Karnataka 2nd PUC Question Bank for Maths Basic

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

QUESTION BANK

The revised Question Bank for 2025–26 has been prepared in adherence to the syllabus prescribed by the Pre-
University Board of Karnataka. The objective of this Question Bank is to serve as a comprehensive tool that
enhances students' understanding and preparation for examinations. A great deal of effort has been put into
making this question bank a valuable resource for students.

To assist students, questions are organized chapter-wise. They are divided into one-mark, two-mark, three-
mark, four-mark and five-/six-mark categories. Under the one-mark section, both multiple-choice questions
and very short answer types (VSA) are included, in compliant with the latest blueprint.

By providing a wide range of questions and structured content, this question bank aims to facilitate effective
study habits and reinforce key concepts. We hope that students will leverage this resource to boost their
confidence and achieve academic success. Students are encouraged to utilize this question bank to the fullest.

Wishing all students the best in their studies!

The total weightage from each chapter is given below:

Sl.No Chapter Name Total Marks

1. Matrices and Determinants 13
2. Permutations and Combinations 8
3. Probability 3
4. Binomial Theorem 4
5. Partial Fractions 5
6. Mathematical Logic 6
7. Ratios and Proportions 8
8. Bill Discounting 5
9. Stocks and Shares 3
10. Learning Curve 5
11. Linear Programming Problem 5
12. Sales Tax and VAT 3
13. Heights and Distances 4
14. Compound Angles, Multiple and Submultiple Angles and Transformation 7
Formulae
15. Circles 6
16. Parabola 4
17. Limits ans Continuity 7
18. Differential Calculus 8
19. Application of Derivatives 5
20. Indefinite Integrals 5
21. Definite Integrals and Application to Areas 6
Total marks 120

Chapter 1: MATRICES AND DETERMINANTS

One Mark Questions:

1 2
1. If 𝐴 = [ ] then adj A is
3 −4
1 2 −4 2 −4 −2 −1 −2
a) [ ] b) [ ] c) [ ] d) [ ]
3 4 3 −1 −3 1 3 4
[E]
1 2 4 3 −4 −1
2. If 𝐴 = [ ] and 𝐵 = [ ] then (A+B) is
−1 3 −2 1 5 −2
4 2 −3 4 −2 3 −4 2 3 4 −2 −3
a) [ ] b) [ ] c) [ ] d) [ ]
0 8 4 0 8 −4 0 −8 4 0 −8 −4
[E]
400 404
3. The value of | | is
408 412
a) -23 b) 23 c) 32 d) -32
[A]
1 −3 5
4. If 𝐴 = [ ] then 5𝐴′ will be
6 2 4
6 2 4 5 30 5 −15 25 6 1
a) [ ] c) [ ]
1 −3 5 b) [−15 10] 30 10 20 d) [2 −3]
25 20 4 5
[A]
2 3 −1
5. If 𝐴 = [ ] then 𝐴′ is
1 −1 0
−2 −3 1 2 1
a) [ ]
−1 1 0 b) [ 3 −1]
−1 0
2 3 −1 1 −1 0
c) [ ] d) [ ]
−1 1 0 2 3 −1
[E]
1 −3
6. If 𝐴 = [ ] then the matrix 2A is
2 4
2 −6 2 4 8 −6 2 4
a) [ ] b) [ ] c) [ ] d) [ ]
4 8 −6 8 4 2 1 −3
[A]

3 𝑥
7. If | | = 0 then the value of x is
4 5
15 b) 9 4 d) 13
a) c)
4 13

[D]
1 −2
8. If 𝐴 = [ ] then the matrix 2A is
3 4
2 −4 2 4 2 −6 2 4
a) [ ] b) [ ] c) [ ] d) [ ]
6 8 −6 8 4 2 1 −3
[E]
3200 3201
9. The Value of | | is
3202 3203
a) 4 b) 0 c) -2 d) 2
[A]
2011 2012
10.The value of | | is
2013 2014
a) 2 b) 0 c) -2 d) 1
[A]
𝑥 3
11.If | | = 0 then the value of x is
12 𝑥
a) ±12 b) 12 c) ±6 d) 6
[D]
1 −2
12.If 𝐴 = [ ] then the matrix 3A is
3 4
3 −6 3 4 3 −6 3 4
a) [ ] b) [ ] c) [ ] d) [ ]
6 8 −6 8 9 12 1 −3
[E]
2 −𝑥
13.If 𝐴 = [ ]. Find 𝐴 + 𝐴′ .
𝑥 −7
2 −6 2 4 4 −6 4 0
a) [ ] b) [ ] c) [ ] d) [ ]
6 −7 −6 7 9 −14 0 −14
[A]

1 2 4 3 −4 −1
14.If 𝐴 = [ ] and 𝐵 = [ ] then A-B is
−1 3 −2 1 5 −2
−2 6 5 2 −2 3 −4 2 3 4 −2 −3
a) [ ] b) [ ] c) [ ] d) [ ]
−2 −2 0 0 8 −4 0 −8 4 0 −8 −4
[A]
4
15.If 𝐴 = [ ] and 𝐵 = [3 1 5] then AB is
−3
−12 4 20 12 −2 3 1 2 3 12 4 20
a) [ ] b) [ ] c) [ ] d) [ ]
−9 −2 0 9 8 −4 −9 3 4 −9 −3 −15
[D]
1
16.If [2 𝑥 2] [4] = [3] then x is
2
1 −3 3 −1
a) b) c) d)
4 4 4 4

[D]
3 −1
17.If 𝐴 = [ ] then 𝐴2 is
2 −4
7 1 9 1 9 −1 7 −1
a) [ ] b) [ ] c) [ ] d) [ ]
−2 14 4 16 4 16 2 −14
[A]
−4 −3
18.If 𝐴 = [ ] then adj A is
−2 −1
1 2 −4 2 1 −2 −1 3
a) [ ] b) [ ] c) [ ] d) [ ]
3 4 3 −1 −3 1 2 −4
[E]
1 2
19.If 𝐴 = [ ] then 𝐴2 is
0 −3
1 −4 1 4 1 2 1 −1
a) [ ] b) [ ] c) [ ] d) [ ]
0 9 0 −9 4 16 2 6
[A]
𝑥
4
20.If | | = 0 then the value of x is
𝑥4
a) ±4 b) 4 c) ±2 d) 2
[D]

