluate 𝑖)(96)3 𝑖𝑖) (98)5 𝑖𝑖𝑖)(99)5 𝑖𝑣)(102)5 𝑣)(101)4
5
2 x 𝑥 1 5 1 5. 3
2. Expand 𝑖)  −  𝑖𝑖) (3 + 𝑥) 𝑖𝑖𝑖) (2𝑥 − 3)6 iv) (1 − 2𝑥)5. , v) (𝑥 + 𝑥) 𝑣𝑖) (𝑥 2 + 𝑥) 4
 x 2
3. Find an approximation of (0.99)5 using the first three terms of its expansion. (K)
4. Show that 9𝑛+1 − 8𝑛 − 9 is divisible by 64, whenever n is a positive integer. (A)
5. Using Binomial theorem, prove that 6𝑛 − 5𝑛 always leaves remainder 1 when divided by
25. (S)
6. Find 𝑎𝑖𝑓𝑡ℎ𝑒17 𝑎𝑛𝑑 18 𝑡𝑒𝑟𝑚𝑠𝑜𝑓𝑡ℎ𝑒𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛(2 + 𝑎) 𝑎𝑟𝑒𝑒𝑞𝑢𝑎𝑙.
𝑡ℎ 𝑡ℎ 50
(U)
7. Find a positive value of m for which the coefficient of 𝑥 2 in the expansion (1 + 𝑥)𝑚 is 6(U)
8. In the expansion of (1 + 𝑎)𝑚+𝑛 , prove that coefficients of 𝑎𝑚 𝑎𝑛𝑑 𝑎𝑛 are equal. (A)
9. Which is larger (1.01)100000 or 10,000? (K)
𝑛
10. Prove that∑𝑟=0 3 . 𝑛𝐶𝑟 = 4 .
𝑟 𝑛
(K)
4 4
11. Find the value of (𝑎2 + √𝑎2 − 1) + (𝑎2 − √𝑎2 − 1) . (A)
4 4
12. Find (𝑎 + 𝑏)4 − (𝑎 − 𝑏)4 . 𝐻𝑒𝑛𝑐𝑒, 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒(√3 + √2) − (√3 − √2) . (S)
13. Find (𝒙 + 𝟏)𝟔 + (𝒙 − 𝟏)𝟔.
6
14. Find (√2 + 1)6 + (√2 − 1) .
𝑥 2 4
15. Expand using Binomial Theorem (1 + 2 − 𝑥) , 𝑥 ≠ 0.
16. If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a
positive integer.

Question Bank: Department of School Education (Pre University ) 43

CHAPTER-8
SEQUENCES AND SERIES
MCQ /FB
1. The 20th term of the sequence defined by 𝑎𝑛 = (n– 1) (2 –n) (3 +n) is
(A) 7856 (B) 7866 (C) -7866 (D)-7856.
2. The 4 term of the sequence defined by 𝑎1 = 1, 𝑎𝑛 = 𝑎𝑛−1 + 2 , 𝑛 ≥ 2 is
th

(A) 3 (B) 5 (C) 7 (D) 9.
𝑛−3
3. The first three terms of the sequences defined by 𝑎𝑛 = 4 is
1 −1 1 1 1 −1 1 −1
(A) − 2 , 4 ,1 (B) − 2 , 4 ,0 (C) − 2 , 4 ,0 (D) − 2 , 4 , −1.
4. The 9 term of the sequence defined by 𝑎𝑛 = (– 1) 𝑛 is
th 𝑛−1 3

(A) 36 (B) 33 (C) -33 (D)-36
5. The 5th term of the sequence defined by 𝑎1 = 𝑎2 = 2, 𝑎𝑛 = 𝑎𝑛−1 − 1, 𝑛 > 2 is
(A) -1 (B) 0 (C) 1 (D) -2.
6. The 4 term of the sequence defined by 𝑎1 = 𝑎2 = 1, 𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2 , 𝑛 > 2 is
th

(A) 2 (B) 3 (C) 5 (D) 8.
𝑎𝑛+1
7. The value of 𝑎 , if n = 5 of the sequence defined by 𝑎1 = 𝑎2 = 1, 𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2 , 𝑛 > 2
𝑛
is
5 8 13 5
(A) 8 (B) 5 (C) 8 (D) 3.
8. The sum of the first five terms of the series of the sequence defined by
𝑎1 = −1, 𝑎𝑛 = , 𝑎𝑛−1 /𝑛 , 𝑛 ≥ 2 is
206 206 205 205
(A)− 120 (B) 120 (C) − 120 (D) -120 .
𝑛(𝑛−2)
9. The 20th term of the sequences defined by 𝑎𝑛 = is
𝑛+3
180 360 380 400
(A) 23 (B) 23 (C) (D) .
3 3
𝑛2
10. The 7th term of the sequences defined by 𝑎𝑛 = 2𝑛 is
49 7 14 49
(A) 64 (B) 128 (C) 128 (D) 128.
11. If 3,n,8 are the three consecutive terms of the Fibonacci sequence, then the value of n is
(A) 1 (B) 2 (C) 4 (D) 5.
12. If a = the first term, r = the common ratio, l = the last term, n = the numbers of
terms, and Sn = the sum of first n terms of a GP. If r =1,then Sn =
𝑎(1−𝑟 𝑛 ) 𝑎(𝑟 𝑛 −1) 𝑎
(A) 𝑎𝑛 (B) (C) (D) 1−𝑟
1−𝑟 𝑟−1
13. If 0.01, 0.0001, 0.000001,......... are in GP, then its common ratio is
1 1
(A) 10 (B) 10 (C) 100 (D) 100.
14. The 10th terms of the sequence 5, 25,125,… is
(A) 59 (B) 510 (C) 58 (D) 511 .
15. The 𝑛𝑡ℎ term of the G.P, 2,8,32, ... … is 131072,then n is
(A) 9 (B) 8 (C) 14 (D) 16.
16. In a G.P., the 3 𝑡erm is 24 and the 6th term is 192.then common ratio is
𝑟𝑑

(A) 2 (B) 6 (C) 4 (D) 8.
17. The first term of a G.P. whose 6th term is 192 and the common ratio is 2
(A) 2 (B) 6 (C) 4 (D) 8.
18. A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. then the
number of his ancestors during the ten generations preceding his own is
(A) 1023 (B) 1024 (C) 2048 (D) 2046.

Question Bank: Department of School Education (Pre University ) 44

19. If 1, G1,G2,G3 ,256 are in G.P, then G2 is
(A) 2 (B) 4 (C) 16 (D) 8.
20. If A and G be A.M. and G.M. of two given positive real numbers a and b, then
(A) 𝐴 ≤ 𝐺 (B) 𝐴 < 𝐺 (C) 𝐴 > 𝐺 (D) A ≥ 𝐺.
2 7
21. The values of x, the numbers − 7 , 𝑥 − 2 are in G P.
(A) 1 (B) −1 (C) 4 (D) 7.
22. The 𝑛 term of the sequences 2, 2√2, 4……is 128 ,then n is
𝑡ℎ

(A) 7 (B) 12 (C) 13 (D) 14.
23. The sum to n term of the G.P., 3, 3 , 3 , ... … is 120,then n is
2 3

(A) 2 (B) 3 (C) 4 (D) 5.
24. In the G.P. 2, 2√2, 4... if an = 128, then the value of n is
(A) 11 (B) 12 (C) 13 (D) 14
25. The sum to n term of the progressions 𝑥 3 , 𝑥 5 , 𝑥 7 , ... …(if x≠1 )
𝑥 3 (1−𝑥 2𝑛 ) 𝑥 3 (1−𝑥 2𝑛 ) 𝑥3
(A) 𝑛𝑥 3 (B) (C) (D) 1−𝑥 2
1−𝑥 3 1−𝑥 2
26. Two numbers between 3 and 81 so that the resulting sequence is G.P.
(A) 6 𝑎𝑛𝑑 36 (B) 9 𝑎𝑛𝑑 18 (C) 9 𝑎𝑛𝑑 27 (D) 27 𝑎𝑛𝑑 54.
27. A.M and G.M. of roots of a quadratic equation are 8 and 5, respectively, then the
quadratic equation is
(A) 𝑥 2 + 13𝑥 + 40 = 0 (B) 𝑥 2 − 16𝑥 + 25 = 0
(C) 𝑥 2 + 16𝑥 + 25 = 0 (D) 𝑥 2 − 16𝑥 − 25 = 0.
28. The 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + ... + n terms.
(A) 840 (B) 1680 (C) 1600 (D) 420
5
29. The value of ∑𝑘=1(3 + 2 ) is
𝑘

(A) 77 (B) 62 (C) 124 (D) 72.
30. The 3 term of the sequence an =(−1) (𝑛 − 1)
rd 𝑛 3
(K)
31. If 3,n,8 are the three consecutive terms of the Fibonacci sequence, then the value of n is
32. The fifth term of the G.P. 2, 6,18, is -------- (K)
33. The common ratio of the G.P. 2, 2√3, 6, is -------- (K)
34. The arithmetic mean of the numbers 6 and 10 is -------- (K)
35. The geometric mean of the numbers 4 and 9 is -------- (K)
36. The 4th term of the sequence an = 4n-3 is -------------
𝑛2
37. The 4th term of the sequence an = 𝑛 is -------------
2
38. SATEMENT 1: The 3rd term of a GP is 9, then the product of its first 5 terms is 310 .
𝑎 𝑎
SATEMENT 2: First 5 terms in GP are 𝑟 2 , 𝑟 , 𝑎 , 𝑎𝑟 , 𝑎𝑟 2 ,then their product is 𝑎5 .
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct
explanation of the STATEMENT 1
(C) STATEMENT 1is true and STATEMENT 2 is false.
(D) STATEMENT 1is false and STATEMENT 2 is false.
39. Assertion (A): Sequence 1, 1,2,3,4, is a Fibonacci sequence.
Reason(R): The sequence defined by a1 = a2 = 1, an = an-1 + an-2 𝑛 > 2 is a Fibonacci
Sequence.
(A) Both Assertion (A) and Reason (R) are true.
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
Question Bank: Department of School Education (Pre University ) 45

40. 1. If the first term of a geometric sequence is 7 and the common ratio is 2, then its 5th
term is 224
2. The nth term of a geometric progression is a𝑟 𝑛−1
𝑎
3. In a GP common ratio r = 𝑎 𝑛
𝑛−1
𝑊ℎ𝑖𝑐ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡𝑠 𝑎𝑟𝑒 𝑐𝑜𝑟𝑟𝑒𝑐𝑡?
(A) 1 𝑎𝑛𝑑 2 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝐵) 1 𝑎𝑛𝑑 3 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝐶) 3 𝑜𝑛𝑙𝑦 (𝐷) 𝐴𝑙𝑙 1, 2 𝑎𝑛𝑑 3.
41. SATEMENT 1: Let a1 , a2 ,a1 , a3 ,……be a GP and a i>0 and a1 + a2 =4 and a3 + a4 = 16, then
r =2.
SATEMENT 2: In GP a3 + a4 = 𝑟 2 (a1 + a2 )
(A) Both STATEMENT 1 and 2 are true
(B) Both STATEMENT 1 and 2 are false.
(C) STATEMENT 1is true and STATEMENT 2 is false.
(D) STATEMENT 1is false and STATEMENT 1is true.
42. Match List I with List II
List I List II
𝑎
a) sum to n term of a GP, r=1 i)1−𝑟
b) sum to infinite term of the GP 𝑎(1−𝑟 𝑛 )
ii) 1−𝑟
c)sum to n term of a GP, r≠1 iii)𝑎𝑛

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.
43. In the G.P 3, 6, 12 ,... …….. Match List I with List II
List I List II
a) common ratio i)45
b) 6th term of the G.P ii)2
c)sum to 4term of the G.P iii) 96

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.
2 3
44. SATEMENT 1: If − 3 , 𝑥, − 2 are in GP ,then the value of x is ±1.
SATEMENT 2: If 𝑎, 𝑏, 𝑐 are in GP, then b is geometric mean of a and b.
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct
explanation of the STATEMENT 1
(C) STATEMENT 1is true and STATEMENT 2 is false.
(D) STATEMENT 1is false and STATEMENT 2 is false.
45. A sequence is called ___________________ if an+1 = an * r.
A) Arithmetic progression B) Geometric Progression
C) Fibonacci sequence D) Non3e of these.
1 1
46. The sum of series 1+ + + … … … . 𝑢𝑝 𝑡𝑜 6 𝑡𝑒𝑟𝑚𝑠.
2 4
63 32 26 53
𝐴) 32 𝐵) 63 𝐶) 53 𝐷) 26

Question Bank: Department of School Education (Pre University ) 46

47. The age of three children’s 𝛼, 𝛽, 𝛾 are in a G.P and the sum of their age is 7 and the
product is 8. What is the difference between 𝛼 𝑎𝑛𝑑 𝛽 age?
1 1
(A) 2 (B) 2 (C) 4 (D) 2 𝑜𝑟 2
48. The ratio between 7th term to that of 5rd term for the GP with the common ratio of 2 is
(A) 2 ∶ 1 (B) 4 ∶ 1 (C) 1 ∶ 4 (D) 1 ∶ 2
49. The 5th term of a geometric progression is 81 and the common ratio is 3. What is the
first term?
(A) 3 (B) 9 (C) 27 (D) 1.
50. If the first term and second term of a geometric sequence is 5 and 15 respectively, what
is the 5th term
(A) 81 (B) 1215 (C) 135 (D) 405
51. If the 3rd term of a geometric progression is 18 and the 6th term is 486, then the
common ratio is
A) 2 B) 3 C) 4 D) 5
52. If the sum of the first n terms of a sequence is given by Sn= n2+2n, then the common
ratio is
A) 2 B) 3 C) 4 D) 5
53. Which of the following is the explicit formula for the geometric progression
5,10,20,40,….......
A) 2× 5𝑛 B) 2× 5𝑛−1 C) 5× 2𝑛 D) 5× 2𝑛−1
54. A girl puts 1 grain of rice in the first square of an 8 by 8 chess board. In the subsequent
square, she puts twice that of the previous square, and she continues until she fills all
the squares. How many total grains does she need?
A) 264 B) 264−1 C) 264 − 1 D) 64!
55. If a1 , a2 ,a3 ,…… … .. an are in AP and also GP, then
A) common difference a = 1 and common ratio r = 1
B) common difference a = 0 and common ratio r = 1
C) common difference a = 1 and common ratio r = 0
D) common difference a = 0 and common ratio r = 0
Two / Three mark questions
1. Find the first three terms of the sequence an =(−1)𝑛−1 5𝑛+1 (K)
𝑛(𝑛2 +5)
2. Find the first three terms of the sequence an = 4 (K)
3. Find the sum of the first three terms of the sequence defined by a1= 3, an=3an-1+2
(when n>1). (K)
4. In a G.P., the third term is 24 and the 6 term is 192. Find the 10 term.
th th (U)
5. Find the sum of first n terms and the sum of first 5 terms of the geometric series
2 4
1+ 3 +9 + … ….
3 3 3069
6. In the G.P. 3,2 , 4,… if Sn= 512 , find the value of n. (U)
7. In the G.P. 3, 32, 33 ,… if Sn=120, find the value of n. (U)
8. In the G.P. 2, 2√2, 4... if an = 128, find the value of n. (U)
9. In the G.P. √3, 3,3√3... if an = 729, find the value of n. (U)
1 1 1 1
10. In the G.P. 3 , 9 , 27 , … 𝑖𝑓𝑎𝑛 = 729 , find the value of n. (U)
−2 −7
11. Find the values of x, if the numbers , 𝑥, are in G.P. Also find the common ratios (U)
7 2
13
12. The sum of first three terms of a G.P. is 12 and their product is -1. Find the common ratios.
39
13. The sum of first three terms of a G.P. is 10 and their product is 1. Find the common ratios

Question Bank: Department of School Education (Pre University ) 47

14. Find the 3 numbers to be inserted between 1 and 256 such that the resulting sequence is
an G.P. (U)
15. Find the 3 numbers to be inserted between 1 and 243 such that the resulting sequence is
an G.P. (U)
16. Find the 12 term of the G.P. whose 8 term is 192 and common ratio is 2. (U)
th th

17. Find the 3 numbers to be inserted between 1 and 256 such that the resulting sequence is
an G.P.
18. The 5th, 8th and 11th terms of a G.P. are p, q and s respectively. Show that q2=ps. (A)
19. The 4th term of a G.P. is square of its second term, and the first term is -3. Determine its
7th term. (U)
20. The sum of first three terms of a G.P. is 16 and the sum of next three terms is 128.
Determine the first term and the common ratio. (A)
21. Given a G.P with a=729 and 7 term is 64, Determine S7.
th (U)
22. If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in
G.P. (A)
23. Show that the products of the corresponding terms of the sequence, ar, ar ,...arn-1 and A,
2

AR, AR2, ... ARn-1 form a G.P. and find the common ratio. (U)
24. Find the sum of the products of the corresponding terms of the sequence 2,4,8,16,32 and
1
128, 32, 8, 2, 2. (U)
25. If the first and n term of a G.P. are a and b, respectively, and if P is the product of first n
th

terms, Prove that P2 = (ab)n. (S)
26. If a, b, c and d are in G.P. show that (𝑎2 + 𝑏 2 + 𝑐 2 )(𝑏 2 + 𝑐 2 + 𝑑 2 ) =(𝑎𝑏 + 𝑏𝑐 + 𝑐𝑑)2. (S)
27. The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria
present in the culture originally, how many bacteria will be present at the end of 8th hour?
28. What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual
interest rate of 10% compounded annually? (A)
29. The sum of some terms of G.P is 315 whose first term and the common ratio are 5 and 2
respectively. Find the last term and the number of terms. (U)
30. The first term of a G.P is 1. The sum of third and fifth term is 90.Find the common ratio of
G.P. (U)
31. A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the
amount in 15th year since he deposited the amount and also calculate the total amount
after 20 years. (A)
32. A manufacturer reckons that the value of a machine, which costs him Rs 15625, will
depreciate each year by 20%. Find the estimated value at the end of 5 years. (A)
33. Find four numbers forming a G.P. in which the third term is greater than the first term by
9, and the second term is greater than the fourth by 18.
34. Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n+1)th to
1
(2n)th term is 𝑟 𝑛 .
35. If f is a function satisfying f(x + y) = f(x).f(y) for all x, y ∈ N such that f(1)=3 and ∑𝑛𝑥=1 𝑓(𝑥) =
120, find the value of n.

Four mark questions
1. If A and G respectively represents the Arithmetic mean and Geometric mean of two
positive real numbers , then show that A≥G. (S)
2. If A.M. and G.M. of two positive numbers ‘a’ and ‘b’ are 10 and 8 , respectively, find the
numbers. (A)
3. In a G.P. if sum of the first two terms is -4 and the fifth term is 4 times the third term.
Find the first terms of the G.P.

Question Bank: Department of School Education (Pre University ) 48

4. Find four numbers forming a G.P. in which the third term is greater than the first term by
9, and the second term is greater than the fourth by 18. (S)
5. If m , p and q terms of a G.P are x, y and z respectively. Prove that xp-q yq-m zm-p =1 (S)
th th th

6. If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the
quadratic equation. (S)
7. Find the sum of n terms of the sequence 7,77,777,7777,….to n terms.
8. Find the sum of the series up to n terms 5 + 55 +555 + … …
9. Find the sum of the series up to n terms 0 .6 +0. 66 +0. 666+…
10. If A and G be A.M. and G.M., respectively between two positive numbers,
11. prove that the numbers are A ±√(𝐴 + 𝐺 )(𝐴 − 𝐺 ) .

Question Bank: Department of School Education (Pre University ) 49

CHAPTER-9
STRAIGHT LINES
MCQ /FB
1. Which of the following is incorrect
(A) The slope of a line whose inclination is 90° is not defined.
(B) The slope of x- axis is zero.
(C) The slope of y- axis is not defined.
𝑥 𝑥
(D) The slope m of the line through the points (x1, y1) and (x2, y2) is given by m= 𝑦2−𝑦1 .
2− 1
2. Which of the following is incorrect?
(A) Two non vertical lines l1 and l2 are parallel if and only if their slopes are equal..
(B) Two non-vertical lines are perpendicular to each other if and only if their slopes are
negative reciprocals of each other,
(C) Three points A, B, C are collinear, if and only if slope of AB = slope of BC.
(D) Equation of the horizontal line having distance a from the x-axis is either
x = a or x = – a.
3. Which of the following correct, if a line its slope is negative?
(A) 𝜃 is acute angle (B) 𝜃 is an obtuse angle
(C) Either the line is x-axis or it parallel to x-axis
(D) Either the line is y-axis or it parallel to y-axis
4. The line if its slope is zero is
(A) Either the line is y-axis or it parallel to y-axis (B) 𝜃 is an obtuse angle
(C) Either the line is x-axis or it parallel to x-axis (D) 𝜃 is an acute angle
5. The slope of the line through the points (3, -2) and (-1, 4) is
2 3 3
(A) 3 (B) 2 (C) − 2 (D) not defined.
6. The slope of the line through the points (3, -2) and (7, -2) is
(A) 0 (B) 4 (C) −1 (D) not defined.
7. The slope of the line through the points (3, -2) and (3, 4) is
(A) 0 (B) 4 (C) −1 (D) not defined.
8. The slope of the line making inclination is 60° with positive direction of x-axis is
2 1 √3
(A) (B) √3 (C) (D) 2 .
√3 √3
9. Line through the points (–2, 6) and (4, 8) is perpendicular to the line through the points
(8, 12) and (x, 24),then the value of x.
1
(A) 0 (B) 4 (C) −1 (D) 4.
10. The slope of the line which makes angle 300 with positive direction of y axis measured
anticlockwise
1
(A)√3 (B)−√3 (C) 1 (D) .
√3
11. The value of x for which the points (x, – 1), (2,1) and (4, 5) are collinear
(A)2 (B)−2 (C) 1 (D) −1.
12. The angle between the x-axis and the line joining the points (3,–1) and (4,–2).
(A) 450 (B) 1350 (C) -450 (D) -1350
13. If three points (h, 0), (a, b) and (0, k) lie on a line, then
𝑎 𝑏 𝑎 𝑏 ℎ 𝑘 ℎ 𝑘
(A) ℎ + 𝑘 = 1 (B) ) ℎ − 𝑘 = 1 (C) ) 𝑎 + 𝑏 = 1 (D) 𝑎 − 𝑏 = 1.
14. Equation of the line parallel to x-axis and passing through (-2,3) is
(A) 𝑥 = 3 (B) 𝑥 = −2 (C) 𝑦 = 2 (D) 𝑦 = 3.
15. Equation of the line parallel to y-axis and passing through (-2,3) is
(A) 𝑥 = 3 (B) 𝑥 = −2 (C) 𝑦 = 2 (D) 𝑦 = 3.

Question Bank: Department of School Education (Pre University ) 50

16. Equation of the line passing through (-2,3) with slope -4 is
(A) 4𝑥 + 𝑦 − 11 = 0 (B) 4𝑥 + 𝑦 + 5 = 0 (C) 𝑥 + 4𝑦 − 11 = 0 (D) +4𝑦 + 5 = 0 .
17. Equation of the line through the points (1,-1) and (3,5) is
(A) 𝑦 = 3𝑥 + 4 (B) 𝑦 = 3𝑥 − 4 (C) 3𝑥 + 𝑦 − 4 = 0 (D) 3𝑥 + 𝑦 + 4 = 0 .
18. Equation of the passing through (0, 0) and slope m is
(A) y =mx +c (B) x=my +c (C) y=mx (D) x=my.
19. Equation of the lines for which tan 𝜃 = 1/2 where θ is the inclination of the line and y-
intercept is 4 is
(A) x-2y+8=0 (B) 𝑦 = 𝑥 + 8 (C) x+2y+8= 0 (D) −𝑥 + 2𝑦 + 4 = 0 .
20. 20.Equation of the line passing through the point (2,3)with inclined with x-axis at an angle
of 450 𝑖𝑠
(A) x-y+5=0 (B) x-y-1=0 (C) x+y+8= 0 (D) 𝑥 − 𝑦 + 1 = 0 .
21. The Equations for x and y axes are
(A) 𝑥 = 1 , 𝑦 = 1 (B) 𝑦 = 1 (C) 𝑥 = 0 𝑎𝑛𝑑 𝑦 = 0 (D) 𝑥 = 1 𝑎𝑛𝑛 𝑦 = 0.
22. The equation of the line passing through the point (2, 3) with slope 2 is
(A) 2x+y-1=0 (B) 2x-y+1=0 (C) 2x-y-1=0 (D) 2x+y+1=0
23. Equation of the line intersecting the x-axis at a distance of 3 units to the left of origin
with slope –2 is
(A) 2x-2y-6=0 (B) 𝑦 = 2𝑥 + 6 (C) 2x-y+6= 0 (D) 2𝑥 + 𝑦 + 6 = 0 .
3
24. A line cutting off intercept -3 from the y-axis and slope of tangent is , its equation
5
(A) 5y - 3x+15=0 (B) 3y-5x +15=0 (C) 5y-3x-15 =0 (D) None of these
25. The values of k for which the line (k–3)x–(4–𝑘 )y+𝑘 –7k+ 6 = 0 is parallel to the y-axis
2 2

(A) 4 (B) ±2 (C) 3 (D) 6 𝑜𝑟 1.
26. Perpendicular distance from the origin to line √3𝑥 + 𝑦 + 𝑘 = 0 is 5 units , then k is
(A)6 (B)10 (C)5 (D)-10
27. The vertices of ∆PQR are P (2, 1), Q (–2, 3) and R (4, 5), then equation of the median
through the vertex R.
(A) 3 x +4 y-8=0 (B) 3𝑥 + 4𝑦 + 8 = 0 (C) 3x - 4y +8= 0 (D) 3𝑥 − 4𝑦 − 8 = 0 .
28. The slope of the line ax +by +c =0
a −a −c c
(A) (B) (C) (D)
b b b b
29. The slope of the line 3x– 4y+ 10 = 0 is
3 3 10 5
(A) − (B) (C) - (D) .
4 4 3 2
30. The x-intercept of the line 3x– 4y+ 10 = 0 is
(A) 5/2 (B) −10/3 (C) 3/4 (D) 2 .
31. The y-intercept of the line 3x– 4y+ 10 = 0 is
(A) 5/2 (B) −10/3 (C) 3/4 (D) 2 .
32. Slope of a line which cuts off intercepts of equal lengths on axes is
(A) -1 (B) 0 (C) 2 (D) 3
33. The inclination of the line x-y+3=0 with the positive direction of x-axis is
(A) 450 (B) 1350 (C) -450 (D) -1350
34. The values of k for which the line (k–3)x–(4–𝑘 2 )y+𝑘 2 –7k+ 6 = 0 is parallel to the x-axis,
(A) 4 (B) ±2 (C) 3 (D) 6 𝑜𝑟 1.
35. The values of k for which the line (k–3)x–(4–𝑘 )y+𝑘 –7k+ 6 = 0 is passing through the
2 2

origin.
(A) 4 (B) ±2 (C) 3 (D) 6 𝑜𝑟 1.

Question Bank: Department of School Education (Pre University ) 51

36. The two lines ax+ by =c and 𝑎′ 𝑥 + 𝑏 ′ 𝑦 = 𝑐 ′ are perpendicular if
𝑎 𝑏 𝑐
(A) aa'+bb'=0 (B) ab'=ba' (C) ab + a'b'=0 (D) a′ = b′ = 𝑐 ′
37. The two lines ax+ by =c and 𝑎′ 𝑥 + 𝑏 ′ 𝑦 = 𝑐 ′ are parallel if
𝑎 𝑏 𝑎 𝑏 𝑐
(A) a′ = b′ B) aa'=bb' (C) aa'+bb'=0 (D) a′ = b′ = 𝑐 ′
38. The perpendicular distance of the point P (1,-3) from the line 2y-3x=4 is
7
(A) 13 (B) 13 √13 (C) √13 (D) None of these
39. The distance of the point (–1, 1) from the line 12(x+ 6) = 5(y– 2).
55 60
(A) 13 (B) 5 (C) 13 (D) 13.
40. The distance between the lines 3x-4y +7=0 and 3x-4y +5=0 is
2 5 12 12
(A) (B) (C) (D)
5 2 25 5
41. The equation of the straight line passing through the point (3,2) and perpendicular to the
line 𝑦 = 𝑥 is
(A) x-y=5 (B) x+y=5 (C) x+y=1 (D) x-y = 1.
42. The distance between the lines y= mx +c1and y= mx+c2 is
c1 − c2 c −c c2 − c1
(A) (B) 1 2 (C) (D) 0
m2 + 1 1 + m2 1 + m2
43. Equation of the line passing through (1,2) and parallel to y=3x-1 is
(A) y+2 = x+1 (B) y+2= 3(x+1) (C) y-2 = 3(x-1) (D) y-2 = x-1
44. The equation of the straight line passing through the point (1,2) and perpendicular to the
line x+y+1=0 is
(A) y-x+1=0 (B) y-x-1=0 (C) y-x+2=0 (D) y-x-2 = 0
45. The distance between parallel lines 15x+ 8y– 34= 0 and 15x+ 8y+ 31 = 0
65 3 65
(A) (B) (C) 5 (D)
17 17 19
46. Find the distance between P (x1, y1) and Q (x2, y2) when PQ is parallel to the y-axis
(A) y2 (B) x2 (C) y2 – y1 (D) x2- x1
47. Find the distance between P (x1, y1) and Q (x2, y2) when PQ is parallel to the x-axis.
(A) y2 (B) x2 (C) y2 – y1 (D) x2- x1
48. The slope of the x – axis is ……
𝜋
49. If a line makes an inclination of 3 with the positive direction of x–axis , then the slope of
the line is …….
𝜋
50. The slope of the line which makes angle 4 with positive direction of y-axis is…
51. The slope of the line passing through the points (2, 3) and (-5, 8) is…..
52. The slope of the line parallel to the line passing through the points (-2, 5) and (4,9) is …
53. The slope of the line perpendicular to the line passing through the points (6,3) and (4,8)
is …..
54. The equation of the horizontal line passing through the point (3,5) is y =…
55. The equation of the vertical line passing through the point (-4,8) is x = ….
56. The equation of the horizontal line intercepting the y- axis 3 units above the origin is y=.
57. The equation of the horizontal line intercepting the y- axis 6 units below the origin is y=.
58. The equation of the vertical line intercepting the x- axis 1units right of the origin is x =..
59. The equation of the vertical line intercepting the x- axis 5 units left of the origin is x = ….
60. The equation of x axis is ….
61. The equation of y axis is …..
62. The slope of the line 2x+7y+9 = 0 is ……
63. The x-intercept of the line 5x – 3y = 6 is ……
Question Bank: Department of School Education (Pre University ) 52

64. The y-intercept of the line 8x-y+6 = 0 is ……
65. Equation of line in the figure is

A) 5𝑥 + 3𝑦 = 15
B) 3𝑥 + 5𝑦 = 15
C) 3𝑥 + 5𝑦 + 15 = 0
D) 5𝑥 + 3𝑦 + 15 = 0
66. For the figure given below the slope of altitude AD is

5 3
A) 1 B) −1 C) − 3 D) 5.

67. For the figure given below the equation of line AB is

A) 𝑥 + 𝑦 = 5 B) 𝑥 + 𝑦 = 2 C) 𝑥 + 𝑦 = 1 D) 𝑥 + 𝑦 = 3
68. For the figure given below, consider the following statements 1 and 2

Statement 1: The slope of the line AC = −√3
Statement 2: The equation of line DB is y = −√3 (𝑥 + 2)
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.

Question Bank: Department of School Education (Pre University ) 53

69. For the figure given below, consider the following statements 1,2 and 3

Statement 1: Distance PQ =|𝑦1 − 𝑦2 |
Statement 2: The equation of line PQ is x=a.
Statement 3: The slope of the line PQ = 0.
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
70. Statement I: The point (0,2) is at 2 units distance from X-axis above the origin
Statement II: The point (2,0) is at 2 units distance from the Y- axis left of origin
(A)Both statements are true
(B)Statement I is true and statement II is false
(C)Statement I is false and statement II is true
(D)Both statements are false.
71. The two lines 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 and 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 = 0 where 𝑏1 , 𝑏2 ≠ 0 are
𝑎 𝑏
Statement I: Parallel if 𝑎1 = 𝑏1
2 2
Statement II: Perpendicular if 𝑎1 . 𝑎2 − 𝑏1 𝑏2=0
(A) Both I and II are true (B) only I is true
(C) only II is true (D) Both I and II are false
72. SATEMENT 1:
Equation of vertical line to the right of y-axis at 5 units from y-axis is 𝑥 = 5
SATEMENT 2: Equation of y-axis is x=0. Vertical line is parallel to y-axis
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct
explanation of the STATEMENT 1
(C) STATEMENT 1is true and STATEMENT 2 is false.
(D) STATEMENT 1is false and STATEMENT 2 is false.
73. Assertion (A): If x-intercept of a line is 4 and its y-intercept is 2 then the equation of
line is x+2y-4=0
Reason(R): If x-intercept of a line is a and y-intercept of line is b so, equation of line
𝑥 𝑦
is 𝑎 + 𝑏 = 1.
(A) Both Assertion (A) and Reason (R) are true.
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.

Question Bank: Department of School Education (Pre University ) 54

74. 𝐼𝑓 3𝑥 + 4𝑦 + 60 = 0 equation of a straight line. Match List I with List II
List I List II
a) The slope of the line i)12
b) The y-intercept of the line ii) − 4
3

c) 𝑇ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (0, 0) 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 iii)−15

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.
75. 1. If area of triangle formed by three points is zero then the three points must be
collinear.
2. If a line is parallel to x-axis then its inclination is 90°
3. If a line is parallel to y-axis then its inclination is zero.
Which of the above statements are correct?
(A) 1 𝑎𝑛𝑑 2 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝐵) 1 𝑎𝑛𝑑 3 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝐶) 1 𝑜𝑛𝑙𝑦 (𝐷) 𝐴𝑙𝑙 1, 2 𝑎𝑛𝑑 3.
Two mark questions
1. Find the slope of a line, which passes through origin, and the midpoint of the line segment
joining the points (0, -4) and (8, 0). (U)
2. Line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points
(8, 12) and (x, 24). Find the value of x. (U)
3. The line through the points (h, 3) and (4, 1) is perpendicular to the line 7x + 9y + 19 = 0.
Find the value of h. (U)
4. Line through the points (5, -6) and (7,-3) is parallel to the line through the points (x, 8) and
(5, 24). Find the value of x. (U)
5. Find the value of x for which the points (x, -1), (2, 1) and (4, 5) are collinear. (K)
6. Find the value of k for which the points (1, k), (3, 1) and (4, -1) are collinear. (K)
7. Find the angle between the x- axis and the line joining the points (3, -1) and (4, -2). (A)
8. Find the angle between the lines x - √3y + 5 = 0 and √3𝑥 − 𝑦 + 7 = 0. (K)
9. Find the angle between the lines x – y + 9 = 0 and 𝑥 + 𝑦 + 7 = 0. (K)
10. Find the angle between the lines √3x + y =1 and 𝑥 + √3𝑦 =1. (K)
11. Find the tangent of the angle between the lines 2x+3y-8 = 0 and 5x-y+7 = 0. (K)
12. Find the equation of the line passing through the point (-2, 3) with slope -4. (K)
1
13. Find the equation of the line passing through the point (-4, 3) with slope 2. (K)
14. Find the equation of the line passing through the point (0,0) with slope 8. (K)
15. Find the equation of the line passing through the point (-2, -9) and inclined with x axis at an
angle 45°. (U)
16. Find the equation of the line passing through the point (2, 2√3) and inclined with x axis at
an angle 75°. (U)
17. Find the equation of the line intersecting x axis at distance 3 units to the left of origin with
slope -2. (U)
18. Find the equation of the line intersecting y axis at a distance 2 units above the origin and
making an angle 30° with positive direction of x axis. (U)
19. Find the equation of the line passing through the points (1, -1) and (3, 5). (K)
20. Find the equation of the line passing through the points (-1, 1) and (2, -4). (K)
21. Find the equation of the line with slope 2 and y- intercept 3. (K)
1
22. Find the equation of the line for which tan𝜃 = 2, where 𝜃 is the inclination of the line and y-
−3
intercept 2 . (K)