3 𝑥
21.If | | = −2 then the value of x is
4 5
17 −17 15 13
a) b) c) d)
4 4 4 4

[D]
1 −1
22.If 𝐴 = [ ] then the value of |𝐴| is
2 4
a) 6 b) 7 c) 8 d) 9
[A]
23.A square matrix A is singular matrix if |𝐴 | is
a) 1 b) 2 c) 3 d) 0
[E]
1 −2
24.If 𝐴 = [ ] then find adj A
3 4
4 2 4 2 −4 −2 4 2
a) [ ] b) [ ] c) [ ] d) [ ]
3 1 −3 1 3 1 3 −1
[A]
6 𝑥+2 6 1
25.If [ ] = [ ] then the value of x is
2 4 2 4
a) -2 b) -1 c) 1 d) 2
[A]
3 𝑥
26.If [ ] is symmetric matrix then the value of x is
4 7
a) 3 b) 4 c) 7 d) 0
[A]
0 −7
27.If [ ] is a skew-symmetric matrix then the value of x is
7 𝑥
a) 7 b) -7 c) 1 d) 0
[D]
3 𝑥
28.If [ ] is symmetric matrix then the value of x is
7 9
a) 3 b) 7 c) 9 d) 0
[A]

Two Marks Questions:

1 3 −1 4 −1 2
1. If 𝐴 = [ ] and 𝐵 = [ ] Find 𝐴 − 3𝐵. [E]
−1 0 2 1 3 −2
2
2. If 𝐴 = [−1] and 𝐵 = [1 4 2]. Find AB. [D]
3
2 3 1 1 −2 4
3. If 𝐴 = [ ] and 𝐵 = [ ] Find 2𝐴 − 3𝐵. [E]
1 −2 0 1 3 2
2 3 2 𝑥−2 4 1
4. If [ ]+[ ]=[ ]. Find x and y. [A]
7 5 𝑦−1 5 7 10
4 5 2 𝑥−3 6 1
5. If [ ]+[ ]=[ ]. Find x and y. [A]
3 2 𝑦−4 1 2 3
2 3 1 −1
6. If 𝐴 = [ ] and 𝐵 = [ ] Find 𝐴′ 𝐵. [A]
−1 4 2 4
2 −1 𝑥 10
7. If [ ] [𝑦] = [ ] Find x and y. [A]
3 1 2
2 3 −1
8. If 𝐴 = [ ] Find 𝐴𝐴′ . [A]
1 −1 0
3 2
9. If 𝐴 = [ ]. Prove that 𝐴2 − 4𝐴 − 5𝐼 = 0, where I is unit matrix and 0 is the null matrix.
4 1
[D]
1 3
10.If 𝐴 = [ ]. Prove that 𝐴2 − 𝐴 − 3𝐼 = 0 [D]
1 0
2
11.Solve for x: [𝑥 3] + [5𝑥 1] = [6 4]. [D]
1 2 4 3 5 5
1 −1 3 2 3 1
12.If 𝐵 + 𝐴 = [ ] and 𝐵 − 𝐴 = [ ]. Find B [E]
2 3 4 3 4 2
3 −2 4 1 −1 4
13.If 𝐴 = [ ],𝐵 = [ ] verify that (𝐴 − 𝐵)′ = 𝐴′ − 𝐵′ . [A]
2 1 0 6 2 5
2 −2 3 −1
14.If 𝐴 = [ ] and 𝐵 = [ ]. Find AB. [E]
3 −1 −2 0
1
2 1 3
15.If 𝑋 = [ ] , 𝑌 = [0] 𝑎𝑛𝑑 𝑍 = [2 1]. Verify X(YZ)=(XY)Z. [D]
0 1 4
3

Three Marks Questions:

1. Solve by Cramer’s rule: 2x+3y=11; x-y=3. [A]
2. Solve by Cramer’s rule: 3x+4y=7; 7x-y=6. [A]
3. Solve by Cramer’s rule: 3x+2y=8; 4x-3y=5. [A]
4. Solve by Cramer’s rule: 2x+y=1; x-3y=4. [A]
5. Solve by Cramer’s rule: 5x-7y=3; 7x-5y=9. [A]
1 2 4 −3 1 0
6. If𝐴 = [ ] and 𝐵 = [ ] and 𝐶 = [ ] Verify that A(BC)=(AB)C. [D]
1 4 2 1 −2 4
2 3 −1 5
7. If 𝐴 = [ ] and 𝐵 = [ ]. Show that (𝐴𝐵)′ = 𝐵′ 𝐴′ . [D]
−4 1 6 2
2 3 1 −3
8. If A+B = [ ] and 2A-B = [ ]. Find A and B. [D]
4 1 2 2
7 0 3 0
9. Find matrix A and B if 𝐴 + 𝐵 = [ ] and 𝐴 − 𝐵 = [ ]. [E]
2 5 0 3
2 −1 4 −3
10.If 𝐴 = [ ] and 𝐵 = [ ]. Show that (𝐴𝐵)′ = 𝐵′ 𝐴′ . [E]
1 4 2 1
1 2 4 −3 1 0
11.If𝐴 = [ ] and 𝐵 = [ ] and 𝐶 = [ ] Verify that A(B+C)=AB+AC. [E]
1 4 2 1 −2 4
3
12.Find x if [𝑥 1] + [−2𝑥 2 3] = [−𝑥 4]. [D]
3 3 1 4 4 7
1 2
13.If 𝐴 = [ ] and 𝐴 + 2𝐵 = 𝐴2 , find B. [A]
3 4
1 5
14.Find x and y if [𝑥 𝑦] [ ] = [2 0]. [D]
−2 3
3 −1 4 2
15.If 2A+B = [ ] and A-2B = [ ]. Find A and B. [A]
−2 5 −1 5
2 −1
16.If 𝐴 = [ ] 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 that 𝐴2 − 4𝐴 + 3𝐼 = 0. [D]
−1 2
Five Marks Questions:

1. Solve by matrix method x+y+z = 5; 2x+y-z=2; 2x-y+z = 2. [A]
2. Solve by matrix method x+y-z = 1; 3x+y-2z=3; x-y-z = -1. [A]
3. Solve by matrix method 3x+y+2z = 3; 2x-3y-z=-3; x+2y+z = 4. [A]
4. Solve by matrix method x-y+2z = 3; 2x+z=1; 3x+2y+z = 4. [D]
5. Solve by matrix method x-y-2z = 3; 2x+y+z=5; 4x-y-2z = 1. [A]