Question Bank: Department of School Education (Pre University ) 55

−1
23. Find the equation of the line with slope 3 and x- intercept -3. (U)
24. Find the equation of the line, which makes intercepts -3 and 2 on the x and y axes
respectively. (K)
25. Find the equation of the median of the triangle PQR through the vertex R whose vertices are
given by P(2,1), Q(-2,3), R(4,5). (U)
26. Find the equation of the line passing through (-3, 5) and perpendicular to the line through
the points (2,5) and (-3,6). (U)
27. The perpendicular from the origin to a line meets it at the point (-2, 9), find the equation of
the line. (A)
28. Show that the points (3,0), (-2, -2) and (8,2) are collinear. (U)
29. Find the equation of the line parallel to the line 3x-4y+2 = 0 and passing through the point
(-2,3) (U)
30. Find the equation of the line perpendicular to the line 6x+5y+2 = 0 and passing through the
point (5, 2). (U)
31. Find the equation of the line perpendicular to the line 3x-5y+9 = 0 and passing through the
point (-1,8). (U)
32. Find the equation of the line perpendicular to the line x-7y+5 = 0 and having x-intercept 3.
33. Find the equation of the line parallel to the line 5x+3y+1 = 0 and having y-intercept 8.(U)
34. Reduce the equation 6x+3y-5=0 into slope–intercept form and find the slope and y-intercept
of the line. (U)
35. Reduce the equation 3x-4y-5=0 into slope–intercept form and find the slope and y-intercept
of the line. (U)
36. Reduce the equation x+7y =0 into slope–intercept form and find the slope and y-intercept of
the line. (U)
37. Reduce the equation 3x+2y-12=0 into intercept form and find the values of x and y
intercepts. (U)
38. Reduce the equation 4x-3y=6 into intercept form and find the values of x and y intercepts.
39. Find the distance of the point (-1,1) from the line 12x-5y+82=0. (K)
40. Find the distance of the point (3, -5) from the line 3x-4y-26=0. (K)
41. Find the distance between the parallel lines 3x-4y+7=0 and 3x-4y+5=0. (K)
42. Find the distance between the parallel lines 15x+8y-34=0 and 15x+8y+30 = 0. (K)
43. Find the equation of right bisector(perpendicular bisector) of the line segment joining the
points(3,4) and (-1,2). (A)
44. Find the point of intersection of the lines 2x+3y-7 = 0 and x-4y+7 = 0. (U)
Three mark questions
1. If the angle between two lines is π/4 and slope of one of the lines is 1/2 find the slope of the
other line.
2. The slope of a line is double of the slope of another line. If tangent of the angle between them
is 1/3 , find the slopes of the lines.
3. Find the equation of the line which cuts off equal intercepts on the coordinate axes and
passes through the point (2,3).
4. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the
axes whose sum is 9.
5. The perpendicular from the origin to a line meets it at the point (-2, 9), find the equation of
the line.
6. If P(a,b) is the midpoint of line segment between the axes . Show that equation of the line is
𝑥 𝑌
+𝑏=1
𝑎
7. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

Question Bank: Department of School Education (Pre University ) 56

𝑥 𝑌
8. Find the points on the x-axis, whose distances from the line 3 + 4 = 1 are 4 units
9. The line through the points (h, 3) and (4, 1) intersects the line 7x +9y -19 =0. at right angle.
Find the value of h.
10. Two lines passing through the point (2, 3) intersects each other at an angle of 600. If slope of
one line is 2, find equation of the other line.
11. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line
3x – 4y – 16 = 0.
12. The perpendicular from the origin to the line y = mx + c meets it at the point (–1, 2). Find
the values of m and c
13. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length
of altitude from the vertex A.
14. Find the equation of the line which cuts off equal intercepts on the coordinate axes and
passes through the point (2,3).
15. If P(2,4) is the midpoint of line segment between the axes, find the equation of the line. (A)
16. In the triangle ABC with vertices A(2,3), B(4,-1) and C(1,2), find the equation of the altitude
from the vertex A. (U)
17. If p is the length of perpendicular from the origin to the line whose intercepts on the axes
1 𝑦 1
are a and b, then show that 𝑎2 + 𝑏2 = 𝑝2
18. Find the equations of the lines, which cut-off intercepts on the axes whose sum and product
are 1 and – 6, respectively.
𝑥 𝑌
19. Find the equation of a line drawn perpendicular to the line 4 + 6 = 1 through the point,
where it meets the y-axis.
20. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and 2x – y – 3 = 0
may intersect at one point
21. Find the equation of the lines through the point (3, 2) which make an angle of 450 with the
line x – 2y = 3.
22. Find the distance of the line 4x + 7y + 5 = 0 from the point (1, 2) along the line 2x – y = 0.

Five mark questions
1. Derive an expression for the acute angle between two lines having slopes m1 and m2 and
hence find the acute angle between the lines x+y-6=0 and x-y -5=0. (U)
2. Derive the equation of the line having slope ‘m’ and passing through the point (x0 ,y0) and
hence find the equation of the line having slope 3 and passing through the point (3,-1). (U)
3. Derive the equation of the line passing through the points (x1 ,y1) and (x2 , y2) hence find the
equation of the line passing through the points (4,7) and (-3, 8). (U)
4. Derive the equation of the line having slope ‘m’ and y- intercept ‘c’ and hence find the
equation of the line having slope -2 and y- intercept 4. (U)
5. Derive the equation of the line having x and y- intercept values as ‘a’ and ‘b’ respectively and
hence find the equation of the line having x and y- intercept values as 2 and 6 respectively.
6. Derive an expression for the perpendicular distance between a point and a line. (U)

Question Bank: Department of School Education (Pre University ) 57

CHAPTER - 10
CONIC SECTIONS
MCQ /FB
1. If α be the angle between axis and generator of the cone .Let β be the angle made by
the intersecting plane with the vertical axis of the cone, then which of the following is
incorrect
A) When β > 90o, the section is a circle B) When α < β < 90o, the section is an ellipse
C) When β = α; the section is a parabola D) When 0 ≤ β < α; the section is a hyperbola
2. When the plane cuts at the vertex of the cone . If α be the angle between axis and
generator of the cone .Let β be the angle made by the intersecting plane with the
vertical axis of the cone, then which of the following is incorrect
A) When α < β ≤ 90o, then the section is a point
B) When β = α, the plane contains a generator of the cone and the section is a straight
line
C) When 0 ≤ β < α, the section is a pair of intersecting straight lines
D) When β=900 the section is a circle.
3. Which of the following is incorrect
(A) A circle is the set of all points in a plane that are equidistant from a fixed point in the
plane.
(B) parabola is the set of all points in a plane that are equidistant from a fixed line
and a fixed point in the plane
(C) An ellipse is the set of all points in a plane, the difference of whose distances from
two fixed points in the plane is a constant.
(D) A hyperbola is the set of all points in a plane, the difference of whose distances
from two fixed points in the plane is a constant.
4. Which of the following is incorrect
A)The equation of a circle with centre (h, k) and the radius r is (x – h)2 + (y – k)2 = r2.
B) The equation of the parabola with focus at (a, 0) a > 0 and directrix x = – a is y2= 4ax.
𝑥2 𝑦2
C) The equation of an ellipse with foci on the x-axis is 𝑎2 + 𝑏2 = 1
𝑦2 𝑥2
D) The equation of a hyperbola with foci on the x-axis is 𝑎2 − 𝑏2 = 1.
5. Which of the following is incorrect
A) Length of the latus rectum of the parabola y2= 4ax is 4a.
B) Length of the latus rectum of the parabola x2= 4ay is 2a.
𝑥2 𝑦2 2𝑏 2
C) Length of the latus rectum of an ellipse 𝑎2 + 𝑏2 = 1 is 𝑎
𝑥2 𝑦2 2𝑏 2
D) Length of the latus rectum of a hyperbola 𝑎2 − 𝑏2 = 1 is 𝑎 .
6. Which of the following is correct
A) When the axis of symmetry is along the x-axis the parabola opens to the right if a<0
B) When the axis of symmetry is along the x-axis the parabola opens to the left if a>0
C) When the axis of symmetry is along the y-axis the parabola opens upwards if a<0
D) When the axis of symmetry is along the y-axis the parabola opens downwards if a<0
7. Equation of the circle with centre at (0,0) and radius r.
A ) 𝑥2 + 𝑦2 = 𝑟2 B) (𝑥 − 1)2 + (𝑦 − 1)2 = 𝑟 2
C ) (𝑥 + 1) + (𝑦 + 1) = 𝑟
2 2 2
D) 𝑥 2 + 𝑦 2 = 1.
8. Equation of the circle with centre at (-3,2) and radius 4.
A) 𝑥 2 + 𝑦 2 + 6𝑥 − 4𝑦 + 3 = 0 B) 𝑥 2 + 𝑦 2 − 6𝑥 + 4𝑦 − 3 = 0
C ) 𝑥 2 + 𝑦 2 − 6𝑥 + 4𝑦 + 3 = 0 D) 𝑥 2 + 𝑦 2 + 6𝑥 − 4𝑦 − 3 = 0

Question Bank: Department of School Education (Pre University ) 58

9. Equation of the circle with centre at (1,1) and radius √2.
A) 𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 = 0 B) 𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 + 2 = 0
C) 𝑥 + 𝑦 + 2𝑥 + 2𝑦 = 0
2 2
D) 𝑥 2 + 𝑦 2 + 2𝑥 + 2𝑦 + 2 = 0
10. Equation of the circle with centre at (0,2) and radius 2.
A) 𝑥 2 + 𝑦 2 − 4𝑦 = 4 B) 𝑥 2 + 𝑦 2 − 4𝑥 + 4 = 0
C ) 𝑥 2 + 𝑦 2 − 4𝑦 = 0 D) 𝑥 2 + 𝑦 2 − 4𝑥+= 0
11. The radius of a circle with centre (2,2) and passes through the point (4,5).
A) 13 B) √13 C) 5 D) √5
12. The centre of the circle with radius 5 whose centre lies on x- axis and passes through
the point (2,3).
A) (6, 6) B) (−2, 0)&(6,0) C) (4, 0) D) (5, 0)
13. The centre and radius of the circles (𝑥 + 5) + (𝑦 − 3) = 36
2 2

A) (5, −3) ,6 B) (−5, 3) ,6 C) (5, −3) ,36 D) (−5, 3) ,36
14. The centre and radius of the circles 𝑥 + 𝑦 − 4𝑥 − 8𝑦 − 45 = 0
2 2

A) (−2, −4) ,5 B) (2, 4) ,5 C) (−2, −4) ,25 D) (2, 4) ,√65 .
15. The centre and radius of the circles 2𝑥 2 + 2𝑦 2 − 𝑥 = 0
1 1 1 1 1 1 1 1
A) (4 , 0) , 4 B) (4 , 0) , 16 C) (0, 4) , 4 D) (2 , 0) , 4
16. Which of the following is correct
A)The point (–2.5, 3.5) lie inside the circle 𝑥 2 + 𝑦 2 = 25
B)The point (–2.5, 3.5) lie outside the circle 𝑥 2 + 𝑦 2 = 25
C) The point (–2.5, 3.5) lie on the circle 𝑥 2 + 𝑦 2 = 25
D)The point (–2.5, 3.5) lie inside the circle 𝑥 2 + 𝑦 2 = 16.
17. The coordinates of the focus and the equation of the directrix of the parabola 𝑦 2 = 8x.
A) (2, 0) , 𝑥 = 2 B) (2, 0) , 𝑥 = −2 C) (0, 2) , 𝑦 = 2 D) (0, 2) , 𝑦 = −2.
18. The length of the latus rectum of the parabola 𝑦 2 = -9x
9 9
A) 9 B) −9 C) 4 D) 2 .
19. The coordinates of the focus and the length of the latus rectum of the parabola
𝑥 2 = -16y.
A) (−4, 0) , 16 B) (0, 4) , 16 C) (4, 0) , 16 D) (0, −4) ,16.
20. The axis and the length of the latus rectum of the parabola 𝑥 2 = 6y.
6 6
A) 𝑥 = 0 𝑎𝑛𝑑 6 B) 𝑦 = 0 𝑎𝑛𝑑 6 C) 𝑥 = 0 𝑎𝑛𝑑 4 D) 𝑦 = 0 𝑎𝑛𝑑 4.
21. The equation of the parabola with vertex at (0, 0) and focus at (0, 2) is
A) 𝑥 2 = 4y B) 𝑥 2 = 8y C) 𝑦 2 = 8x. D) 𝑦 2 = 4x.
22. The equation of the parabola with focus (6,0); directrix x = – 6 is
A) 𝑥 2 = -24y B) 𝑥 2 = 24y C) 𝑦 2 = 24x. D) 𝑦 2 = -24x.
23. The equation of the parabola with vertex (0,0) passing through (1,2) and axis is along
x-axis.
A) 𝑥 2 = ½ y B) 𝑥 2 = 4y C) 𝑦 2 = 2x. D) 𝑦 2 = 4x.
𝑥2 𝑦2
24. The coordinates of the foci and length of major axis of the ellipse 25 + 9 = 1.
A) (±4, 0) and 10 B) (±5, 0) and 10 C) (0, ±4) and 10 D) (0, ±3) and 10
𝑥2 𝑦2
25. The coordinates of the vertices and the length of minor axis of the ellipse 25 + 9 = 1
A) (±4, 0) and 6 B) (±5, 0) and 6 C) (0, ±3) and 6 D) (0, ±4) and 6
𝑥2 𝑦2
26. The the eccentricity of the ellipse 25 + 9 = 1
4 4 18 3
A) 5 B) 3 C) 5 D) 4 .

Question Bank: Department of School Education (Pre University ) 59

𝑥2 𝑦2
27. The latus rectum of the ellipse 25 + 9 = 1
4 10 18 25
A) 5 B) 3 C) 5 D) 3 .
28. The coordinates of the foci of the ellipse 16𝑥 + 𝑦 = 16
2 2

A) (±√15, 0) B) (0, ±√15) C) (0, ±√17) D) (±√17 ,0)
29. The coordinates of the vertices of the ellipse 16𝑥 + 𝑦 = 16
2 2

A) (±4, 0) B) (0, ±4) C) (0, ±1) D) (±1 ,0)
30. The length of major axis and the minor axis of the ellipse 16𝑥 2 + 𝑦 2 = 16
A) 8 𝑎𝑛𝑑 2 B) 2 𝑎𝑛𝑑 8 C) 8 𝑎𝑛𝑑 1 D) 1 𝑎𝑛𝑑 8 .
31. The eccentricity of the ellipse 16𝑥 + 𝑦 = 16
2 2

√15 √17 1
A) 4 B) √15 C) 4 D) 2 .
32. The length of latus rectum of the ellipse 16𝑥 2 + 𝑦 2 = 16
A)1/2 B) 16 C) 32 D) 8 .
33. The equation of the ellipse, with Ends of major axis (± 3, 0), ends of minor axis (0, ± 2)
𝑥2 𝑦2 𝑥2 𝑦2 𝑥2 𝑦2 𝑥2 𝑦2
A) 16 + 36 = 1 B) 36 + 16 = 1 C) 9 + 4 = 1 D) 4 + 9 = 1
𝑥2 𝑦2
34. The coordinates of the foci and the vertices of the hyperbolas: 9 − 16 = 1
A) (±5, 0) and (±4, 0) B) (±5, 0) and (±3, 0)
C) (0, ±5) and (±4, 0) D) (0, ±5) and ( 0, ±4)
𝑥2 𝑦2
35. The eccentricity of the hyperbolas: 9 − 16 = 1
5 5 18 32
A) 4 B) 3 C) 4 D) 3 .
𝑥2 𝑦2
36. The length of latus rectum of the hyperbolas: 9 − 16 = 1
5 5 18 32
A) 4 B) 3 C) 4 D) 3 .
37. The coordinates of the foci of the hyperbolas: 16𝑦 − 𝑥 = 16
2 2

A) (±√15, 0) B) (0, ±√15) C) (0, ±√17) D) (±√17 ,0)
38. The coordinates of the vertices of the hyperbolas: 16𝑦 − 𝑥 = 16
2 2

A) (±4, 0) B) (0, ±4) C) (0, ±1) D) (±1 ,0)
39. The eccentricity of the hyperbolas: 16𝑦 − 𝑥 = 16
2 2

√17 √15
A) 4 B) √17 C) 4 D) √15 .
40. The latus rectum of the hyperbolas: 16𝑦 2 − 𝑥 2 = 16
1 1
A) 16 B) 8 C) 2 D) 4 .
41. The eccentricity of the an equilateral hyperbola is
√5
A) 2 B) √2 C) 2 D) √3 .
42. The eccentricity is never less than one
A) 𝑐𝑖𝑟𝑐𝑙𝑒 B) 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 C) 𝑒𝑙𝑙𝑖𝑝𝑠𝑒 D) ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑎 .
43. The equation of the parabola which is symmetric about the y-axis, and passes through
the point (2,–3).
A) 3𝑥 2 = 4𝑦 B) 4𝑥 2 = −3𝑦 C) 3𝑥 2 = −4𝑦 D) 3𝑦 2 = −4𝑥
44. The length of the minor axis of the ellipse, whose length of the major axis is 20 and
foci are (0, ± 5).
A) 75 B) 10√3 C) 5√3 D) 10 .
45. The length of the minor axis of the ellipse, whose Vertices (± 5, 0) and foci (± 4, 0)
A) 6 B) 3 C) 4 D) 8
.