Chapter 2: PERMUTATIONS AND COMBINATIONS

One mark Questions:
1. The number of ways 9 flowers of different can be strung together to form a garland is
9! 8!
a) 9! b) 8! c) d) [A]
2 2
2. The number of ways in which 10 people ca be seated around a table is
9! 10!
b) 9! b) 10! c) d) [E]
2 2
3. The number of straight lines and triangles that can be formed from 12 points in a plane on
which 5 are collinear is
a) 57 ,210 b ) 57 , 211 c) 56 , 210 d) 56 , 211 [A]
4. The number of diagonals in a decagon is
a) 45 b) 35 c) 55 d)10 [E]
5. The number of ways in which 6 people be selected out of 12 so that two
particular members are always excluded is
a) 12P6 b) 12P4 c) 10P 4 d) 10P6 [D]
6. 5P3: 5P2=
a ) 3:1 b) 1:3 c) 2:3 d) 3:2 [E]
8
7. The value of P3 is
a) 336 b) 56 c) 24 d) 8 [E]
5
8. If Pr = 60 then r
a) 4 b) 3 c) 2 d)5 [A]
9. 𝐼𝑓 𝑛𝐶4 = 𝑛𝐶5 find the value of n.

a ) 20 b) 9 c) 8 d) 10 [E]

10.Find the value of 8P3 [E]
11. In how many ways can 9 soldiers stand in a queue? [E]
12. In how many ways can four people occupy 6 vacant chairs. [A]
13. If 5Pr=60 find r. [A]
14. If nP3=210 Find n. [A]
15. If nP3: nP2=3:1 Find n. [A]
16. How many 3 letter words can be formed using only consonants but each only once. [E]

17. Find the number of permutations of the letters of the word MONDAY. [E]
18. How many 4 digit numbers can be formed using the digits 1, 2, 4, 5, 7, 8, 9 no digit being
repeated in any number. [E]
19. Find the total number of ways in which 8 different coloured beads can be strung together
to form a necklace. [A]
20. In how many ways can 9 flowers of different colours be strung together to form a
garland. [A]
21. In how many ways can 10 different precious stones be set to form a necklace. [A]
22. In how many ways can 10 people be seated around the table. [E]
23. In how many ways a committee of 5 can be chosen from 10 students. [E]
24. How many triangles can be formed from 8 non collinear points. [E]
25. How many straight lines can be formed from 10 points if no three of them
are collinear? [A]
26. Find the number of permutations of the letters of the word “MATHEMATICS”. [A]
27. If 𝑛𝐶 𝑛𝐶 [E]
4= 5 find the value of n.

28. Find x if 99C40+99Cx=100C59 [D]
29. If 10 = 𝑛𝐶15 find the value of n. [E]
30. Find x if 9Cx+9C7=10C7 [E]
31. In how many ways can the letters of the word “HOPPER” be arranged? [E]
32. In how many ways can 5 letters be posted in 5 letter boxes if each box has one letter? [A]
33. If there are 4 routes to go from A to B and 3 routes to go from B to C, in how many ways
can you go from A to C via B. [E]
34. In how many ways can you go from A to B and return to A from a different route, if there
are 10 ways of travelling from A to B. [E]
35. How many 3 digit numbers can be formed using digits 1,2,5,6,8. [E]
36. How many different signals can be made by taking 3 different coloured flags at a time
from 7 different coloured flags? [E]
37. How many 6 digit numbers can be formed from 2,7,6,1,9,8. [E]

Two Marks Questions:

1. In how many ways can 7 students and 4 teachers be seated in a row so that no two teachers
are together. [D]
2. A box contains 5 red 4 black and 3 white shirts. How many selections of 8 shirts can be
made if the selection contains exactly 4 red 2 black and 2 white shirts. [E]
3. In how many ways can 6 people be chosen out of 10 people if one particular person is
always included? [A]
4. Find the number of triangles and straight lines that can be formed out of 20 points in which
8 are collinear. [A]
5. How many words can be formed from the letters of the word FRIEND which begin with F
and end with D. [E]
6. Find the number of diagonals in a decagon. [A]
7. Find the number of diagonals of a polygon of 20 sides. [A]
8. Find the number of parallelograms that can be formed from 6 parallel lines intersecting
another set of 4 parallel lines. [A]
9. If a convex polygon has 170 diagonals, find the number of sides of the polygon. [A]
10.In how many ways can 7 people be seated for a photograph if two particular people always
want to be together. [A]
11.In how many ways can 7 math books 8 statistics books and 4 accountancy books be
arranged in a shelf if books of the same subject are always together? [E]
12.In how many ways can 9 boys and 6 girls be arranged in a row if no two girls
are together? [D]
13.Find the number of words that can be formed using all the letters of the word
ENGINEERING having all the 3 E’s together. [E]
14.How many 4 digit even numbers can be formed from the digits 1, 2, 3,4,5,6 no digit being
repeated. [A]

15. In how many ways 10 beads of different colours are strung into a necklace if the red green
and yellow beads are always together. [A]
16.Find the number of ways in which 8 men be arranged around a table so that two particular
men may not be next to each other. [D]
17.In how many ways 6 gentlemen and 4 ladies are seated round a table so that no two ladies
are together. [D]
18.In how many ways can 6 boys be arranged in a circle if 2 particular boys always want to be
together? [A]
19.If nPr = 3024 nC r= 126 find r. [A]
20.If nPr = 24 nC4 find n. [A]
21.If 8 = 𝑛𝐶12 find the value of nC5. [E]
22.A person has 9 friends .In how many ways can he invite one or more of them to dinner. [D]
23.Find the number of parallelograms that can be formed from a set of 5 parallel lines
intersecting another set of 6 parallel lines. [A]
24.In how many ways can 6 people be selected out of 12 people so that a) two particular people
must be included b) Two particular members must be excluded. [A]
25.In a party each person shakes hand with everyone else. If there are 25 members in the party,
calculate the number of handshakes. [E]
26.In how many ways can 3 boys and 4 girls be arranged in a row so that all the three boys are
together? [E]
27.In how many ways can 5 men and 4 women be seated in a row such that women occupy the
even places? [A]
28.In how many ways can 7 persons be seated in a row if two persons always occupy the end
seats? [A]
29.In how many ways can 6 red and 4 white marbles be chosen from a bag containing 10 red
and 6 white marbles. [E]

30.In how many ways can 6 people be chosen out of 10 people if one particular person is
always included? [A]
31.In how many ways can 7 persons be seated in a row if two persons always occupy the end
seats? [E]
32.In how many ways the word “CARROM” be arranged such that the 2R’s are always
together. [E]
33.How many words can be formed from the letters of the word “FRIEND” which begin with F
and end with D.? [E]
34.How many four digit numbers can be formed with the 0, 2,3,5,7 such that no digits are
repeated. [A]
35.How many three digit numbers ending with 7 can be formed using the digits 1, 2,5,7,8 digits
cannot be repeated. [E]
36.How many 6 digit numbers can be formed from the digits 1, 2, 3,4,5,6 (no digit being
repeated) which are divisible by 5. [A]
37.In how many ways can 7 students and 4 teachers be seated in a row such that no two
teachers are together? [D]
38.In how many ways can 7 books on art, 5 books on fiction, 4 books on adventure be arranged
so that books of the same category are always together. [E]
39.Find the number of ways in which 4 boys and 5 girls can be arranged in a row so that no two
boys are together and no two girls are together. [D]
40.In how many ways 10 examination papers are arranged so that best and worst papers never
come together. [D]