Question Bank: Department of School Education (Pre University ) 60

46. The equation for the ellipse whose ends of major axis (± 3, 0) and ends of minor axis
(0, ± 2)
𝑥2 𝑦2 𝑥2 𝑦2 𝑥2 𝑦2 𝑥2 𝑦2
A) 9 + 4 = 1 B) 4 + 9 = 1 C) 6 + 4 = 1 D) 3 + 4 = 1
47. The equation for the ellipse whose ends of major axis (0,± 5 ) and ends of minor axis
(±1, 0)
𝑥2 𝑦2 𝑥2 𝑦2 𝑥2 𝑦2 𝑥2 𝑦2
A) 25 + 1 = 1 B) 1 + 25 = 1 C) 25 + 4 = 1 D) 2 + 25 = 1
48. The equation for the hyperbola whose Vertices (± 2, 0) and foci (± 3, 0)
𝑥2 𝑦2 𝑥2 𝑦2 𝑥2 𝑦2 𝑦2 𝑥2
A) 4 − 9 = 1 B) 4 − 5 = 1 C) 5 − 4 = 1 D) 4 − 5 = 1
49. The equation for the hyperbola whose . Vertices (0, ± 5) and foci (0, ± 8)
𝑥2 𝑦2 𝑦2 𝑥2 𝑥2 𝑦2 𝑦2 𝑥2
A) 25 − 64 = 1 B) 25 − 64 = 1 C) 25 − 39 = 1 D) 25 − 39 = 1
50. The equation for the hyperbola whose Foci (± 5, 0) and the transverse axis is of length 8
𝑥2 𝑦2 𝑦2 𝑥2 𝑥2 𝑦2 𝑦2 𝑥2
A) 16 − 9 = 1 B) 9 − 16 = 1 C) 16 − 25 = 1 D) 25 − 16 = 1
51. The equations of the hyperbola whose Foci (0, ±13) and the conjugate axis is of length
24.
𝑥2 𝑦2 𝑦2 𝑥2 𝑥2 𝑦2 𝑦2 𝑥2
A) 25 − 144 = 1 B) 144 − 25 = 1 C) 144 − 25 = 1 D) 25 − 144 = 1
5
52. Statement 1:The eccentricity of hyperbola 9𝑥 2 − 16𝑦 2 = 144 is 4
𝑥2 𝑦2 √𝑎2 +𝑏 2
Statement 2:The eccentricity of hyperbola 𝑎2 − 𝑏2 = 1 is 𝑎 .
(A) Statement 1 is true and Statement 2 is false.
(B) Statement 1 is false and Statement 2 is false.
(C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation
for Statement 1
(D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct
explanation for Statement 1
53. Statement 1: The equation of circle which pass through (5, 9) and center at (2, 5) is
(𝑥 − 2)2 + (𝑦 − 5)2 = 25
Statement 2: Equation of circle with center at (a, b) and radius r units is
(x-a)2 + (y-b)2 = r2.
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
54. Statement 1: The center of ellipse is same as a vertex.
Statement 2: An ellipse has two vertices and two foci.
𝑥2 𝑦2
Statement 3: Length of major axis of an ellipse 𝑎2 + 𝑏2 = 1 is 2a.
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

Question Bank: Department of School Education (Pre University ) 61

55. Assertion (A): When the axis of symmetry is along the x-axis the parabola opens to the
right if a<0
Reason(R): When the axis of symmetry is along the y-axis the parabola opens
downwards if a>0
(A) Both Assertion (A) and Reason (R) are true.
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
56. The equation of the parabola is 𝑦 2 = −4𝑥
Statement I: length of the latus rectum is - 4
Statement II: The directrix of the parabola is x= 1
(A) Both I and II are true (B) only I is true
(C) only II is true (D) Both I and II are false
57. 1.Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola,
through the focus and whose end points lie on the parabola.
2.The eccentricity of a hyperbola is the ratio of the distances from the centre of the
hyperbola to one of the foci and to one of the vertices of the hyperbola.
3. An ellipse is the set of all points in a plane, the difference of whose distances from two
fixed points in the plane is a constant.
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.
58. For the figure given below, consider the following statements 1 and 2

Statement I: Figure 1 ellipse and figure 2 parabola
Statement II: Figure 1 circle and figure 2 hyperbola
(A) Both I and II are true (B) only I is true
(C) only II is true (D) Both I and II are false
59. For the figure given below, radius is

A) 25 B) √125 C) 5 D) √5

Question Bank: Department of School Education (Pre University ) 62

60. For the figure given below

which of the following is incorrect
A) When β = 90o, the section is a circle B) When α < β < 90o, the section is an ellipse
C) When β = α; the section is a parabola D) When 0 ≥ β > α; the section is a hyperbola.
61. For the figure given below, consider the following statements 1 and 2

Statement I: Figure (b) equation of parabola is 𝑦 2 = −4𝑎𝑥
Statement II: Figure (c) equation of parabola is 𝑥 2 = −4𝑎𝑦
(A) Both I and II are true (B) only I is true
(C) only II is true (D) Both I and II are false
62. For the figure given below

which of the following is correct
A) directrix x = 6 B) vertex (0,0) and length of the latus rectum is 6
C) equation of parabola is 𝑦 2 = 16𝑥 D) axis is y=0 and focus (0,6)
𝑥2 𝑦2
63. The equation of the ellipse + 9 = 1.
4
Match List I with List II
List I List II
a) The length of major axis i)4
b) The length of minor ii) 6
c) The length of latus rectum iii)9

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.
Question Bank: Department of School Education (Pre University ) 63

64. Match Column I and Column II
Column I Column II
a) Equation of an equilateral hyperbola. 𝑥2 𝑦2
i) 𝑎2 + 𝑏2 = 1
b) Equation of an ellipse with centre of 𝑦2 𝑥2
ii) 𝑏2 − 𝑎2 = 1
the origin and major axis along the x-axis
c) 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 of ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑎𝑠 𝑥2 𝑦2
iii)𝑎2 − 𝑎2 = 1
Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.
65. The area of circle which pass through (1, 3) and center at (1, 0) is ........................
66. Length of latus rectum of parabola 𝑦 2 = 4𝑎𝑥 which passes from (3,2) is ..............
67. Chord AB is diameter of circle with centre (4,2 ).If point A (3,2) and point B (5,k), then k
is... ........
68. If length of latus rectum of an ellipse is half then minor axis ,then its eccentricity is
...........
69. length of the conjugate axis of 12𝑥 2 − 3𝑦 2 = −36 is ........
70. If ratio of major and minor axis of ellipse is 5:3 then eccentricity e = ................

Two marks questions
1. Find the centre and radius of the following equation of the circles
1) 𝑥 2 + 𝑦 2 + 8𝑥 + 10𝑦 − 8 = 0. 2) 𝑥 2 + 𝑦 2 − 8𝑥 + 10𝑦 − 12 = 0.
3) 𝑥 2 + 𝑦 2 − 4𝑥 − 8𝑦 − 45 = 0. 4) 2𝑥 2 + 2𝑦 2 − 𝑥 = 0. (U)
2. Find the coordinates of the focus, axis, and the equation of the directrix of the
parabola 𝑦 2 = 8𝑥. (U)
3. Find the coordinates of the focus and the equation of the directrix of the parabola
𝑦 2 = 12𝑥. (U)
4. Find the coordinates of the focus and the equation of the directrix of the parabola
𝑦 2 = 10𝑥. (U)
5. Find the coordinates of the focus, and latus rectum of of the parabola 𝑦 = −8𝑥. (U)
2

6. Find the coordinates of the focus, axis, and latus rectum of of the directrix of the
parabola 𝑥 2 = 6𝑦. (U)
7. Find the coordinates of the axis, and the equation of the directrix of the parabola
𝑥 2 = −16𝑦. (U)
8. Find the coordinates of the axis, and the equation of the directrix of the parabola
𝑥 2 = −9𝑦. (U)
9. Find the coordinates of the focus, the equation of the directrix and latus rectum of the
parabola 𝑦 2 = 8𝑥.
10. Find the equation of the directrix and latus rectum of the parabola 𝑦 2 = 12𝑥. (U)
11. Find the equation of the directrix and latus rectum of the parabola 𝑦 2 = 10𝑥. (U)
12. Find the coordinates of the focus, and latus rectum of the parabola 𝑦 2 = −8𝑥.(U)
13. Find the coordinates of the focus, and latus rectum of the parabola 𝑥 2 = 6𝑦. (U)
𝑥2 𝑦2
14. Find the foci and Latus rectum of the ellipse 4 + 25 = 1. (U)
𝑥2 𝑦2
15. Find the eccentricity and Latus rectum of the ellipse25 + 100 = 1. (U)
𝑥2 𝑦2
16. Find the foci and eccentricity of the ellipse100 + 400 = 1. (U)
17. Find the eccentricity and Latus rectum of the ellipse9𝑥 2 + 4𝑦 2 = 36. (U)
18. Find the foci and Latus rectum of the ellipse36𝑥 2 + 4𝑦 2 = 144. (U)
19. Find the length of major axis and minor axis of the ellipse 16𝑥 2 + 𝑦 2 = 16. (U)
Question Bank: Department of School Education (Pre University ) 64

20. Find the vertices, and eccentricity of the ellipse 4𝑥 2 + 9𝑦 2 = 36. (U)
21. Find the equation of the ellipse given that Vertices(±5,0) and foci(±4,0). (U)
22. Find the equation of the ellipse given that Vertices(±13,0) and foci(±5,0). (U)
23. Find the equation of the ellipse given that Vertices(±6,0) and foci(±4,0). (U)
24. Find the equation of the ellipse given that Vertices(0, ±13) and foci(0, ±5). (U)
25. Find the equation of the ellipse given that Length of major axis is 26 and foci (±5,0). (U)
26. Find the equation of the ellipse given that Length of major axis is 20 and foci(0, ±5). (U)
27. Find the equation of the ellipse given that Length of major axis is 16 and foci(0, ±6). (U)
28. Find the equation of the ellipse given that Foci(±3,0) and
length of semi major axis is 4. (U)
𝑥2 𝑦2
29. Find the foci and the eccentricity of the latus rectum of the hyperbola − 16 = 1. (U)
9
𝑥2 𝑦2
30. Find the foci and the length of the latus rectum of the hyperbola − = 1. (U)
16 9
31. Find the foci and the eccentricity of the hyperbola 16𝑥 − 9𝑦 = 576.
2 2
(U)
𝑦2 𝑥2
32. Find the eccentricity and the length of the latus rectum of the hyperbola − 27 = 1.
9
𝑥2 𝑦2
33. Find the foci, vertices and the length of the latus rectum of the hyperbola − 16 = 1.
9
𝑥2 𝑦2
34. Find the foci , vertices and the length of the latus rectum of the hyperbola 16 − 9 = 1(U)
35. Find the vertices and the length of the latus rectum of the hyperbola 16𝑥 2 − 9𝑦 2 = 576.
𝑦2 𝑥2
36. Find the foci and vertices the hyperbola 9 − 27 = 1. (U)
37. Find the vertices and the length of the latus rectum of the hyperbola 9𝑦 2 − 4𝑥 2 = 36.
38. Find the vertices and the length of the latus rectum of the hyperbola 5𝑦 2 − 9𝑥 2 = 36. (U)
39. Find the vertices and the length of the latus rectum of the hyperbola
49𝑦 2 − 16𝑥 2 = 784. (U)
40. Find the foci , vertices and the length of the latus rectum of the hyperbola
𝑦 2 − 16𝑥 2 = 16. (U)
41. Find the equation of the hyperbola given that Vertices (±2,0)and foci(±3,0). (U)
42. Find the equation of the hyperbola given that Vertices (0, ±5)and foci(0, ±8). (U)
43. Find the equation of the hyperbola given that Vertices (0, ±3) and foci (0, ±5). (U)
44. Find the equation of the hyperbola given that Foci (±5,0) and the transverse axis is of
length 8. (U)
45. Find the equation of the hyperbola given that Foci (0, ±13) and conjugate axis is of
length 24. (U)
46. Find the equation of the hyperbola given that Foci (±3√5, 0)and latus rectum is of length 8. (U)
47. Find the equation of the hyperbola given that Foci (±4,0)and the latus rectum is of length 12(U)
4
48. Find the equation of the hyperbola given that Vertices(±7,0), 𝑒 = . (U)
3
49. Find the equation of the hyperbola given that Foci(0, ±√10), passing through (2, 3). (U)
50. Find the area of the triangle formed by the lines joining the vertex of the parabola
𝑥 2 = 12𝑦 to the ends of its latus rectum.
Three marks questions
1. Find the equation of the circle which passes through (2, -2) and (3, 4) and whose
centre lies on the line x + y=2. (U)
2. Find the equation of the circle which passes through (4, 1) and (6, 5) and whose
centre lies on the line 4x+y=16. (U)
3. Find the equation of the circle which passes through (2, 3) and (-1, 1) and whose
centre lies on the line x-3y-11=0. (U)
4. Find the equation of the circle with radius 5 whose centre lies on x-axis and passes
through the point (2, 3).

Question Bank: Department of School Education (Pre University ) 65

5. Find the equation of the circle passing through (0, 0) and making intercepts ‘a’ and ‘b’
on the coordinate axes. (U)
6. Find the coordinates of the focus, the equation of the directrix and latus rectum of the
parabola 𝑥 2 = −16𝑦. (U)
7. Find the coordinates of the focus, , the equation of the directrix and latus rectum of
the parabola 𝑥 2 = −9𝑦. (U)
8. Find the equation of the parabola given that vertex (0, 0), passing through (2, 3) and
axis is along x-axis. (U)
9. Find the equation of the parabola given that vertex (0, 0), passing through (5, 2) and
symmetric with respect to y-axis. (U)
𝑥2 𝑦2
10. Find the foci, eccentricity and Latus rectum of the ellipse 25 + 9 = 1. (U)
𝑥2 𝑦2
11. Find the foci, eccentricity and Latus rectum of the ellipse + = 1. (U)
36 16
𝑥2 𝑦2
12. Find the foci, eccentricity and Latus rectum of the ellipse 16 + 9 = 1. (U)
𝑥2 𝑦2
13. Find the foci, eccentricity and Latus rectum of the ellipse 49 + 36 = 1. (U)
14. Find the foci, eccentricity and Latus rectum of the ellipse 16𝑥 + 𝑦 = 16.
2 2
(U)
15. Find the foci, eccentricity and Latus rectum of the ellipse 4𝑥 + 9𝑦 = 36.
2 2
(U)
16. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
𝑥2 𝑦2
+ 9 = 1. (U)
25
17. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
𝑥2 𝑦2
+ 16 = 1. (U)
36
18. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
𝑥2 𝑦2
+ 9 = 1. (U)
16
19. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
𝑥2 𝑦2
+ 36 = 1. (U)
49
20. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
𝑥2 𝑦2
+ 25 = 1. (U)
4
21. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
𝑥2 𝑦2
+ 100 = 1. (U)
25
22. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
𝑥2 𝑦2
+ 400 = 1. (U)
100
23. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
9𝑥 2 + 4𝑦 2 = 36. (U)
24. Find the vertices, length of major axis, minor axis, and eccentricity of the ellipse
36𝑥 2 + 4𝑦 2 = 144. (U)
25. Find the equation of the ellipse given that Centre at (0, 0), major axis on the x-axis
and passing through the points (4, 3) and (-1, 4). (U)
26. Find the equation of the ellipse given that Centre at (0, 0) , major axis on the x-axis
and passing through the points (4, 3) and (6,2). (U)
27. Find the equation of the ellipse given that Centre at (0, 0), major axis on the y-axis
and passing through the points (3, 2) and (1, 6).
28. Find the foci , the eccentricity and the length of the latus rectum of the hyperbola
9𝑦 2 − 4𝑥 2 = 36. (U)
29. Find the foci , the eccentricity and the length of the latus rectum of the hyperbola
5𝑦 2 − 9𝑥 2 = 36. (U)
Question Bank: Department of School Education (Pre University ) 66

30. Find the foci, the eccentricity and the length of the latus rectum of the hyperbola
49𝑦 2 − 16𝑥 2 = 784. (U)
31. Find the foci, the eccentricity and the length of the latus rectum of the hyperbola
𝑦 2 − 16𝑥 2 = 16. (S)
Six marks questions
𝑥2 𝑦2
1. Define the Ellipse and derive the equation of the ellipse in the form 𝑎2 + 𝑏2 = 1. (𝑎 > 𝑏)
(K)
𝑥2 𝑦2
2. Define the Hyperbola and derive the equation of the hyperbola in the form 𝑎2 − 𝑏2 = 1.
(K)
3. Define the Parabola and derive the equation of the parabola in the form 𝑦 = 4𝑎𝑥. Find
2

the length of the Latus Rectum. (K)

Question Bank: Department of School Education (Pre University ) 67

CHAPTER 11
Introduction to three Dimensional Geometry
MCQ /FB
1. The three coordinate planes divide the space into eight parts is called is
(a) Quadrants (b) Octants (c) Eighths (d) None of these
2. The coordinates of any point on the x-axis will be as
(a) (0,x,0) (b) (0,y,z) (c) (x,0,0) (d) (0,0,x)
3. The coordinates of any point in the YZ-plane will be as
(a) (x,y,0) (b) (0,y,z) (c) (x,0,0) (d) (x,0,z)
4. Any point on y-axis is of the form
(a) (x,y,0) (b) (0,y,0 ) (c) (0,y,z) (d) (x,0,z)
5. Any point on z-axis is of the form
(a) (x,0,0) (b) (0,y,0 ) (c) (0,0,z) (d) (x,y,0)
6. The coordinates of any point in the XY-plane is
(a) (x,y,0) (b) (0,y,z) (c) (x,0,0) (d) (x,0,z)
7. The coordinates of any point in the XZ-plane is
(a) (x,y,0) (b) (0,y,z) (c) (x,0,0) (d) (x,0,z)
8. A point is on the x-axis, then its y –coordinate is
(a) x (b) y (c) 0 (d) z
9. A point is on the x-axis, then its z –coordinate is
(a) x (b) y (c) 0 (d) z
10. A point is in the XZ-plane ,then its y –coordinate is
(a) x (b) y (c) 0 (d) z
11. The x-axis and y-axis taken together determine a plane known as
(a) xy-plane (b) yz-plane (c) zx-plane (d) z-plane
12. The perpendicular distance from the point P( x, y, z) to xy-plane is
(a) x (b) y (c) z (d) √𝑥 2 + 𝑦 2
13. The perpendicular distance from the point P( 6, 7, 8) to zx- plane is
(a) 8 (b) 7 (c) 6 (d) 10
14. If P (2,4,5) point in the space and PF perpendicular xy-plane, then coordinate of point F
is
(a) (0,0,5) (b) (0,4,5) (c) (2,0,5) (d) (2,4,0)
15. The locus of a point for which y = 0, z = 0?
(a) equation of x-axis (b) equation of y-axis (c) equation of z-axis (d) none of these
16. The x − axis is the intersection of two planes
(a) xy and xz (b) yz and zx (c) xy and yz (d) none of these.
17. Equation of x-axis is considered as
(a) x = 0, y = 0 (b) y = 0, z = 0 (c) z = 0, x = 0 (d) none of these
18. The shortest distance of the point (a,b,c) from the x-axis is
(a) (a + b )
2 2
(b) (b + c )
2 2
(c) (c + a )
2 2
(d) (b + c + a )
2 2 2

19. The octant in which the points (–3,1,2 lie.
(a) I octant (b) II octant (c) III octant (d) IV octant
20. The octant in which the points (–3,1,-2) lie.
(a) V octant (b)VI octant (c) VII octant (d) VIII octant
21. The octant in which the points (–2,-4,-7 ) lie.
(a) V octant (b)VI octant (c) VII octant (d) VIII octant
22. The octant in which the points (–3, –1, 6)lie.
(a) I octant (b) II octant (c) III octant (d) IV octant

Question Bank: Department of School Education (Pre University ) 68

23. From which of the following the distance of the point (1,2,3) is 10
(a)Origin (b)x-axis (c) y-axis (d)z-axis
24. Perpendicular distance of the point (3,4,5) from the y-axis, is
(a) 34 (b) 41 (c) 4 (d) 5
25. The distance between the points (1,-3,4) and (-4,1,2) is
(a)√29 (b) √45 (c)√17 (d) 7
26. The distance between the points (2, 3, 5) and (4, 3, 1) is
(a)√6 (b)√108 (c)√20 (d)√56
27. The distance between the points (2, –1, 3) and (–2, 1, 3).is
(a)√20 (b)√18 (c)√6 (d)√56
28. The distance of the point (3, 4, 5) from the origin.is
(a)5√2 (b) 5√3 (c) √5 (d) √12
29. The coordinates of the centroid of a triangle whose vertices are (2,-3, 1), (1, 0, -1)
and (3, 6, 0).
(a) (2,1,1) (b) (2,1,0) (c) (2,3,0) (d) (2,-1,0)
30. The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are
(3, –5, 7) and (–1, 7, – 6), respectively, find the coordinates of the point C.
(a) (1,1,2) (b) (3,1,2) (c) (1,2,2) (d) (1,-1,2)
31. The distance of the point (1,2,3) from the coordinate axes are
(a) √5,√10,√13 (b) 1,2,3 (c) 5,10,13 (d) 3,4,5.
32. Coordinates of the point on X-axis at distance 4 units from origin will be
(a) (0,4,0) (b) (0,4,4) (c) (4,0,0) (d) (0,0, 4)
33. A point is on negative direction of Y-axis at 5 units from origin will be
(a) (0,5,0) (b) (0,-5,0) (c) (5,0,5) (d) (-5,0, -5)
34. Statement 1: The perpendicular distance from the point P( 6, 7, 8) to zx- plane is 7
Statement 2 : The shortest distance of the point (a, b, c) from the x-axis is √𝑏 2 + 𝑐 2
A) Statement 1 is true and Statement 2 is false.
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
D) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
35. Statement 1: The point (−5,4,3) lie in II Octant.
Statement 2 : The x-coordinate is negative but y and z coordinates are positive. point lie
in octant II.
a) Statement 1 is true and Statement 2 is false.
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
36. Statement 1:The point (4, 0, 0) lie on yz-plane.
Statement 2:a point with y and z coordinate zero and x-coordinate having non-zero
value must lie on x-axis.
a) Both statements are true b) Statement I is true and statement II is false
c) Statement I is false and statement II is true d) Both statements are false

Question Bank: Department of School Education (Pre University ) 69

37. Assertion (A): The point (4, 2, 0) lie on xy-plane.
Reason(R): In 3d coordinate system, a point with z-coordinate zero and x & y coordinate
having non-zero value must lie on XY.
a) Both Assertion (A) and Reason (R) are true .
b) Both Assertion (A) and Reason (R) are false.
c) Assertion (A) is true and Reason (R) is false.
d) Assertion (A) is false and Reason (R) is true.
38. 1. The distance of point (2, 3, 5) from X-Y plane is √13 units.
2. The distance of point (2, 3, 5) from Y-Z plane is 2 units.
3. The point (1, 5, 7) lies in first octant .
Which of the above statements are correct?
(𝑎)1 𝑎𝑛𝑑 2 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝑏) 1 𝑎𝑛𝑑 3 𝑏𝑢𝑡 𝑛𝑜𝑡 3 (𝑐) 3 𝑜𝑛𝑙𝑦 (𝑑) 𝐴𝑙𝑙 1, 2 𝑎𝑛𝑑 3.
39. Statement 1: A point is in XY plane. Its distance measured parallel to X axis is 10 units
and that of parallel to Y axis is 20 units. Then its coordinates is (10,20,0).
Statement 2: The coordinates of any point in the XZ-plane is (0,b,c).
a)Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
b) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct
explanation of the STATEMENT 1
c) STATEMENT 1 is true and STATEMENT 2 is false.
d) STATEMENT 1 is false and STATEMENT 1 is true.
40. For the figure given below coordinates of point P is

(a) (2,4,3) (b) (3,4,2) (c) (4,3,2) (d) (4,2,3)
41. For the figure given below coordinates of point P is

(a) (2,3,2) (b) (2,2,3) (c) (2,3,3) (d) (2,2,2)

Question Bank: Department of School Education (Pre University ) 70

42. For the figure given below, consider the following statements 1 and 2

Statement 1: The coordinates of point D is (1,-2,8)
Statement 2: The length of diagonal is AB= 20
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.
43. The three coordinate planes divide the space into ------ pats (K)
44. A point is on the X-axis. Then its y is ------- (U)
45. A point is in the XZ plane, then its y coordinate is -------- (U)
46. The point (1,-2,-3) lie in --------- octants (K)
47. The distance of the point (3, 4, 5) from the origin is ------ (K)
48. If the distance of the point (2, 1, k) from the origin is 3, then k is ------ (K)
49. The coordinates of the centroid of a triangle whose vertices are (2,-3, 1), (1, 0, -1) and
(3, 6, 0). is ------ (K)
50. If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6),Q (– 4, 3b, –10)
and R(8, 7, 2c).
Match Column I and Column II
Column I Column II
a) value of a i) −3
b) value of b ii) −2
c) value of c iii)2

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.

Two marks questions:
1. Find the distance between the points P (1,-3,4) and Q (-4,1,2). (K)
2. Find the coordinates of the centroid of a triangle whose vertices are (3,-2, 1), (2, 3, -1)
and (4, 5, 0).
3. The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are
(3, –5, 7) and (–1, 7, – 6), respectively, find the coordinates of the point C.
4. Find the distance distance between the points (1,-3,4) and (-4,1,2) .
5. Find the distance between the points (2, –1, 3) and (–2, 1, 3).
Three marks questions:
1. Derive the Distance between Two Points P(x1 , y1, z1) and Q ( x2, y2, z2)
2. Find the equation of the set of points P such that its distances from the points A (3, 4,-5)
and B (-2, 1, 4) are equal.

Question Bank: Department of School Education (Pre University ) 71

3. Find the equation of the set of points P such that PA2 + PB2 = K2, where A and B are the
points (3, 4, 5) and (-1, 3, -7) respectively.
4. Show that the points P (-2, 3, 5), Q (1, 2, 3) and R (7, 0, -1) are collinear.
5. Are the points A(3,6,9), B(10,20,30) and C(25,-41,5),the vertices of a right angled
triangle
6. Show that the points (-1,2,1),(1,-2,5),(4,-7,8) and (2,-3,4) are the vertices of a
parallelogram.
7. Show that the points (0,7,-10),(1,6,-6) and (4,9,-6) are the vertices of an isosceles
triangle.
8. Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and
B (-4, 0, 0) is equal to 10.
9. Show that the points (0, 7, 10), (–1, 6, 6) and (– 4, 9, 6) are the vertices of a right angled
triangle.
10. Show that the points A (1, 2, 3), B (–1, –2, –1), C (2, 3, 2) and D (4, 7, 6) are the vertices
of a parallelogram ABCD, but it is not a rectangle.
11. If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6),Q (– 4, 3b, –10)
and R(8, 14, 2c), then find the values of a, b and c.
12. If A and B be the points (3, 4, 5) and (–1, 3, –7), respectively, find the equation of the set
of points P such that PA2 + PB2 = k2, where k is a constant.
13. Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and
B (-4, 0, 0)is equal to 10
14. centroid of a triangle ABC is (1, 1, 1).If the coordinates of A and B are (3,-5, 7) and
(-1, 7,-6) respectively, find the coordinates of the point C. (S)
15. Find the coordinates of the point which trisect the line segment joining the points
P(4,2,-6) and Q(10,-16,6). (S)

Question Bank: Department of School Education (Pre University ) 72

CHAPTER -12
LIMITS AND DRIVATIVES
MCQ /FB
1. If f(x) = sinx, then lim
𝜋−
𝑓(𝑥) is
𝑥→
2
(A) 0 (B) 1 (C) -1 (D) 2.
2. If f(x) = x + cos x, then lim+ 𝑓(𝑥) is
𝑥→0
(A ) 0 (B) 1 (C) -1 (D) 2.
1
3. lim ( 2 ) is
𝑥→0𝑥
(A)1 (B) 0 (C) 1 (D) Limit does not exists.
2
4. lim (𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 + ⋯ . +𝑎𝑛−1 𝑥 𝑛−1
+ 𝑎𝑛 𝑥 ) is
𝑛
𝑥→𝑎
(A) a (B) f(a) (C) 0 (D) 𝑎𝑛 𝑎𝑛 .
5. lim (𝑥 4 − 𝑥 2 + 1) is
𝑥→1
(A) -1 (B) 0 (C) 1 (D) 2.
6. lim ( x ( x + 1) ) =
x →3
(A) 12 (B) 9 (C) 6 (D) 3
7. lim (𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 1) is
10 9 8 7 6 5 4 3 2
𝑥→−1
(A) 11 (B) 0 (C) 1 (D) -1.
𝑥 2 +1
8. lim𝑥→1 ( ) is
𝑥+100
1 2 1 3
(A) (B) (C) (D) .
50 101 100 101
𝑥 − 2 ,𝑥 < 0
9. If f(x) = ,then lim 𝑓(𝑥) is
𝑥 + 2, 𝑥 > 0 ⃑
𝑥→0
(A) 0 (B) 2 (C) -2 (D) Limit does not exists.
𝑥 − 2 ,𝑥 < 0
10. If f(x) = ,then lim+ 𝑓(𝑥) is
𝑥 + 2, 𝑥 > 0 𝑥→0
(A) 0 (B) 2 (C) -2 (D) Limit does not exists.
𝑥 + 2 ,𝑥 ≠ 0
11. If f(x) = ,then lim− 𝑓(𝑥) is
0, 𝑥 = 0 𝑥→0
(A) 0 (B) 2 (C) -2 (D) Limit does not exists.
sin 𝑥
12. lim𝜋 is equal to
𝑥→ 𝑥
2
𝜋 2
(A)1 (B) 2 (C)0 (D) 𝜋.
22
13. log 𝑥→𝜋 (𝑥 − ) is
7
22
(A) 0 (B) 𝜋 − 7 (C) 1 (D) 𝜋.
𝑥 10 +𝑥 5 +1
14. lim𝑥→−1 ( ) is
𝑥−1
1 1
(A) (B) 0 (C) − 2 (D) Limit does not exists.
2
 ax + bx + c 
2
15. lim  =
 cx + bx + a 
x →1 2

(A) 𝑎 + 𝑏 + 𝑐 (B) 0 (C) 1 (D) -1/2.
𝑐𝑜𝑠𝑥
16. lim𝑥→0 ( ) is
𝜋−𝑥
1
(A) (B) 0 (C) 1 (D) Limit does not exists.
𝜋

Question Bank: Department of School Education (Pre University ) 73

17. lim (𝑥 𝑠𝑒𝑐𝑥) is
𝑥→0
(A) 1 (B) 0 (C) -1 (D) Limit does not exists.
18. lim(𝝅𝒓𝟐 ) is
𝑟→1
(A) 𝝅 (B) 0 (C) 1 (D) Limit does not exists.
4𝑥+3
19. lim [ ] is
𝑥→4 𝑥−2
𝟏𝟗 19
(A) (B) 2 (C) 0 (D) Limit does not exists
𝟒
sin 𝑎𝑥
20. lim [ ] is
𝑥→0 𝑏𝑥
𝑏 𝒂
(A) 𝑎 (B) 1 (C) 0 (D) 𝒃
tan 𝑥
21. lim [ ] is
𝑥→0 𝑥
𝜋
(A) 0 (B) 1 (C)2 (D) Limit does not exists.
22. The derivative at x = 2 of the function f(x) = 3x.
(A) 2 (B) 6 (C) 3 (D) 0.
23. The derivative of f(x) = 3 at x = 0
(A) 3 (B) 1 (C) 6 (D) 0.
24. The derivative of f(x) = 10x w. r. t x
(A) 10𝑥 (B) x (C) 10 (D) 0.
1
25. The derivative of f(x) = w. r. t x
𝑥
1 −1
(A) 𝑥 (B) 𝑥 2 (C) −𝑥 2 (D) -x.
26. The derivative of f(x) = 𝑥 w. r. t x
2

(A) 𝑥 (B) 2x (C) 𝑥 2 (D) 0.
27. The derivative of f(x) = 𝑥 at x =100
(A) 100 (B) 99 (C) 1 (D) 0.
28. The derivative of f(x) = 6𝑥 100
− 𝑥 + 𝑥 w. r. t x
55

(A) 6𝑥 99 − 𝑥 54 + 1 (B) 600𝑥 99 − 55𝑥 54 + 1
(C) 600𝑥 99 − 55𝑥 54 + 𝑥 (D) 600𝑥 99 + 55𝑥 54 − 1
29. The derivative of f(x) = 99𝑥 at x =100
(A) 100 (B) 99 (C) 1 (D) 9900
30. The derivative of f(x) = 1 + 𝑥 + 𝑥 + ⋯ . +𝑥
2 50
at x =1.
(A) 50 (B) 1200 (C) 1 (D) 1275.
3
31. The derivative of f(x) = (2𝑥 − ) w. r. t x
2
1
(A) 𝑥 (B) 2x (C) 2 (D) 2.
32. The derivative of f(x) = 𝑥 (5 + 3x) w. r. t. x
−3

(A) −15𝑥 −4 − 6𝑥 −3 (B) −15𝑥 −4 (C) −10𝑥 −3 (D) −15𝑥 −2 − 10𝑥 −1 .
33. The derivative of f(x) = (−𝑥) w. r. t x
−1
1 −1
(A) 𝑥 2 (B) 𝑥 2 (C) −𝑥 2 (D) -x.
1 1
+
34. lim 𝑥 2
equals
𝑥→2 𝑥+2
1 −1
(A)−1 (B) 4 (C) 4 (D) -1/8.
sin 𝑎𝑥
35. lim equals
𝑥→0 𝑏
𝑎 𝑏
(A)0 (B) 𝑏 (C)𝑎 (D) 1
36. lim− 𝑓(𝑥), 𝑤ℎ𝑒𝑟𝑒 𝑓(𝑥) = |𝑥 − 5|
𝑥→5
(A)0 (B) 5 (C) 1 (D) does not exists

Question Bank: Department of School Education (Pre University ) 74

𝑑𝑦
37. If 𝑦 = 4√𝑥 then 𝑑𝑥 =
−1⁄ 1 −1⁄ 1 −1⁄
(A)𝑥 2 (B) 2 𝑥 2 (C) 2𝑥 ⁄2 (D)2𝑥 2
𝑥 3 −23
38. lim equals
𝑥→2 𝑥−2
(A)3 (B) 12 (C) 8 (D) 4.
𝑥 5 +32
39. lim equals
𝑥→−2 𝑥+2
(A)16 (B) 80 (C) 64 (D) does not exists
1 1
+ −1
40. Statement 1: lim ( 𝑥 3
) =9
𝑥→−3 𝑥+3
𝑥 𝑛 −𝑎𝑛
Statement 2: lim 𝑥−𝑎 = 𝑛 𝑎𝑛−1 .
𝑥→𝑎
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
𝑥
41. Assertion (A): lim [ ] = 1. Is
𝑥→0 𝑠𝑖𝑛𝑥
𝑓(𝑥) lim 𝑓(𝑥)
Reason(R): lim [𝑔(𝑥)] = lim 𝑔(𝑥) 𝑥→𝑎
𝑥→𝑎 𝑥→𝑎
(A) Both Assertion (A) and Reason (R) are true.
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
|𝑥|
,𝑥 < 0
𝑥
42. If f(x) = {|𝑥| }
,𝑥 > 0
𝑥
Statement I: lim+ 𝑓(𝑥) is -1
𝑥→0
Statement II: lim− 𝑓(𝑥) is 1
𝑥→0
(A) Both I and II are true (B) only I is true
(C) only II is true (D) Both I and II are false
43. 1. Limit of sum of two functions is sum of the limits of the functions
2. Limit of product of two functions is product of the limits of the functions
3. Limit of a function at a point is the common value of the left and right hand limits, if
they coincide.
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
𝑓(𝑎+ℎ)−𝑓(𝑎)
44. 1.The derivative of a function f at a is defined by 𝑓 (𝑎) = lim ( )
ℎ→0 ℎ
𝑓(𝑥+ℎ)−𝑓(𝑥)
2. Derivative of a function f at any point x is defined by 𝑓 / (𝑥)
= lim ( )
ℎ→0 ℎ
𝑑(𝑢𝑣) 𝑑𝑢 𝑑𝑣
3. 𝐹𝑜𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑢 𝑎𝑛𝑑 𝑣 𝑑𝑥 = 𝑢 𝑑𝑥 + 𝑣 𝑑𝑥
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

Question Bank: Department of School Education (Pre University ) 75

5 2 5 −1
45. Statement 1: The derivative of f(x) = √𝑥 2 w. r. t x is ( √𝑥 3 )
5
1 𝑛
Statement 2: The derivative of 𝑥 𝑛 w. r. t x is − 𝑥 𝑛+1
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
46. Assertion (A): The derivative of 𝑓(𝑥) = 𝑥 at x = 1 is 0.
Reason(R): The derivative of x −n w. r. t x is − nx −n−1
(A) Both Assertion (A) and Reason (R) are true.
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
47. Match Column I and Column II

Column I Column II
a) lim [
𝑥
]. 0
𝑥→0 𝑐𝑜𝑠 𝑥
b) The derivative of |𝑥| at x =-2 ii) −1
c) The derivative of the function 2𝑥 at 𝑥 = 1 iii)2

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-ii, c-iii.
48. For the figure given below, consider the following statements 1 and 2

Statement 1: The given function limit exists at x =1,x = 2 and x = 4
Statement 2: : lim 𝑓(𝑥) is 2
𝑥→2
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
49. For the figure given below, consider the following statements 1 and 2

Statement 1: lim 𝑓(𝑥) is 2 Statement 2: lim 𝑓(𝑥) dose not exists.
𝑥→0 𝑥→2
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

Question Bank: Department of School Education (Pre University ) 76

50. For the figure given below, consider the following statements 1 and 2

Statement 1: lim− 𝑓(𝑥) =1 Statement 2: lim+ 𝑓(𝑥) = -1.
𝑥→0 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
51. For the figure given below

which of the following is correct
A) lim− 𝑓(𝑥) =0 B) lim+ 𝑓(𝑥) =0 C) lim 𝑓(𝑥) =0. D) lim 𝑓(𝑥) does not exists
𝑥→0 𝑥→0 𝑥→0 𝑥→0
52. For the figure given below

which of the following is incorrect
A) lim(𝑦) =1 B) lim (𝑦) =2 C) lim (𝑦) =0. D) lim (𝑦) =-2
𝑥→1 𝑥→2 𝑥→−1 𝑥→−2
53. The derivative of the function f(x) = 𝑥 3 − 27 at x = –1 is ..............
𝑥
54. lim [ ] =................
𝑥→0 𝑡𝑎𝑛𝑥
𝑠𝑖𝑛𝑥 3
55. lim [ ] =................
𝑥→0 𝑥
3𝑥 2 +2𝑥+4
56. lim𝑥→0 ( )=........
4𝑥 2 +2𝑥+2
57. lim𝑥→0 (𝑥 𝑐𝑜𝑠𝑒𝑐𝑥)=........
58. The derivative of f(x) = 10 at x = 10 is .........
1
−
59. The derivative of f(x) = 𝑥 𝑘 𝑤 . 𝑟 𝑡 𝑥 𝑖𝑠 𝑘𝑥 2 , 𝑡ℎ𝑒𝑛 𝑘 = ⋯ … ….