41.In how many ways can the letters of the word “LAUGHTER” be arranged so that the vowels
are always together? [A]
42.In how many ways can the letters of the word “HEXAGON” be arranged so that the vowels
are in even places [D]

Three Marks Questions:
Note: The questions with multiple sub questions can be asked for two or three marks. For
two mark questions only two sub questions will be asked and for three mark questions
only 3 sub questions will be asked.
1. A family of 4 brothers and 3 sisters is to be arranged for a photograph in one row. In how
many ways can they be seated if
a) all the sisters sit together
b) no two sisters sit together [D]
2. Find the number of ways in which 4 boys and 5 girls can be arranged in a row so that no two
boys are together and no two girls are together. [D]
3. In how many ways can 6 boys and 6 girls be arranged in a row so that
a) All girls are together
b) All boys are not together
c) Boys and girls are arranged alternatively.
d) Two particular girls are always together
e) No two girls are together [D]
4. In how many ways 7 math books 8 statistics books and 4 accountancy books can be arranged
in a shelf if
a) Maths books are always together
b) Books of the same subject are always together. [E]
5. Find the number of permutations of the letters of the word MISSISSIPPI, In how many of
these
a) The 4 S’s are together
b) The 4 S’s are not together
c) Begin with MISS
d) Begin with SIP [E]
6. Find the number of permutations of the letters of the word ‘COMMITTEE’
a) How many of them begin with T and end with T
b) In how many all the vowels are together

C) In how many no two vowels are together
d) How many of them end with MITE. [E]
7. Find the number of permutations of the letters of the word ASSASSINATION. In how many
of these
a) the vowels are in even places
b) vowels are in odd places
c) The word NATION is always present together.
d) begin with ‘AS’ and end with ‘AS’ [E]
38. In how many ways can 7 English books, 8 Kannada books and 5 Hindi books be arranged
in a shelf? In how many of these
a) All Kannada books are together
b) No two English books are together
c) Books of the same language are together
d) All Kannada books are together and all Hindi books are together. [E]
39.How many 4 digit numbers can be formed using digits 0, 1, 2, 3, 4, 5, 6 (No digit can be
repeated. How many of them are
a) Even b) Odd c) divisible by 5 [A]
40.A man has 10 relatives, 4 of them are ladies 3 gentlemen and 3 children. In how many ways
can he invite 7 relatives to a dinner party so that
a. There are exactly 2 ladies, 3 gentlemen and 2 children
b. There are exactly 2 gentlemen and at least 3 ladies.
c. There are exactly 3 children, at least 1 lady and at least two gentlemen. [E]
41.From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition,
including at least 4 boys and at least 4 girls. The 2 girls who won prize last year should be
included. In how many ways can the selection be made? [D]
42.A box contains 5 red, 4 black and 3 white balls. How many ways selection of 8 balls can be
done if the selection contains
a. Exactly 4 red, 2 black and 2 white balls.
b. Atleast 3 red, atleast 3 black and atleast 1 white balls. [E]

43.An examination paper consists of 12 questions divided into part A and B contains 7
questions in part A and part B contains 5 questions. A candidate is required to answer 8
questions selecting atleast 3 from each part. In how many ways can the candidate select the
questions? [E]
44.A team of eleven is to be chosen out of 16 cricket players of whom 4 are bowlers and 2
wicket keepers. In how many ways can the team be chosen so that
a) There atleast 3 bowlers and 1 wicket keeper.
b) There atleast 3 bowlers and atleast one wicket keeper. [E]
45.Out of 4 officers and 10 clerks in an office a committee consisting of 2 officers and 3 clerks
is to be formed. In how many ways can this be done if
a) any officer and any clerk can be included
b) one particular clerk must be on committee
c) one particular officer can be on the committee. [E]
46.A committee of 5 are to be formed from 8 Americans and 5 Anglo- Indians. In how many
ways can this be done when the committee contains
a) exactly two Anglo-Indians
b) at least two Anglo-Indians. [E]
47.A candidate is required to answer 6 out of 12 questions which are divided into 2 groups
containing 6 questions in each group. Find the number of choices he has if he cannot answer
more than five questions from any group. [A]
48. Find the number of permutations of the letters of the word “ENGINEERING” How many
of these
a) Begin with E and end with E
b) Have all the 3 E’s together
c) Begin with GIN and end with GRIN
d) All the vowels are not together
e) No two vowels are together [E]

Chapter 3: PROBABILITY
One Mark Questions:

1. If E is an event of sample space 𝑆 then 𝑃(𝐸𝑈𝐸1 ) = [E]
a) 𝑃(𝐸) b) 𝑃(𝐸1 ) c) 𝑃(𝑆) d) 0
1
2. If 𝑃(𝐴) = , 𝑃(𝐵) = 0, then 𝑃(𝐴|𝐵) is [E]
2
1
a)0 b) c) not defined d) 1
2
3. The probability of obtaining an even prime number on each die, when a pair of dice is
rolled is [E]
1 1 1
a) 0 b) c) d)
3 12 36
4. If 𝐴 and 𝐵 are events such that 𝑃(𝐴|𝐵) = 𝑃(𝐵|𝐴), then [A]
a) 𝐴 ⊂ 𝐵 but 𝐴 ≠ 𝐵 b) 𝐴 = 𝐵
c) 𝐴 ∩ 𝐵 = ∅ d) 𝑃(𝐴) = 𝑃(𝐵)
5. Two events 𝐴 and 𝐵 will be independent, if [E]
a) 𝐴 and 𝐵 are mutually exclusive c) 𝑃(𝐴) = 𝑃(𝐵)
b) 𝑃(𝐴′ 𝐵′ ) = [1 − 𝑃(𝐴)][1 − 𝑃(𝐵)] d) 𝑃(𝐴) + 𝑃(𝐵) = 1
6. Let 𝐸 and 𝐹 be events of a sample space 𝑆 of an experiment, then 𝑃(𝐸 ′ |𝐹 ) + 𝑃(𝐸|𝐹) is
[A]
a) 𝑃(𝐹) b) 𝑃(𝐸) c) 𝑃(𝑆) d) 𝑃(𝐸 ∩ 𝐹)
7. If the events 𝐴 and 𝐵 independent then 𝑃(𝐴 ∩ 𝐵) = [E]
𝑃(𝐴)
a) 𝑃(𝐴) + 𝑃(𝐵) b) 𝑃(𝐴) = 𝑃(𝐵) c) 𝑃(𝐴) × 𝑃(𝐵) d)
𝑃(𝐵)
7 9 4
8. If 𝑃(𝐴) = , 𝑃(𝐵) = and 𝑃(𝐴 ∩ 𝐵) = then 𝑃(𝐴|𝐵) is [A]
13 13 13
4 4 9 4
a) b) c) d)
13 9 13 7
8 7 2
9. If 𝑃(𝐴) = , 𝑃(𝐵) = and 𝑃(𝐴 ∩ 𝐵) = then 𝑃(𝐵|𝐴) is [A]
15 15 15
2 7 1 2
a) b) c) d)
7 8 4 15
10.If 𝑃(𝐵) = 0.5 and 𝑃(𝐴 ∩ 𝐵) = 0.32 then 𝑃(𝐴|𝐵) is [E]
8 1 4 16
a) b) c) d)
25 2 25 25
11.If 𝑃(𝐴) = 0.5 and 𝑃(𝐵) = 0 then find 𝑃(𝐴|𝐵) is [E]
1
a) b) 0 c) 1 d) not exists
2