Question Bank: Department of School Education (Pre University ) 77

60. Match Column I and Column II

Column I Column II
a)
𝑑9𝑠𝑖𝑛𝑥
. i)−9𝑠𝑒𝑐 2 𝑥
𝑑𝑥
b)
𝑑−9𝑡𝑛𝑥𝑥 ii)- 9𝑥 −10
𝑑𝑥
c) 𝑑𝑥
𝑑𝑥 −9 iii)9cosx

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-ii, c-iii.
Two marks questions:
3𝑥 2 −𝑥−10
1. Evaluate lim [ ]
𝑥→2 𝑥 2 −4
𝑥 4 −81
2. Evaluate lim [2𝑥 2−5𝑥−3]
𝑥→3
1 1
+
3. Evaluate lim [𝑥+2] 𝑥 2
𝑥→−2
𝑥 3 −4𝑥 2 +4𝑥
4. Evaluate lim [ ]
𝑥→2 𝑥 2 −4
𝑥 2 −4
5. Evaluate lim [𝑥 3−4𝑥 2+4𝑥]
𝑥→2
𝑥 3 −2𝑥 2
6. Evaluate lim [𝑥 2−5𝑥+6]
𝑥→2
𝑥 15 −1
7. Evaluate lim [𝑥 10 −1]
𝑥→1
(𝑥+1)5 −1
8. Evaluate lim [ ]
𝑥→0 𝑥
1
𝑧 ⁄3 −1
9. Evaluate lim [ 1⁄ ]
𝑧→1 𝑧 6 −1
√1+𝑥−1
10. Evaluate lim [ 𝑥 ]
𝑥→0
sin 4𝑥
11. Evaluate lim [sin 2𝑥]
𝑥→0
sin 𝑎𝑥
12. Evaluate lim [sin 𝑏𝑥]
𝑥→0
sin (𝜋−𝑥)
13. Evaluate lim [ 𝜋(𝜋−𝑥) ]
𝑥→𝜋
cos 2𝑥−1
14. Evaluate lim [ ]
𝑥→0 cos 𝑥−1
𝑎𝑥+𝑥 cos 𝑥
15. Evaluate lim [ ]
𝑥→0 𝑏 sin 𝑥
sin 𝑎𝑥+𝑏𝑥
16. Evaluate lim [𝑎𝑥+sin 𝑏𝑥]
𝑥→0
17. Evaluate lim[𝑐𝑜𝑠𝑒𝑐 𝑥 − cot 𝑥]
𝑥→0
1−cos 𝑥
18. Evaluate lim [ ].
𝑥→0 𝑥

Three marks questions:
2𝑥 + 3, 𝑥 ≤ 0
1. Find lim 𝑓(𝑥) and lim 𝑓(𝑥), where 𝑓(𝑥) = {
𝑥→0 𝑥→1 3(𝑥 + 1), 𝑥 > 0
2
𝑥 − 1, 𝑥 ≤ 1
2. Find lim 𝑓(𝑥), where 𝑓(𝑥) = {
𝑥→1 −𝑥 2 − 1, 𝑥 > 1
|𝑥|
, 𝑥≠0
3. Find lim 𝑓(𝑥), where 𝑓(𝑥) = { 𝑥
𝑥→0 0, 𝑥 = 0

Question Bank: Department of School Education (Pre University ) 78

𝑎 + 𝑏𝑥, 𝑥 < 1
4. Suppose 𝑓(𝑥) = {4, 𝑥=1
𝑏 − 𝑎𝑥, 𝑥 > 1
and if lim 𝑓(𝑥) = 𝑓(1) what are possible values of a and b?
𝑥→1
𝑥−2 1
5. Evaluate lim [𝑥 2−𝑥 − 𝑥 3 −3𝑥 2 +2𝑥]
𝑥→1
√4+𝑥−2
6. Evaluate lim [ ].
𝑥→0 𝑥
7. Find the derivative of 𝑓(𝑥) = 𝑥 2 , from first principal.
1
8. Find the derivative of 𝑓(𝑥) = 𝑥 , from first principal.
9. Find the derivative of 𝑓(𝑥) = 𝑠𝑖𝑛 𝑥, from first principal.
10. Find the derivative of 𝑓(𝑥) = 𝑐𝑜𝑠 𝑥, from first principal.
11. Find the derivative of 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥, from first principal.
12. Find the derivative of 𝑓(𝑥) = 𝑥 3 − 27, from first principal.
13. Find the derivative of 𝑓(𝑥) = (𝑥 − 1)(𝑥 − 2), from first principal.
1
14. Find the derivative of 𝑓(𝑥) = 𝑥 2 , from first principal.
𝑥+1
15. Find the derivative of 𝑓(𝑥) = 𝑥−1, from first principal.
16. Find the derivative of 𝑓(𝑥) = 𝑠𝑖𝑛2 𝑥, w.r.t x.
17. Find the derivative of 6𝑥100 − 𝑥 55 + 𝑥, w.r.t x.
18. Find the derivative of 𝑓(𝑥) = 𝑥 5 (3 − 6𝑥 −9 ), w.r.t x.
19. Find the derivative of 𝑓(𝑥) = 𝑥 −4 (3 − 4𝑥 −5 ), w.r.t x.
20. Find the derivative of f(x)= sin x cos x
1
21. Find the derivative of (𝑖) 𝑓(𝑥) = 𝑥 + 𝑥 , (𝑖𝑖) 𝑓(𝑥) = 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 (𝑖𝑖𝑖) 𝑓(𝑥) = 𝑥𝑠𝑖𝑛𝑥 from first
principal.
22. Find the derivative of 𝑓(𝑥) = sec 𝑥 w.r.t x.
23. Find the derivative of 𝑓(𝑥) = 5sec 𝑥 + 4cos 𝑥 w. r. t x.
24. Find the derivative of 𝑓(𝑥) = cosec 𝑥 w.r.t x.
25. Find the derivative of 𝑓(𝑥) = 3cot 𝑥 + 5cosec 𝑥 w. r. t x.
26. Find the derivative of 𝑓(𝑥) = 5sin 𝑥 − 6cos 𝑥 + 7 w. r. t x.
27. Find the derivative of 𝑓(𝑥) = 2tan 𝑥 − 7sec 𝑥 w.r.t x.
28. Find the derivative of 𝑓(𝑥) = 𝑐𝑜𝑡𝑥 w.r.t x.
29. Find the derivative of 𝑓(𝑥) = 𝑠𝑖𝑛2𝑥 w. r. t x.
30. Find the derivative of 𝑓(𝑥) = (𝑎𝑥 2 + 𝑠𝑖𝑛𝑥)(𝑝 + 𝑞𝑐𝑜𝑠𝑥) w.r.t x.
31. Find the derivative of 𝑓(𝑥) = sec 𝑥+cosec x w.r.t x.
Four marks questions:
1. Find the derivative of 𝑓(𝑥) = (5𝑥 3 + 3𝑥 − 1)(𝑥 − 1)
2. Find the derivative of 𝑓(𝑥) = 𝑥 −3 (5 + 3𝑥)
𝑥+1
3. Find the derivative of 𝑓(𝑥) = 𝑥−1
2𝑥+3
4. Find the derivative of 𝑓(𝑥) = 𝑥−2
𝑥 5 −𝑐𝑜𝑠 𝑥
5. Find the derivative of 𝑓(𝑥) = 𝑠𝑖𝑛 𝑥
𝑥+𝑐𝑜𝑠 𝑥
6. Find the derivative of 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥
𝑎+𝑏 𝑠𝑖𝑛 𝑥
7. Find the derivative of 𝑓(𝑥) = 𝑐+𝑑 𝑐𝑜𝑠 𝑥
𝑥
8. Find the derivative of 𝑓(𝑥) = 1+𝑡𝑎𝑛 𝑥
2 𝑥2
9. Find the derivative of 𝑓(𝑥) = 𝑥+1 − 3𝑥−1.
𝑠𝑖𝑛𝑥+𝑐𝑜𝑠 𝑥
10. Find the derivative of 𝑓(𝑥) = 𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠𝑥

Question Bank: Department of School Education (Pre University ) 79

4𝑥+𝑠𝑖𝑛𝑥
11. Find the derivative of 𝑓(𝑥) = 3𝑥+7𝑐𝑜𝑠𝑥
𝑠𝑒𝑐𝑥−1
12. Find the derivative of 𝑓(𝑥) = 𝑠𝑒𝑐𝑥+1
sin (𝑥+𝑎)
13. Find the derivative of 𝑓(𝑥) = 𝑐𝑜𝑠𝑥
Five marks questions:
sin 𝑥
1. Prove that lim [ 𝑥 ] = 1, where x in radian.
𝑥→0
𝑥 𝑛 −𝑎𝑛
2. Prove that ,for any positive integer n, lim [ 𝑥−𝑎 ] = 𝑛𝑎𝑛−1 , and hence
𝑥→0
𝑥 15 −1
evaluate lim [𝑥 10 −1]
𝑥→1

Question Bank: Department of School Education (Pre University ) 80

CHAPTER -13
STATSTICS
MCQ/FB
1. The Mean, Median, and Mode are called
A) Measures of Dispersion B) Measure of central tendency
C) Measure of variability D) Co-efficient of variation.
2. Range is the one of the
A) Measures of Dispersion B) Measure of central tendency
C) Variation D)None of these.
3. If 𝑥1 , 𝑥2 , 𝑥3 , − − − − 𝑥𝑛 are n observations, then mean is
𝑛+1 𝑡ℎ 𝑛 𝑡ℎ ∑𝑥
A) ( 2 ) 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 B) ( 2) 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 C) 𝑛 𝑖 D) 𝑥𝑛 − 𝑥1
4. The mean of the scores given as 6, 7, 10, 12, 13, 4, 8, 12.
A) 12 B) 9 C) 8 D) 10.
5. Mean of the first n natural Numbers is
n +1 n( n + 1 ) n2 − 1 1 − n2
(A) (B) (C) (D)
2 2 2 n
6. If 𝑥1 , 𝑥2 , 𝑥3 , − − − − 𝑥𝑛 are n observations in ascending order. If the number of
observations is odd, then the median is
𝑛 𝑡ℎ 𝑛+3 𝑡ℎ
A) (2) 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 B) ( 2 ) 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛
𝑛+2 𝑡ℎ 𝑛+1 𝑡ℎ
C) ( 2 ) 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 D) ( 2 ) 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛.
7. Median for the data 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21. is
(A) 8 (B) 10 (C) 9 (D) 9.5
8. Range, Mean deviation, Quartile deviation, Standard deviation are commonly used as
(A) Measures of central tendency (B) Measures of the nature of variates
(C) Measures of dispersion (D) Measures of the size of the variates.
9. The mean of the absolute values of the deviations of the observations from central
value ‘a is
(A) Range (B) Quartile deviation
(C) Mean deviation (D) Standard deviation.
10. Mean deviation for n observations 𝑥1 , 𝑥2 , 𝑥3 , − − −𝑥𝑛 from their mean 𝑥̅ is given by

( ) ( ) ( )
n
1 n n
1 n
(A)  xi − x (B)  xi − x (C)  xi − x  xi − x
2
(D)
i =1 n i =1 i =1 N i =1
11. Mean deviation about mean for the following data a, a, a,.......a (n times). Is
𝑎
(A) a (B) na (C) 𝑛 (D) 0
12. Mean of the squares of the deviations from mean is
(A) Standard deviation (B) Quartile deviation
(C) Mean deviation (D) Variance
13. The mean deviation of the data 5,7,9,11,13,15 from mean is
A) 2 B) 2.57 C)3 D) 3.75
14. The positive square-root of the mean of the squares of the deviations from mean is
(A) Standard deviation (B) Quartile deviation
(C) Mean deviation (D) Variance
15. Let 𝑥1 , 𝑥2 , 𝑥3 , − − −𝑥𝑛 be n observations and 𝑥̅ be their arithmetic mean. The formula
for the standard deviation is given by

( x − x) ( x − x)
2

(
(A)  xi − x )
2
(B)
i
(C)
i
(D)
x + x 2
i 2

n n n
Question Bank: Department of School Education (Pre University ) 81

16. If 𝑥1 , 𝑥2 , 𝑥3 , − − − − 𝑥𝑛 are n observations with mean 𝑥̅ , mean of the following n
observations (𝑥1 + 𝑎), (𝑥2 + 𝑎),---------(𝑥𝑛 + 𝑎) is
x x +a
(A) x + a (B) x − a (C) + a (D) n
n n
17. The sum of Mean deviation from the mean is
(A) less than the sum of the mean deviation from the median
(B) more than the sum of the mean deviation from the median
(C) equal to the sum of the mean deviation from the median
(D) None of these.
(
18. If  xi − x ) is small , this indicates that the observations x1, x2, x3,...,xn are
2

(A) close to the mean 𝑥̅ and there is a lower degree of dispersion
(B) close to the mean 𝑥̅ and there is a higher degree of dispersion
(C) not close to the mean 𝑥̅ and there is a higher degree of dispersion
(D) None of these.
19. If each observation is multiplied by a constant k, the variance of the resulting
observations becomes
(A) k times the original variance (B) 𝑘 2 times the original variance
(C) original variance (D) k+ times the original variance
20. If each observation is adding (or subtracting) a positive number k, the variance of the
resulting observations becomes
(A) k times the original variance (B) 𝑘 2 times the original variance
(C) original variance (D) k + original variance
21. The variance of n observations is 𝜎 , if each observation is multiplied by k, then new
2

variance is
A) k 2 2 B) k 2 C) k D) None of these.
22. The variance of 20 observations is 5. If each observation is increased by 2, then the new
variance of the resulting observations is
(A) 7 (B) 20 (C) 5 (D) 10
23. If the sum of squares of the deviations of 10 observations taken from their mean is 4.9,
then their Standard Deviation is
(A) 4.9 (B) 0.7 (C) 0.5 (D) 5.3
24. The variance of 20 observations is 5. If each observation is multiplied by 2, then the
new variance of the resulting observations is
(A) 7 (B) 20 (C) 5 (D) 10
25. The mean of 100 observations is 50 and their standard deviation is 5.The sum of all
squares of the observations is
(A) 50000 (B) 250000 (C) 252500 (D) 255000
26. Let 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 be the observation with mean m and standard deviation s. The
standard deviation of the observations 𝑘𝑥1 , 𝑘𝑥2 , 𝑘𝑥3 , 𝑘𝑥4 , 𝑘𝑥5
s
A) k+s B) C) ks D) s
k
27. Variance for first 10 natural numbers is
99 100
(A) 99 (B) 12 (C) 9.9 (D) 12
28. Most powerful measure of dispersion is
A) Mean deviation about mean B) Mean deviation about median
C) Standard deviation D) Variance

Question Bank: Department of School Education (Pre University ) 82

29. The variance is
(A) Positive square of Standard deviation
(B) Square of the Mean deviation about mean
(C) Square root of Mean deviation
(D) Square of standard deviation
30. The standard deviation of certain data is 4, then the variance is
(A) 2 (B) 4 (C) 8 (D) 16
31. A study in Statistics that helps to interpret the variability of data is known’s as .....
(A) Standard Deviation (B) The measurer of Central tendency
(C) The measurer of dispersion (D) None of the above
32. Statement 1. In absolute Measure of Dispersion, the Square root of variation is known
as Standard Deviation
Statement 2 . The Arithmetic average of the Absolute Deviation of a series called Mean
Deviation
Statement 3 . Variance measure of dispersion ignores the signs of deviations
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
33. Statement 1: There is a set of 5 numbers with variance = 10. If each number is
divided by 2, then the new Variance is 2.5.
Statement 2: if each observation is multiplied by a constant k, the variance of the
resulting observations becomes k2 times the original variance.
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
34. Assertion (A): There is a set of 10 observation s with variance = 20. If each observation
is subtracted by 5, then the new Variance is 15
Reason(R): Adding (or subtracting) a positive number to each observation of a group
does not affect the variance.
(A) Both Assertion (A) and Reason (R) are true.
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
35. Statement 1: The mean of 10 observations were calculated as 35, by a student who
took by mistake 40 instead of 60 for one observation. The correct mean is 37.
1
Statement 2: Let x1, x2, x3, ..., xn be n observations. Then mean 𝑥̅ = 𝑛 ∑𝑛𝑖=1 𝑥𝑖
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
Question Bank: Department of School Education (Pre University ) 83

36. Statement I: The 𝑅𝑎𝑛𝑔𝑒 𝑜𝑓 6 observations 60, 70, 30, 80, 65, 55. 𝑖𝑠 50
Statement II: Range of a series = Maximum value – Minimum value
Which of the above statements are correct?
(A) Both I and II are true (B) only I is true
(C) only II is true (D) Both I and II are false
37. Assertion (A): The mean deviation about the mean for the 8 observations 13, 9, 7, 11,
16, 4, 8, 12 with Mean 10 is 3
Sum of absolute values of deviations from central value a
Reason(R): M.D.(a) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
(A) Both Assertion (A) and Reason (R) are true.
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
38. 1. The measure of dispersion which uses only two observations is Range.
2. The measure of dispersion which uses all observations is Mean deviation.
Which of the above statements are correct?
(A) Both I and II are true (B) only I is true
(C) only II is true (D) Both I and II are false
39. The exact Mean deviation of the values 12, 15, and 18 is_______
40. The sum of the deviations from mean ( 𝑥̅ ) is zero. The mean deviation about the mean
is_______
41. The sum of the absolute values of the deviations from mean of 8 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 is 32.
The mean deviation about the mean is_______
42. The mean deviation for the 10 observations is 5 and every observations are decreased
by 2,then mean Deviation of new observation is _____
43. The mean deviation from median for 6,6,6,6,6 is _____
44. Mean and Standard deviation of 10 observations are 40 and 4respectivly ,then sum of
squares of the Observations is ______
45. The deviation taken from median are -6, -2 ,3, 4, 5. The mean deviation from median
is _____
46. The variance of 19, 21, 23, 25 and 27 is 8. The variance of 14, 16, 18, 20 and 22
is_____
47. If the maximum value in a series is 25 and its range is 15, the maximum value of the
series is_____
48. Standard deviation of 3 observations a, b and c are 2.5, then a+3,b+3 and c+3 is _____
49. If 2,5,8,12,13 are 5 observations.
Match Column I and Column II

Column I Column II
a) 𝑀𝑒𝑎𝑛 i)0
b) mean deviation ii) 8
c) sum of the values of the deviations iii)3.6

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-ii, c-iii.

Question Bank: Department of School Education (Pre University ) 84

50. If 𝑥1 , 𝑥2 , 𝑥3 , − − − − 𝑥𝑛 are n observations. Match Column I and Column II

Column I Column II
a) Variance
i) 
(x − x ) i
2

n
b) mean deviation 1 n
ii)  xi − x
n i =1
c ) Standard deviation
( )
n
iii)𝑛  xi − x
1 2

i =1

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-ii, c-iii.

TWO MARKS QUESTION:
1. Write the mean of the given data: 6, 7, 10, 12, 13, 4, 8, 12. (U)
2. Write the mean of the given data: 4, 7, 8, 9, 10, 12, 13, 17. (U)
3. Write the mean of the given data: 38,70,48,40,42,55,63,46,54,44. (U)
4. Compute the variance and Standard deviation of the following observations of marks of 5
students. Class: marks out of 25: 8, 12, 13, 15, 22. (U)
5. Co-efficient of variation and the standard deviation of certain distribution is 60 and 21
respectively. Find the arithmetic mean of the arithmetic mean. (A)
6. Find the variance of 6, 8, 10, 12, 14. (U)
7. The standard deviation of certain data is 4. Find the variance. (U)
8. The mean of 200 scores is 48 and their standard deviation n is 3. Find the sum and sum of
the squares of scores. (A)
9. Find the variance and standard deviation of the five observations: 11, 14,15,17,18 (U)
10. Find the mean and variance for the following data: 2, 4,5,6,7,8,17. (U)
11. Find the mean and variance for the following data:6,8,10,12,14,16,18,20,22,24. (U)
12. The scores of a batsmen in 10 matches are 38,70,48,34,42,55,63,46,54,44. Find standard
deviation and variance. (A)
13. Find the mean and variance for the following data: 6, 7, 10, 12, 13, 4, 8, 12. (U)
14. Find the mean deviation about the mean for the following data:6,7,10,12,13,4,8,12.(U)
15. Find the mean deviation about the mean for the following
data:12,3,18,17,4,9,17,19,20,15,8,17,2,3,16,11,3,1,0.5.
16. Find the mean deviation about the mean for the following data:4,7,8,9,10,12,13,17. (U)
17. Find the mean deviation about the mean for the following
data:38,70,48,40,42,55,63,46,54,44. (U)
FIVE MARKS QUESTION:
1. Find the mean deviation about the median for the following
data:13,17,16,14,11,13,10,16,11,18,12,17. (U)
2. Find the mean deviation about the median for the following
data:36,72,46,60,45,53,46,51,49. (U)
3. Find the mean deviation about the median for the following
data:3,9,5,3,12,10,18,4,7,19,21. (U)
4. Find the mean deviation about the mean for the following data: (U)
xi 2 5 6 8 10 12
fi 2 8 10 7 8 5

Question Bank: Department of School Education (Pre University ) 85

5. Find the mean deviation about the median for the following data: (A)

xi 3 6 9 12 13 15 21 22
fi 3 4 5 2 4 5 4 3
6. Find the mean deviation about the mean for the following data: (A)
Marks obtained 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Number of
students 2 3 8 14 8 3 2

7. Calculate the mean deviation about median for the following data: (A)
Class 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 6 7 15 16 4 2

8. Find the mean deviation about the mean for the following data: (A)
xi 5 10 15 20 25
fi 7 4 6 3 5
9. Find the mean deviation about the mean for the following data: (A)
xi 10 30 50 70 90
fi 4 24 28 16 8
10. Find the mean deviation about the median for the following data: (A)
xi 5 7 9 10 12 15
fi 8 6 2 2 2 6

11. Find the mean deviation about the mean for the following data: (A)
Marks obtained 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Number of students 2 3 8 14 8 3 2
12. Find the mean deviation about the median for the following data: (A)
xi 15 21 27 30 35
fi 3 5 6 7 8
13. Find the mean deviation about the mean for the following data: (A)
Income per 0-100 100- 200- 300- 400- 500- 600- 700-
day 200 300 400 500 600 700 800
No.of persons 4 8 9 10 7 5 4 3

14. Find the mean deviation about the mean for the following data: (A)
Height in cms 95-105 105-115 115-125 125-135 135-145 145-155

Number of boys 9 13 26 30 12 10
15. Find the mean deviation about the median for the following data: (A)
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of 6 8 14 16 4 2
Girls
16. Calculate the mean deviation about median age for the age distribution of 100 persons given
be Find the mean deviation about the mean for the following data: (A)
Age 16-20 21-25 26-30 31-35 36-40 41-45 46-50 51-55

Number 5 6 12 14 26 12 16 9

17. Find the variance and standard deviation for the following data: (A)
xi 4 8 11 17 20 24 32
fi 3 5 9 5 4 3 1
18. Calculate the mean, variance and standard deviation for the following distribution: (A)
Class 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Frequency 3 7 12 15 8 3 2
Question Bank: Department of School Education (Pre University ) 86

19. Find the Variance of the following data: 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 (A)
20. Find the standard deviation for the following data: (A)
xi 3 8 13 18 23
fi 7 10 15 10 6
21. Find the mean and variance for First n natural numbers. (S)
22. Find the mean and variance for first 10 multiples of 3. (A)
23. Find the mean and variance for the following data: (A)
xi 6 10 14 18 24 28 30
fi 2 4 7 12 8 4 3
24. Find the mean and variance for the following data: (A)
xi 92 93 97 98 102 104 109
fi 3 2 3 2 6 3 3

25. Find the mean and standard deviation using short-cut method: (A)
xi 60 61 62 63 64 65 66 67 68
fi 2 1 12 29 25 12 10 4 5

26. Calculate the mean, variance and standard. (A)
Class 30-40 40-50 50-60 60-70 70-80 80-90 90-100
Frequency 3 7 12 15 8 3 2
27. Find the mean, variance for the following frequency distributions : (A)
Class 0-30 30-60 60-90 90-120 120-150 150-180 180-210
Frequency 2 3 5 10 3 5 2
28. Find the mean, variance for the following frequency distributions : (A)
Class 0-10 10-20 20-30 30-40 40-50
Frequency 5 8 15 16 6
29. Find the mean, Variance and standard deviation . (A)
Height in ms 70-75 75-80 80-85 85-90 90-95 95-100 100-105 105-110 110-115

Number of boys 3 4 7 7 15 9 6 6 3

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

Question Bank: Department of School Education (Pre University ) 87

CHAPTER-14
Probability
MCQ/FB
1. Two coins are tossed once. The number of possible outcomes.
(A) 2 B) 4 C) 8 D) 6 .
2. If E is an event “Number of tails is atleast one” associated with a sample space “two
coins are tossed once”. The number of outcomes in E is
A) 4 B) 3 C) 2 D) 1 .
3. If E is an event “Second toss is not head ” associated with a sample space “two coins
are tossed once”. The number of outcomes in E is
A) 4 B) 3 C) 2 D) 1 .
4. If E is an event “Number of tails is atmost two ” associated with a sample space “two
coins are tossed once”. The event E is
A) {TT} B) {HT, TH, TT} C) {HH, HT, TH, TT} D) φ .
5. The number of simple events corresponding to the sample space “two coins are tossed
once” is
A) 1 B) 2 C) 3 D) 4 .
6. Which of the following is incorrect
A) An event which has one sample points of a sample space is called impossible event.
B) An event which has all sample point of a sample space is called sure event.
C) Let two events A and B associated with sample space S is said to be mutually
exclusive events if 𝐴 ∩ 𝐵 = 𝜙.
D) Events 𝐸1 , 𝐸2 , . . . . .. 𝐸𝑛 are said to Mutually exhaustive events
if 𝐸1 ∪ 𝐸2 ∪ . . . . ∪ 𝐸𝑛 = 𝑆 , where 𝑆 is the sample space.
7. Which of the following event is simple event
A) a number less than 7 B) a number greater than 7
C) an even number greater than 4 D) a number not less than 3.
8. If sample space 𝑆 = {1,2,3,4,5,6} and events 𝐴 = {2,4,6}𝑎𝑛𝑑 𝐵 = {1,3,5}, 𝑡ℎ𝑒𝑛 events A and B
are
A) Mutually exhaustive but not exclusive events
B) Mutually exclusive but not exhaustive events
C) Mutually exclusive and exhaustive events
D) None of these
9. A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even
Number , then E and F are
A) Mutually exhaustive but not exclusive events
B) Mutually exclusive but not exhaustive events
C) Mutually exclusive and exhaustive events
D) Neither mutually exclusive nor exhaustive events
10. Which of the following is incorrect
A) Complementary event or ‘not event’ : The set 𝐴 or S – A.
B) Event A or B: The set A ∪ B
C) Event A and B: The set A ∩ B
D) Event A and not B: The set B – A
11. Let S be the sample space of a random experiment. The probability P is a real valued
function whose domain is
A) Set of real number B) [ 0, 1 ]
C) Power set of S D) Set of integers.

Question Bank: Department of School Education (Pre University ) 88

12. Let S be the sample space of a random experiment. The probability P is a real valued
function whose range is
A)Set of real number B) [ 0, 1 ] C) Power set of S D) Set of integers.
13. If A is any event associated with a sample space 𝑆 then
A) 0 ≤ 𝑃(𝐴) ≤ 1 B) 0 < 𝑃(𝐴) < 1 C) 𝑃(𝐴) ≥ 1 D) 𝑃(𝐴) ≤ 0.
14. If A and 𝐵 are mutually exclusive events, then
A) P(A ∪ B) = 1 B) P(A ∩B ) = 0
C) P(not A) = 1 – P(B) D) P(A ∩B) = P(A)+ P(B).
15. If A and 𝐵 are mutually exhaustive events, then
A) P(A ∪ B) = 1 B) P(A ∩B ) = 0
C) P(not A) = 1 – P(B) D) P(A ∩B) = P(A)+ P(B).
16. Which of the following is incorrect
A) If 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑎𝑛𝑦 𝑡𝑤𝑜 𝑒𝑣𝑒𝑛𝑡𝑠 𝑡ℎ𝑒𝑛 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
B) 𝑃(𝐴′ ) = 1 − 𝑃(𝐴)
C) 𝑃(𝐴 ∩ 𝐵 ′ ) = 𝑃(𝐴) − 𝑃(𝐴 ∩ 𝐵)
D) 𝑃(𝐴′ ∩ 𝐵 ′ ) = 1 + 𝑃(𝐴 ∪ 𝐵)
17. Which of the following cannot be valid assignment of probabilities for outcomes of
sample spaces S = {𝜔1, 𝜔2 , 𝜔3 , 𝜔4 , 𝜔5 , 𝜔6 }
Assignment 𝜔1 𝜔2 𝜔3 𝜔4 𝜔5 𝜔6
1 1 1 1 1 1
A)
6 6 6 6 6 6
1 2 1 1 1 1
B) − −
8 3 3 3 4 3
1 1 1 1 1 3
C)
12 12 6 6 6 2
D) 0.1 0.2 0.3 0.4 0.5 0.6

2
18. If 11 is the probability of an event A, what is the probability of the event ‘notA’.
10 9 11 11
A) B) C) D) .
11 11 9 10
3 1
19. Given P(A)= 5 and P(B) = 5. Then P(A or B), if A and B are mutually exclusive events.
2 3 4 1
A) B) C) D) .
5 5 5 5
1 1 1
20. If P(A) = 4 , P(B) = 2 and P(A 𝑎𝑛𝑑 B) = 8, then P(A or B) is
3 7 5 1
A) 4 B) 8 C) 8 D) 4
21. A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) =0.16.
Then P(not A)
A) 0.58 B) 0.52 C) 0.72 D) 0.90.
22. A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16,
then P(A or B) is
A) 0.74 B) 0.90 C) 1.6 D) 0.64.
23. A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35,
then P(A but not B) is
A) 0.34 B) 0.19 C) 0.88 D) 1.13
24. A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35,
then P(B but not A) is
A) 0.34 B) 0.19 C) 0.88 D) 1.13
25. Events E and F are such that P(not E or not F) = 0.25,then P(E and F) is
A) 0 B) 1 C) 0.75 D) 0.25

Question Bank: Department of School Education (Pre University ) 89

26. Which of the following is incorrect
A) For any event E, P (E) ≥ 0 B) P (S) = 1
C) P (∅) = 0. D) 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵)
27. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, then the probability that the card will be a diamond
1 1 1 12
A) 4 B) 13 C) 52 D) 13
28. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, then the probability that the card will be a non ace
1 1 1 12
A) 4 B) 13 C) 52 D) 13
29. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, then the probability that the card will be not a black card
1 1 1 12
A) 4 B) 13 C) 2 D) 13.
30. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, then the probability that the card will be not a not a diamond
1 1 1 3
A) B) C) D) .
4 13 2 4
31. A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are
similar in shape and size. A disc is drawn at random from the bag. Then the probability
that it will be either red or blue.
7 4 2 3
A) 9 B) 9 C) 9 D) 9.
32. A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are
similar in shape and size. A disc is drawn at random from the bag. Then the probability
that it will be not blue.
4 2 2 3
A) 9 B) 9 C) 3 D) 4.
1 1 1
33. If P(A) = 3 , P(B) = 5 and P(A 𝑎𝑛𝑑B) = 15, then P(A or B) is
3 7 5 1
A) 4 B) 8 C) 8 D) 4
34. If P(A) = 0.35 , P(A∩B) = 0.25 and P(A ∪B) = 0.6, then P(B) is
A) 0.75 B) 0.5 C) 0.46 D) 0.65
35. If P(A) = 0.5 , P(B) = 0.35 and P(A 𝑜𝑟 B) = 0.7, then P(A and B) is
A) 0.3 B) 0.85 C) 0.15 D) 0.3
36. The number of sample point in impossible event is …………………
37. The number of sample points in elementary event is -------
38. A coin is tossed and a die is thrown. The number of sample points for this random
experiment is ---
39. The number of sample points for the random experiment “a coin is tossed and then a die
is rolled only in case a head is shown on the coin” is ---------
40. A box contains 2 red and 3 identical white balls. Two balls are drawn at random in
succession without replacement. The number of sample points for this random
experiment is --------
41. An experiment consists of recording boy-girl composition of families with two children.
The number of sample points for the random experiment whether it is a boy or a girl in
the family is ----- (U)
42. A die is thrown repeatedly until a six comes up The number of sample points for this
random experiment is –
2
43. If is the probability of an event A, then the probability of the event ‘not A’ is ----(S)
11

Question Bank: Department of School Education (Pre University ) 90

44. If S = { w1,w2 ,w3 ,w4 } is the sample space of an experiment, then find the value of
P ( w1 ) + P ( w2 ) + P ( w3 ) + P ( w4 ) = ---- (U)
1 1
45. If S = { w1,w2 ,w3 ,w4 } is the sample space of an experiment and P( w1 ) = , P( w2 ) = ,
4 3
1
P( w3 ) = , 𝑡ℎ𝑒𝑛 𝑃(𝑤4 ) =---- (U)
6
46. The probability of an impossible event is …………………
47. The probability of a sure event is ………………
1 𝑥
48. If 𝑃(𝐴) = 3 and P(not A) = 6 , then x is ………….
49. Statement 1: A die has two faces each with number ‘1’, three faces each with number ‘3’ and
1
one face with number ‘5’. If die is rolled once, then P(3) is 2 .
Statement 2: Number of outcomes favorable is 3 and total possible outcomes is 6.
(A) Both statements are true
(B) Statement I is true and statement II is false
(C ) Statement I is false and statement II is true
(D) Both statements are false
50. Statement 1: A card is drawn from a well shuffled deck of 52 playing cards. The probability that
1
the card drawn is an ace of hearts is 52.
Statement 2: Out of 52 playing cards 4 ace cards .
(A) Both statements 1 and 2 are true
(B) Statement 1 is true and statement 2 is false
(C ) Statement 1 is false and statement 2 is true
(D) Both statements1 and 2 are false
51. Assertion (A): The probability of drawing a red or a card with a face from a deck of 52 cards is
8
13
Reason(R): The total favorable cards: 26 (red cards) + 12 (face cards) – 6 (red face cards) = 32
favorable cards.
(A) Both Assertion (A) and Reason (R) are true.
(B) Both Assertion (A) and Reason (R) are false.
(C) Assertion (A) is true and Reason (R) is false.
(D) Assertion (A) is false and Reason (R) is true.
52. A bag contains 5 green and 3 blue balls. Two balls are picked at random.
5𝐶2 + 3𝐶
1.The probability that both are of the same colour = 8 𝐶2
2

5𝐶2 × 3𝐶
2 . The probability that both are of the different colours = 8𝐶 2
2

3 . Number of ways in which 2 balls can be picked = 8𝐶2
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
2
53. Statement 1: The probability of getting a king or ace from the deck of 52 cards is 13
Statement 2: The probability of getting a king or ace =
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑘𝑖𝑛𝑔 𝑜𝑟 𝑎𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒𝐟𝐫𝐨𝐦 𝐭𝐡𝐞 𝐝𝐞𝐜𝐤 𝐨𝐟 𝑐𝑎𝑟𝑑𝑠
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
Question Bank: Department of School Education (Pre University ) 91

54. Statement 1: The limit of probability is −1 𝑡𝑜 1
Statement 2: A number is selected from numbers 1 to 25. The probability that it is a prime
9
number is 25
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for
Statement 1
55. Ravi visits three cities (A, B, and C) in a random order. Match Column I and Column II
Column I Column II
a) Probability that A first and B last? 1
i) 2
b)Probability that A before B 2
ii) 3
c ) Probability that A either first or 1
iii)6
second
Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-ii, c-iii.
56. A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35
Match Column I and Column II
Column I Column II
a) P(𝐴/ ) 23
i) 50
b) P(A ∪ B) 22
ii) 25
c ) P(𝐴 ∩ 𝐵)/ 13
iii)20

Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-ii, c-iii.
57. A die is thrown . Match Column I and Column II
Column I Column II
a) Probability that number more than 6 will i) 1
appear
b)Probability that number less than 6 will ii) 0
appear
c ) Probability that prime number will appear 1
iii)2
Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-ii, c-iii.
58. Match Column I and Column II
Column I Column II
a) P(𝐴 )
/
i) 𝑃(𝑆)
b) P(𝐴/ ∪ A) ii) 1 − 𝑃(𝐴)
c ) P(𝐴/ ∩ A) iii)𝑃(∅)
Choose the correct answer from the options given below:
A) a-iii , b-i, c-ii B) a-ii, b-iii, c-i C) a-ii, b-i, c-iii D) a-i, b-ii, c-iii.
Question Bank: Department of School Education (Pre University ) 92