12.If 𝑃(𝐴) = 0.4 , 𝑃(𝐵) = 0.8 and 𝑃(𝐵|𝐴) = 0.6 then 𝑃(𝐴 ∩ 𝐵) is [A]
a) 0.24 b) 0.3 c) 0.48 d) 0.96

13.If 𝑃(𝐴) = 0.3 and 𝑃(𝐵) = 0.4, 𝐴 and 𝐵 are mutually exclusive events then 𝑃(𝐴|𝐵) is
[A]
a) 0 b) 1 c) −1 d) None of these
3 1
14.If 𝑃(𝐴) = and 𝑃(𝐵) = , 𝐴 and 𝐵 are independent events then 𝑃(𝐴 ∩ 𝐵) is [E]
5 5
3 1 4 1
a) b) c) d)
25 5 5 25

15.If 𝑃(𝐸) = 0.6, 𝑃(𝐹) = 0.3 and 𝑃(𝐸 ∩ 𝐹) = 0.2 then 𝑃(𝐸|𝐹) is [E]
1 2
𝑎) b) c) 0 d) 3
3 3

16.If 𝐴 ∩ 𝐵 = ∅ then 𝑃(𝐵|𝐴) is [A]
1
𝑎) b) 1 c) 0 d) not exists
2

17.The chances of getting a doublet with two dice is [A]
3 1 5 5
a) b) c) d)
2 6 6 36
18.If 𝑃(𝐴) = 0.8, 𝑃(𝐵) = 0.5, 𝑃(𝐵|𝐴) = 0.4 then 𝑃(𝐴|𝐵) is [A]
16 4 1 4
a) b) c) d)
25 5 2 25
6 5 7
19.If 𝑃(𝐴) = , 𝑃(𝐵) = and 𝑃(𝐴 ∪ 𝐵) = then 𝑃(𝐵|𝐴) is [A]
11 11 11
5 6 4 2
a) b) c) d)
6 7 5 3
4 7
20.If 𝑃(𝐴) = and 𝑃(𝐴 ∩ 𝐵) = then 𝑃(𝐵|𝐴) = is [E]
5 10
1 1 7 17
a) b) c) d)
10 8 8 20
7 17
21.If 𝑃(𝐴 ∩ 𝐵) = , 𝑃(𝐵) = , then 𝑃(𝐴|𝐵) = is [A]
50 20
14 20 7
a) b) c) d) none of these
85 50 17

22.If 𝑃(𝐴′) = 0.65 then 𝑃(𝐴) = [E]
a) 1 b) 0 c)0.35 d) 0.65

23.If 𝑃(𝑠𝑢𝑟𝑒 𝑒𝑣𝑒𝑛𝑡) = [E]
a) 1 b) 0 c)0.5 d) 0.25

24.If 𝑃(𝑖𝑚𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑒𝑣𝑒𝑛𝑡) = [E]
a) 1 b) 0 c)0.5 d) 0.25
25.If 𝑃(𝐴) = 0.4 then 𝑃(𝐴′) = [E]
a) 0.6 b) 0.5 c)0 d) 1

3
26.If 𝑃(𝐴) = then 𝑃(𝐴′) = [E]
4
1 1 3 4
a) b) c) d)
3 4 2 3

27. Two coins are tossed simultaneously. The probability of getting exactly two heads is
1 3 1
a) b) c) d) 1 [E]
2 4 4
28. The probability of getting a king card from a pack of 52 playing cards is
1 4 2 1
a) b) c) d) [A]
13 13 13 52

29. The probability of getting a multiple of 3, when a die is thrown is
1 1 1
a) 1 b) c) d) [A]
2 4 3

30. The probability of an event A if odds in favour are 3: 5. [A]
a) 3: 8 b) 3: 5 c)8: 5 d) 8: 3
31. The probability of an event A if odds against are 2: 7. [A]

a) 7: 9 b) 9: 7 c)2: 9 d) 9: 7

Two Marks Questions:

1. A box contains 8 red marbles, 6 green marbles and 10 pink marbles. One marble is drawn
at random from the box. What is the probability that the marble drawn is either red or
green? [E]
1 1 7
2. If 𝑃(𝐴) = . 𝑃(𝐵) = , 𝑃(𝐴 ∪ 𝐵) = then find 𝑃(𝐵/𝐴).
2 3 12
[A]
3. Two coins are tossed simultaneously, find the probability of getting a) getting exactly two
heads b) atleast one head. [A]
2 1
4. If A and B are Mutually exclusive events with 𝑃(𝐴) = . 𝑃(𝐵) = ,then find 𝑃(𝐴 ∪ 𝐵).
5 7
[A]
1 1
5. If 𝑃(𝐵) = , 𝑃(𝐴 ∩ 𝐵) = then find 𝑃(𝐴/𝐵) [E]
2 4
6. If 𝑃(𝐴) = 0.65, 𝑃(𝐴𝑈𝐵) = 0.65. Find 𝑃(𝐵), if A and B are Mutually exclusive
events. [A]
2 1
7. If A and B are independent events with 𝑃(𝐴) = . 𝑃(𝐵) = ,then find 𝑃(𝐴 ∪ 𝐵).
5 5
[A]