59. For the figure given below, consider the following statements 1 and 2

Statement 1:From figure 1, 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) – 𝑃(𝐴 ∩ 𝐵)
Statement 2: From figure 2, A and 𝐵 are mutually exclusive events
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

60. For the figure given below.

3
1. P(𝑈𝑛𝑙𝑖𝑘𝑒𝑙𝑦 ) = 4
1
2. 𝑃(𝐸𝑞𝑢𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒𝑙𝑦 ) =
2
1
3. 𝑃(𝐿𝑖𝑘𝑒𝑙𝑦 ) =
4
Which of the above statements are correct?
(A) 2 only (B) 2 and 3 only (C) 1 and 3 only (D) 1, 2 and 3

Two Mark Questions
1. Consider the experiment of rolling a die. Let A be the event ‘getting a prime number’, B
be the event ‘getting an odd number’. Write the sets representing the events
(i) A or B (ii) A and B (ii) A but not B (iv) ‘not A’ (iv) neither A nor B. (S)
2. A die is rolled. Let E be the event, “die shows 4” and F be the event ,”die shows even
number”. Write the events E and F. Check whether E and F mutually exclusive. (S)
3. A die is rolled. Let E be the event, “die shows a prime” and F be the event, ”die shows a
multiple of 3”. Write the events E and F. Check whether E and F mutually exclusive. (S)
4. A coin is tossed twice. Write the events, A: “at least one head appears”, B:”at most one
tail appears”. Check whether they are exhaustive events. (S)
5. A coin is tossed thrice. Write the events, A:”no head appears”, B: “at least two head
appears”. Check whether the events are mutually exclusive. (S)
6. A die is thrown describe the events A: a number less than 7 and C: a multiple of 3.
Find A – C.
7. A die is thrown describe the events A: a number less than 7 and B: a number greater
than 7 find A ∪ B & A ∩ B .
Question Bank: Department of School Education (Pre University ) 93

8. A die is thrown describe the events E: an even number greater than 4 and F: a number
not less than 3 , E ∩ F and E ∩ F′. (S)
9. Three coins are tossed simultaneously. Write the events
(i) A:”getting 3 heads” (ii)B: ”getting a tail on two coins”. Are they mutually exclusive? (S)
10. Three coins are tossed once. Write the events A:”a tail shows on the first coin” and B:”at
least one head shows”. Check whether the events A and B are exhaustive. (S)
11. Three coins are tossed. Write two events which are mutually exclusive. (U)
12. Three coins are tossed. Write three events which are mutually exclusive and exhaustive.
13. Three coins are tossed. Write two events which are not mutually exclusive. (U)
14. Three coins are tossed. Write two events which are mutually exclusive but not
exhaustive. (U)
15. Consider the random experiment “rolling a die once”. Write the events, A: “an even
number greater than 4 appears” and B:”a number not less than 3 appears”. Check
whether they are mutually exclusive. (S)
16. Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤5.
Describe the events (i) A′ (ii) not B (iii) A or B (iv) A and B (v) A but not C (vi) B or C
(vii) B and C (viii) A ∩B′∩C′
3 1
17. Given P (A) = and P (B) = .Find P (A or B), if A and B are mutually exclusive events.
5 5
(U)
2 3
18. Given P (A) = and P (B) = . If A and B are mutually exclusive events, find P(A or B).
7 7
(U)
19. A die is thrown. Find the probability of the event: “A prime number will appear”.
(U)
20. A die is thrown. Find the probability of the event: “An even number less than 4 will
appear”. (U)
21. A die is thrown. Find the probability of the event: “A number not less than 3 will
appear”. (U)
22. A coin is tossed twice. What is the probability that at least one tail occurs? (S)
23. A coin is tossed twice. What is the probability that at most one head occurs? (S)
24. Three coins are tossed at once. Find the probability of getting at least two heads? (S)
25. Two dice are thrown . Find the probability of the event,” the sum of the numbers which
come up on the dice is even”. (S)
Three Mark Question
1. Two dice are thrown .Then write the event, ” the sum of the numbers which come up on
the dice is greater than 8”. Also find the probability of the event. (S)
2. A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed.
find the probability that the sum of numbers that turn up is 3. (S)
3. A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed.
find the probability that the sum of numbers that turn up is 12. (S)
4. A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that
letter is (i) a vowel (ii) a consonant (S)
5. Four cards are drawn from a well – shuffled deck of 52 cards. What is the probability of
obtaining 3 diamonds and one spade? (S)
6. On her vacations Veena visits four cities A, B, C and D in random order. What is the
probability that she visits A before B? (S)
Question Bank: Department of School Education (Pre University ) 94

7. 4 cards are drawn from a well shuffled deck of 52 cards. What is the probability of
obtaining 3 diamonds and one spade? (U)
8. A and B are two events such that P(A)=0.42,P(B)=0.48 and P(A and B )=0.16. Determine
(i) P(not A) (ii)P( A or B). (U)
9. A and B are two events such that P(A)=0.54 ,P(B)=0.69 and P(A∩B)=0.35.
Find (i) P( A  B ) (ii) P( B  A ) (S)
10. The probabilities that at least one of the events A and B occurs is 0.8. If A and B occur
simultaneously with probability 0.1, then find P( A ) + P( B ) . (S)
11. In class XI of a school 40% of the students study Mathematics and 30% study Biology.
10% of the class study both Mathematics and Biology. If a student is selected at random
from the class, find the probability that he will be studying Mathematics or Biology. (A)
1 1 1
12. If E and Fare events such that P( E ) = , P( F ) = and P( EandF ) = then find (i) P( EorF )
3 6 9
(ii)P( not E and not B).
13. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, calculate the probability that the card will be
(i) a spade (ii) a red card (iii) not a King (S)
14. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, calculate the probability that the card will be
(i) a heart (ii) a queen (iii) not a black card (S)
15. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, calculate the probability that the card will be
(i) a diamond (ii) not an ace (iii) not a club. (S)
16. A committee of two persons is to be selected from 2 men and 2 women. What is the
probability that the committee will have (i) no man? (ii) One man? (A)
17. A group of two persons is to be selected from 3 men and 2 women .what is the
probability that the group will have (i) no woman? (ii) at least one woman ? (A)
18. A team of three persons is to be selected from 2 boys and 3 girls. What is the probability
that the team will have (i) three girls (ii) at most one boy? (A)
19. A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed.
Find the probability that the sum of numbers that turn up is ( i ) 3 ( ii ) 12 (S)
Five Mark Questions
1. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, calculate the probability that the card will be (i) a diamond (ii) not an ace card
(iii) a black card (iv) not a diamond (v) not a black card. (A)
2. A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are
similar in shape and size. A disc is drawn at random from the bag. Calculate the
probability that it will be (i) red, (ii) yellow, (iii) blue, (iv) not blue, (v) either red or blue.(A)
3. Two students Anil and Ashima appeared in an examination. The probability that Anil will
qualify the examination is 0.05 and that Ashima will qualify the examination is 0.10. The
probability that both will qualify the examination is 0.02. Find the probability that
(i) Both Anil and Ashima will not qualify the examination.
(ii) At least one of them will not qualify the examination and
(iii) Only one of them will qualify the examination. (A)
4. A committee of two persons is selected from two men and two women. What is the
probability that the committee will have (a) no man? (b) one man? (c) two men? (A)
5. A die is thrown, find the probability of following events:(i) A prime number will appear,
(ii) A number greater than or equal to 3 will appear,(iii) A number less than or equal to
one will appear,(iv) A number more than 6 will appear, (v) A number less than 6 will
appear. (A)
Question Bank: Department of School Education (Pre University ) 95

6. A card is selected from a pack of 52 cards.(a) How many points are there in the sample
space?
(b) Calculate the probability that the card is an ace of spades.(c) Calculate the probability
that the card is (i) an ace (ii) black card. (A)
7. Three coins are tossed once. Find the probability of getting (i) 3 tails (ii) exactly two tails
(iii) no tail (iv) at most two tails (A)
8. Three coins are tossed once. Find the probability of getting (i) 3 heads (ii) 2 heads
(iii) at least 2 heads (iv) at most 2 heads (v) no head (A)
9. A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16.
Determine (i) P(not A), (ii) P(not B) and (iii) P(A or B) (A)
10. In Class XI of a school 40% of the students study Mathematics and 30% study Biology.
10% of the class study both Mathematics and Biology. If a student is selected at random
from the class, find the probability that he will be studying Mathematics or Biology.
11. In an entrance test that is graded on the basis of two examinations, the probability of a
randomly chosen student passing the first examination is 0.8 and the probability of
passing the second examination is 0.7. The probability of passing at least one of them is
0.95. What is the probability of passing both?
12. The probability that a student will pass the final examination in both English and Hindi
is 0.5 and the probability of passing neither is 0.1. If the probability of passing the
English examination is 0.75, what is the probability of passing the Hindi examination?
13. In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC
and NSS. If one of these students is selected at random, find the probability that (i) The
student opted for NCC or NSS.(ii) The student has opted neither NCC nor NSS.(iii) The
student has opted NSS but not NCC.
14. On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the
probability that she visits (i) A before B? (ii) A before B and B before C? (iii) A first and B
last? (iv) A either first or second? (v) A just before B?
15. Find the probability that when a hand of 7 cards is drawn from a well shuffled deck of 52
cards, it contains (i) all Kings (ii) 3 Kings (iii) at least 3 Kings.
16. A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are
drawn from the box, what is the probability that(i) all will be blue? (ii) at least one will be
green?
17. A die has two faces each with number ‘1’, three faces each with number ‘2’ and one face
with number ‘3’. If die is rolled once, determine P (2) (ii) P(1 or 3) (iii) P(not 3)
18. In a certain lottery 10,000 tickets are sold and ten equal prizes are awarded. What is the
probability of not getting a prize if you buy (a) one ticket (b) two tickets (c) 10 tickets.
19. Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are
among the 100 students, what is the probability that (a) you both enter the same
section?(b) you both enter the different sections?
20. A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩B) = 0.35.
Find (i) P(A ∪ B) (ii) P(A´ ∩ B´) (iii) P(A ∩ B´) (iv) P(B ∩ A´) (S)
21. Two friends Ramesh and Arati appeared in a competitive examination. The probability
that Ramesh will qualify the examination is 0.08 and that Arati will qualify the
examination is 0.15. The probability that both will qualify the examination is 0.03. Find
the probability that only one of them will qualify the examination. (A)

Question Bank: Department of School Education (Pre University ) 96

CHAPTER-1
SETS
1 2 3 4 5 6 7 8 9 10
C D D C C B B D C B
11 12 13 14 15 16 17 18 19 20
D B B B B D A D D C
21 22 23 24 25 26 27 28 29 30
D D C D B A C B A C
31 32 33 34 35 36 37 38 39 40
C B A D C C D A A B
41 42 43 44 45 46 47 48 49 50
C C C B C A C A A A
51 52 53 54 55 56 57 58 59 60
B A C D C C D B 9 5
61 62 63 64 65
7 1 4 2 1

CHAPTER-2
RELATIONS AND FUNCTIONS
1 2 3 4 5 6 7 8 9 10
B B D D D C A B D D
11 12 13 14 15 16 17 18 19 20
C D C A B C D D A D
21 22 23 24 25 26 27 28 29 30
A B A B D A B C A 9
31 32 33 34 35 36 37 38 39 40
4 8 8 16 C A C D B C
41 42 43 44 45 46 47 48 49 50
B C D C B B D A A C

CHAPTER-3
Trigonometric Functions

1 2 3 4 5 6 7 8 9 10
C A D A D B B B D D
11 12 13 14 15 16 17 18 19 20
A D A C A C C C B C
21 22 23 24 25 26 27 28 29 30
B A D B D D B A B A
31 32 33 34 35 36 37 38 39 40
C D A B D ½ 0 1 0.34 𝟓
- 𝟏𝟐
41 42 43 44 45 46 47 48 49 50
𝟒𝟒
- 𝟏𝟐𝟓 A D C A A A A A B
51 52 53 54 55 56 57 58 59 60
C D C B B C B D C D

Question Bank: Department of School Education (Pre University ) 97

CHAPTER-4
COMPLEX NUMBER

1 2 3 4 5 6 7 8 9 10
A B A D D A A B B B
11 12 13 14 15 16 17 18 19 20
C C D C A D C A C A
21 22 23 24 25 26 27 28 29 30
C C B C A A C C C B
31 32 33 34 35 36 37 38 39 40
D A C A C 2 4 1 1 1
41 42 43 44 45 46 47
2 3 0 𝟏𝟎𝟗𝟖 D D A
𝟏𝟔

CHAPTER-5
LINER INEQUALITIES

1 2 3 4 5 6 7 8 9 10
D B D C C C C C A B
11 12 13 14 15 16 17 18 19 20
D C B D C A A C B A
21 22 23 24 25 26 27 28 29 30
B A C A D C B C D B
31 32 33 34 35 36 37 38 39 40
B A B 2 0 -1/2 C A D A

CHAPTER-6
PERMUTATIONS AND COMBINATIONS
1 2 3 4 5 6 7 8 9 10
C D B A D C A D C A
11 12 13 14 15 16 17 18 19 20
C D B C B B B C D A
21 22 23 24 25 26 27 28 29 30
B D D B B A D A C D
31 32 33 34 35 36 37 38 39 40
D B C D A A A C C B
41 42 43 44 45 46 47 48 49 50
C D B C 24 36 0 1 66 10
51 52 53 54 55 56 57 58 59 60
D A A B A 4 A C A B
61 62 63 64 65
B C A B B

Question Bank: Department of School Education (Pre University ) 98

CHAPTER-7
BINOMIAL THEOREM
1 2 3 4 5 6 7 8 9 10
B C B C D A A D D D
11 12 13 14 15 16 17 18 19 20
D C C B C B C 7 2 -4
21 22 23 24 25 26 27 28 29 30
121 21 1 A A D A A C C

CHAPTER-8
SEQUENCES AND SERIES
1 2 3 4 5 6 7 8 9 10
C C C A A B B A B D
11 12 13 14 15 16 17 18 19 20
D A C B A A B B C D
21 22 23 24 25 26 27 28 29 30
A C C A C C B B A -8
31 32 33 34 35 36 37 38 39 40
5 162 √3 8 6 13 1 B D B
41 42 43 44 45 46 47 48 49 50
A A B A B A A B D D
51 52 53 54 55
B C D C B

CHAPTER-9
STRAIGHT LINES
1 2 3 4 5 6 7 8 9 10
D D B C C A D B B B
11 12 13 14 15 16 17 18 19 20
C B A D B B B C A D
21 22 23 24 25 26 27 28 29 30
C C B C B B C B B B
31 32 33 34 35 36 37 38 39 40
A A A C D A A C B A
41 42 43 44 45 46 47 48 49 50
B B C B A C D 0 √3 -1
51 52 53 54 55 56 57 58 59 60
𝟓 𝟐 𝟐 5 -4 3 -6 1 -5 y=0
−
𝟕 𝟑 𝟓
61 62 63 64 65 66 67 68 69 70
x=0 𝟐 𝟔 6 B A A C B C
−
𝟕 𝟓
71 72 73 74 75
B A A B C

Question Bank: Department of School Education (Pre University ) 99

CHAPTER - 10
CONIC SECTIONS
1 2 3 4 5 6 7 8 9 10
A D C D B D A D A C
11 12 13 14 15 16 17 18 19 20
B B B D B A B A D A
21 22 23 24 25 26 27 28 29 30
A C D A B A C B B A
31 32 33 34 35 36 37 38 39 40
A C C B B D C C B D
41 42 43 44 45 46 47 48 49 50
B D C B A A B B D A
51 52 53 54 55 56 57 58 59 60
D C C D B C A D C A
61 62 63 64 65 66 67 68 69 70
B A C C 9𝝅 𝟗 2 2√3 𝟒
3
𝟐 √ 𝟓
2

CHAPTER 11
Introduction to three-Dimensional Geometry
1 2 3 4 5 6 7 8 9 10
B C B B C A D C C C
11 12 13 14 15 16 17 18 19 20
A C B D A A B B B B
21 22 23 24 25 26 27 28 29 30
C C C A B C A A B B
31 32 33 34 35 36 37 38 39 40
C C B D D C A B C B
41 42 43 44 45 46 47 48 49 50
A A 8 0 0 8TH 5√𝟐 2 1 C

Question Bank: Department of School Education (Pre University ) 100

CHAPTER -12
LIMITS AND DRIVATIVES
1 2 3 4 5 6 7 8 9 10
B B D B C A C B C B
11 12 13 14 15 16 17 18 19 20
B D B C C A B A B D
21 22 23 24 25 26 27 28 29 30
B C D C B B C B B D
31 32 33 34 35 36 37 38 39 40
D A A B A A D B B C
41 42 43 44 45 46 47 48 49 50
A D D A C D D A A D
51 52 53 54 55 56 57 58 59 60
D C 3 1 1 2 1 0 ½ A

CHAPTER -13
STATSTICS
1 2 3 4 5 6 7 8 9 10
B A C B A D C C C B
11 12 13 14 15 16 17 18 19 20
D D C A C A A A B C
21 22 23 24 25 26 27 28 29 30
A C B B C C B C D A
31 32 33 34 35 36 37 38 39 40
C A C D C A A A 2 0
41 42 43 44 45 46 47 48 49 50
4 5 0 16160 4 8 5 2.5 B A

CHAPTER-14
PROBABILITY
1 2 3 4 5 6 7 8 9 10
B B C C A A C C D D
11 12 13 14 15 16 17 18 19 20
C B A B A D A B C C
21 22 23 24 25 26 27 28 29 30
A A B A C D A D C D
31 32 33 34 35 36 37 38 39 40
A C B B C 0 1 12 7 4
41 42 43 44 45 46 47 48 49 50
4 ∞ 9 1 ¼ 0 1 4 A B
11
51 52 53 54 55 56 57 58 59 60
A C C D A D C C C A

Question Bank: Department of School Education (Pre University ) 101

Study Materials
Notes

Model Papers Class 6 Notes

Sample Papers Class 7 Notes
Half Yearly Sample Papers Class 8 Notes

Class 9 Notes
Important Resources
Class 10 Notes
Periodic Table
Class 11 Notes
Writing Skills / Formats

Maps of India / World Class 12 Notes

Books and Solutions

NCERT Books
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HC Verma Chapter Wise Solutions
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