8. An unbiased die is rolled. Find the probability of (a) getting a multiple of 3. (b) getting a
prime Number. [A]
9. Two cards are drawn at random from a well – shuffled pack of 52 cards, what is the
probability that either both are Queens or both are king cards. [A]
10. A die is thrown twice, what is the probability that at least one of the two numbers is 6.
[A]
1 1
11.The probability of occurrence if two events A and B are and respectively. The
4 2
7
probability of their simultaneous occurrence is . What is the probability that neither A
50
nor B occurs? [D]
12.Two coins are tossed simultaneously. What is the probability of getting (a) atmost 1 tail
(b) atleast 1 tail. [D]
13.Three fair coins are tossed simultaneously. Find the probability of getting atleast one
head and atleast one tail. [D]
14.Two dice are rolled simultaneously. Find the probability of getting a doublet of even
numbers. [A]
15.A box contains 5 defective and 15 non defective bulbs. Two bulbs are chosen at random.
Find the probability that both the bulbs are non-defective. [A]
16.A problem in a question paper is given to 3 students in a class to be solved. The
probability of their solving the problem are 0.5, 0.7 and 0.8 respectively. Find the
probability that the problem will be solved. [D]
17.What is the probability that a randomly chosen two digit positive integer is a multiple of
3. [D]
18.Two cards are drawn from a pack of 52 cards, what is the probability that both are face
cards. [A]
19.Tickets are numbered from 1 to 18 are mixed up together and one ticket is drawn at
random. What is the probability that the ticket has a number which is a multiple of 2 or 3.
[A]
20.If the letters of the word ‘𝑅𝐴𝑀𝐿𝐸𝐸𝐿𝐴’ are arranged in random. What is the probability
that it begins with 𝑅𝐸𝐸𝐿. [D]
21.A committee of 4 has to be selected from 9 boys and 6 girls. What is the probability that
the committee contains 2 boys and 2 girls. [A]
22.If A and B are two events with probability 0.4 and 0.8 correponding to 𝐴 and 𝐴 ∪ 𝐵. Find
𝑃(𝐵), if A & B are mutually exclusive. [A]
23.Three of the six vertices of a regular hexagon are chosen at random. What is the
probability that the triangle formed with 3 vertices is equilateral. [D]

Two marks with sub questions included where needed:

1. A box contains 4 defective and 6 non defective bulbs. Find the probability that atleast 3
bulbs are defective when 4 bulbs are selected at random. [A]
2. A natural number is chosen at random from among the first 300. What is the probability
that the number so chosen is divisible by 3 or 5. [A]
2 1
3. The probability that a MBA aspirant will join IIM is and that he will join XLRI is .
5 3
Find the probability that a) he will join IIM or XLRI b) he will join neither IIM nor
XLRI. [A]
4. A die is thrown twice and sum of the numbers appearing is observed to be 9. What is the
conditional probability that the number 4 has appeared atleast once. [D]
5. A die is rolled. If the outcome is an odd number. What is the probability that it is a
number greater than 1. [A]
6. A couple has two children. Find the probability that both are boys, if it is known that a)
one of the children is a boy b) older child is a boy. [A]
7. A die is rolled twice. If the sum of the numbers facing upwards is even. Find the
probability that both are odd. [A]
8. Seven persons are to be seated in a row. Find the probability that 2 particular persons sit
next to each other. [A]
9. There are 20 girls and 60 boys in a class. Half of the girls and half of the boys are first
class students. A student is selected at random. What is the probability that the student is
either a boy or a first class holder. [A]
10. Among the members of a committee, there are 75% males and 25% females. The
probability that a male member becomes the president is 0.25 and probability that a
female member becomes the president is 0.4. Find the probability that the person selected
at random becomes the president. [Hint: Person selected is Male and becomes President
OR is female and becomes President] [A]
11. Two cards are drawn from a pack of playing cards, one after the other. Find the
probability of getting a heart in first draw and diamond in the second draw if the cards are
drawn. A) without replacement b) with replacement. [A]
12. If three cards are drawn at random from a pack of 52 cards, what is the probability that
atleast two of them are kings. [A]
13.Two dice are rolled together. Find the probability of getting a) a doublet b) sum of 2
numbers on the dice as 11 c) sum of 2 numbers on the dice are more than 11. [A]
14. 3 fair coins are tossed simultaneously. Find the probability of a) getting 1 head
b) getting at the most one head c) getting at least two heads. [A]
15. Two cards are drawn from a pack of playing cards, one after the other. Find the
probability of getting a queen in first draw and second draw, if the cards are drawn. a)
without replacement b) with replacement. [A]

16. A bag contains 10 gold and 8 silver coins. Two successive drawings of 4 coins are made.
Find the probability that the first drawing will get 4 gold coins and second 4 silver coins,
if the coins are replaced before the second trial. [A]
17. One card is drawn from a pack of 52 playing cards. Find the probability that a) card is
either black or jack b) card is red c) card is diamond. [A]
18. A die is thrown. If F is the event “the number appearing is even” and E is the event “the
number appearing is multiple of 3”. Then find whether E and F are independent.
[A]
19. A box contains 8 white and 9 red balls. Two balls are taken at random from the box.
Find the probability that both of them are red, if
a) the two balls are taken together
b) the balls are taken one after the other without replacement
c) the balls are taken one after the other with replacement. [D]

Chapter 4: BINOMIAL THEOREM

Four Marks Questions :

1. Find the value of (1.01)5 using Binomial Theorem up to 5 decimal places. [ A ]
2. Find the value of (99)5 using Binomial Theorem up to 5 decimal places. [A]
3. Find the value of (1.2)5 using Binomial Theorem up to 5 decimal places. [A]
4. Find the value of (0.99)5 using Binomial Theorem up to 5 decimal places. [ A ]
5. Find the value of (98)4 using Binomial Theorem up to 5 decimal places. [A]
4
6. Find the value of (0.99) using Binomial Theorem up to 5 decimal places. [ A ]
7. Find the value of (1.02)6 using Binomial Theorem up to 5 decimal places. [ A ]
8. Find the value of (1.0005)4 using Binomial Theorem up to 5 decimal places. [ A ]
9. Find the value of (1.1)5 using Binomial Theorem up to 5 decimal places. [A]
4
10. Find the value of (1.01) using Binomial Theorem up to 4 decimal places. [ A ]
5 8
11.Find the middle terms in the expansion of (√𝑥 − 2) [A]
𝑥
1 10
12.Find the middle term in the expansion of (𝑥 − ) [A]
2𝑦
𝑎 𝑏 8
13.Find the middle terms in the expansion of ( + ) [A]
3 3
7
𝑥2
14.Find the middle terms in the expansion of (3𝑥 + ) [A]
2
1 11
15.Find the middle terms in the expansion of (2𝑥 2 − ) [ D]
√𝑥

1 17
16.Find the middle terms in the expansion of (2𝑥 − ) [A]
𝑥
10
2𝑥 2 3
17.Find the middle terms in the expansion of ( − ) [D]
3 2𝑥
1 12
18.Find the middle terms in the expansion of (3𝑥 2 − ) [A]
𝑥
1 10
19.Find the middle terms in the expansion of (𝑥 − ) [A]
𝑦
2𝑎 3 6
20.Find the middle terms in the expansion of ( − ) [A]
3 2𝑎
𝑥 3 19
21.Find the middle terms in the expansion of ( + 2) [D]
2 𝑥
2 17
22. Find the co efficient of 𝑥 −11 in (√𝑥 − ) [A]
𝑥
2 12
23. Find the co efficient of 𝑥 −4 in (√𝑥 − ) [D]
√ 𝑥 𝑥
1 17
24.Find the co efficient of 𝑥 5 in (𝑥 + 2) [A]
𝑥
3𝑎 15
25.Find the co efficient of 𝑥 18 in (𝑥 2 + ) [ D]
𝑥
4 21
26. Find the co efficient of 𝑥 −7 in (√𝑥 − 3) [A]
𝑥
1 1 15
27. Find the co efficient of 17
in (𝑥 4 − 3) [A]
𝑥 𝑥
1 10
28. Find the co efficient of 𝑥 8 in (3𝑥 2 − ) [A]
2𝑥
3 25
29. Find the constant term in the expansion of (2𝑥 2 − 3 ) [D]
𝑥
3 1 6
30. Find the term independent of 𝑥 in ( 𝑥 2 − ) [A]
2 3𝑥
4 3 9
31. Find the term independent of 𝑥 in ( 𝑥 2 − ) [A]
3 2𝑥
1 10
32. Find the term independent of 𝑥 in (√𝑥 + ) [D]
3𝑥 2
3 3 15
33. Find the term independent of 𝑥 in (𝑥 − 2) [A]
𝑥
1 18
3
34. Find the term independent of 𝑥 in ( √𝑥 + 3 ) ,𝑥 > 0 [D]
2 √𝑥
2 5
35. Find the term independent of 𝑥 in (𝑥 2 − 3) [A]
𝑥
2 20
36. Find the term independent of 𝑥 in (√𝑥 − 2) [A]
𝑥
1 21
37. Find the term independent of 𝑥 in (𝑥 − 2) [A]
𝑥
10
√𝑥 2
38. Find the term independent of 𝑥 in ( − 2) [ D]
2 𝑥

5 5
39. Simplify (2 + √3) + (2 − √3) [A]
4 4
40. Simplify (3 + √2) + (3 − √2) [A]
5 5
41. Simplify (√3 + 1) + (√3 − 1) [ A]
5 5
42. Simplify (1 + √5) − (1 − √5) [A]
6 6
43. Simplify (√2 + 1) − (√2 − 1) [A]

Chapter 5: PARTIAL FRACTIONS

Five Marks Questions:

Resolve the following into partial fractions.

𝑥+3
1. (𝑥−1)(𝑥+2)(𝑥−2) (E)
𝑥2
2. (𝑥+1)(𝑥+2)(𝑥+3) (E)
𝑥−1
3. (𝑥)(𝑥+2)(𝑥+4) (E)
𝑥 2 −10𝑥+13
4. (𝑥+1)(𝑥 2 (A)
−5𝑥+6)
𝑥+8
5. (2𝑥 2 −𝑥−1)(𝑥+1)
(E)
3𝑥+5
6. (𝑥+2)2(𝑥−3) (E)
1
7. (E)
𝑥(𝑥 2 −9)
2𝑥+21
8. (𝑥+2)(𝑥−1)2 (E)
𝑥
9. (𝑥+1)(𝑥 2 (A)
+3𝑥+2)

𝑥2
10.(𝑥+1)2(𝑥−5) (E)
2𝑥 2 +16𝑥+29
11. (𝑥+3)2 (𝑥+4) (E)
9
12.(𝑥+2)2(𝑥+1) (E)
3𝑥+4
13.(𝑥+1)2(𝑥−1) (E)

𝑥−1
14.(𝑥+2)(𝑥 2 (E)
−𝑥−6)

1+3𝑥+2𝑥 2
15.(1−2𝑥)(1−𝑥 2 ) (A)
𝑥
16.(1+2𝑥)2(1−3𝑥) (D)
9𝑥−27
17.(𝑥+1)(𝑥−2)2 (E)
2𝑥+1
18.(𝑥−1)(𝑥+2)(3𝑥−1) (A)
𝑥
19.(𝑥+2)(𝑥+1)2 (E)
3𝑥+2
20.(𝑥−2)(𝑥+3)2 (E)
2𝑥 2 +10𝑥−3
21. (E)
(𝑥+1)(𝑥−3)(𝑥+3)

Chapter 6 :MATHEMATICAL LOGIC

One Mark Questions:

1. Symbolize the propositions “2 + 5 = 6 or all integers are rationals”. [E]
2. Symbolize the propositions “3x = 9 and x<7”. [E]
3. Symbolize the propositions “If 2 numbers are equal then their squares are not equal”. [E]
4. Write the converse of “If x (x-2) = 0 then x=2”. [E]
5. Write the inverse of “If the questions are easy then students score better marks”. [D]
6. Write the contrapositive of “ If 𝑥 ∈ (𝐴 ∪ 𝐵) 𝑡ℎ𝑒𝑛 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵. [D]
7. Write the converse of “ If 𝑥 2 = 𝑦 2 𝑡ℎ𝑒𝑛 𝑥 = 𝑦. [D]
8. Negate: “4 is an even integer or 7 is a prime number”. [A]
9. Negate: ~𝑝 → 𝑞. [D]
10.Symbolize the propositions “If oxygen is a gas then gold is a compound”. [A]
11.Negate: ~𝑝 ∧ 𝑞.
a) 𝑝 →∼ 𝑞 b) ~𝑝 ∧∼ 𝑞 c) 𝑝 ∨∼ 𝑞 d) 𝑝 ∧ 𝑞 [A]
12. Negate: 𝑝 ∨∼ 𝑞

a) 𝑝 →∼ 𝑞 b) ~𝑝 ∧∼ 𝑞 c) ~𝑝 ∧ 𝑞 d) 𝑝 ∧ 𝑞 [A]
13.If the compound proposition 𝑝 → (𝑞 ∨ 𝑟) is false, then find the truth values of p,q and r.
[E]
14.If the compound proposition 𝑝 → (~𝑞 ∨ 𝑟) is false, then find the truth values of p,q and r.
[E]
15.If (~𝑝 ∨ 𝑞) ∧ ~𝑟 is a false proposition then find the truth values of p,q and r. [A]
16.If the truth value of the proposition p,q and r are F,T and F respectively,then find the truth
values of the compound proposition (~𝑝 → 𝑞) ∨ 𝑟. [D]
17.If the truth value of the proposition p,q and r areT,T and F respectively,then find the truth
values of the compound proposition(𝑝 ∨ 𝑟) ∧ 𝑞 . [D]
18.If the truth value of the proposition p,q and r are F,T and F respectively,then find the truth
values of the compound proposition𝑝 → (𝑞 → 𝑟). [D]
19.Negate: 𝑞 ∨ [~(𝑝 ∧ 𝑟)]. [A]
20.Negate: 14 is a divisor of 48 and 28 is not divisible by 82. [A]
21.Write the converse, inverse and contrapositive of “If x is less than 1 then it is a prime
number”. [A]
22.Write the converse, inverse and contrapositive of “ If oxygen is a gas then accountancy is
easy or the child is brave”. [A]
23.Write the converse, inverse and contrapositive of “If 𝑥 2 = 𝑦 2 then “x = y”. [A]

Five Marks Questions:

1. Show that [~𝑝 ∧ (𝑝 ∨ 𝑞)] → 𝑞 is a Tautology or a Contradiction. [E]
2. Show that [~(𝑝 → ~𝑞)] ∨ (~𝑝 ↔ 𝑞) is a Tautology or a Contradiction. [E]
3. Show that ~(𝑝 ∨ 𝑞) → (~𝑝 ∧ ~𝑞) is a Tautology or a Contradiction. [A]
4. Show that [(∼ 𝑝 ∨ 𝑞)] ↔ (𝑝⋁ ∼ 𝑞) is a Tautology or a Contradiction. [A]
5. Show that (𝑝 ∧ ~𝑞) → ~ (𝑝 ∧ 𝑞) is a Tautology or a Contradiction. [A]
6. Show that(𝑝 ∨ 𝑞)⋀(~𝑝 ∧ ~𝑞)is a Tautology or a Contradiction. [A]

7. Examine whether the propositions ~(𝑝 ↔ 𝑞) ∨ 𝑞 𝑎𝑛𝑑 (𝑝 ∧ ~𝑞) ∨ (𝑞 ∧ ~𝑝) are
logically equivalent or not. [D]
8. Examine whether the propositions (𝑝 ∧∼ 𝑞) ∨ 𝑞 𝑎𝑛𝑑 𝑝 ∨ 𝑞 are logically equivalent or
not. [D]
9. Examine whether the propositions 𝑝 ↔ 𝑞 𝑎𝑛𝑑 [(𝑝 → 𝑞) ∧ (𝑞 → 𝑝)] are logically
equivalent or not. [D]
10.Examine whether the propositions 𝑝 ↔ 𝑞 𝑎𝑛𝑑 (𝑝 → 𝑞) ∧ (𝑞 → 𝑝) are logically
equivalent or not. [D]
11.Verify whether the proposition (~𝑝 ∧ 𝑞) ∧∼ 𝑟 is a Tautology ,Contradiction or neither.
[D]
12.Verify whether the proposition~(𝑝 → 𝑞) ∨ [(∼ 𝑝 ∧ 𝑞) ↔∼ 𝑞] is a Tautology,
Contradiction or neither. [D]

Chapter 7: RATIOS AND PROPORTIONS
One Mark Questions:

1. The ratio of 2hrs to 30mins is (E)
a) 4:1 b) 2:3 c) 2:1 d) None of these
1 1 1
2. The fourth proportional to , , is (E)
2 3 4
1
a) 6 b) 4 c) 2 d)
6

3. The fourth proportional to 1.5,4.5,3.5 is (E)
a) 10 b) 11 c) 10.5 d) 11.5
4. The third proportional to 2.4 and 3.6 is (E)
a) 5.4 b) 5 c) 6 d) 6.4
1 1
5. The mean proportional to 𝑎𝑛𝑑 (E)
16 25

a) 10 b) 12 c) 15 d) None of these

6. The value of x if 5:20 = 3:x (E)
a) 10 b) 12 c) 15 d) None of these
7. The mean proportional of 49 and 169 (E)
a) 91 b) 20 c) 8281 d) None of these
8. The fourth proportional of 6,14,15 (E)
a) 10 b) 20 c) 30 d) None of these
9. The sub triplicate ratio of 729 ∶ 4913 (E)
a) 17:9 b) 9:17 c) 3:17 d) None of these
10.Find the triplicate ratio of 3 : 5. (E)
3 3
a) 27:125 b) 9:25 c) √3 ∶ √5 d) √3: √5
11.The duplicate ratio of 5 : 4 (E)
a) 25:16 b) √5 ∶ √4 c) 125:64 d) None of these
12.The compound ratio of 3 : 4 and 4 : 7 (E)
a) 3:7 b) 7:3 c) 21:16 d) None of these
13.The value of ‘x’ if 32 ∶ 𝑥 = 25 ∶ 50 (E)
a) 64 b) 128 c) 16 d) None of these
14. Find the sub duplicate of 225 ∶ 9. (E)
15.Find the compound ratios of 1 : 2 and duplicate ratio of 2 : 3 and sub triplicate ratio of
125: 64. (A)
16.Find the inverse ratio of 2 : 3. (E)
17. A house consumes 20 kgs of rice and 5 kgs of wheat. Compare the consumption of rice
and wheat in the form of a ratio. (E)
18.The third proportional of 4 and 6 is (E)
19.If a:b = 2:3 , b:c =5:7 , c:d = 3:1 then find a:d. (E)
20.If a:b =3:4 and b:c = 5:12 then find a:b:c (E)
21.The mean proportional of 2 and 8 is (E)
22.Antecedent of 7:5 is (E)
23.The duplicate ratio of 2:7 is (E)

24.The value of x in 5:30 =6:x is (E)
25. The mean proportional of 9 and 16 is (E)
26. The value of x , if 500:x =5:8 is (E)
27.The Compound ratio of 1:2, 2:3 and 3:5 is (E)
28. The 4th proportional of 6, 12 and 15 is (E)

Two Marks Questions:

1. Divide 6000 in the ratio 3 : 4 : 5. (E)
2. 500Workers can finish a work in 8 days. How many workers will finish the same work in
5 days. (A)
3. What must be added to each term in the ratio 2:3 so that it becomes 5:6. (E)
4. What must be added to each term in the ratio 5:6 so that it becomes
11: 12. (E)
5. What must be subtracted from each term in the ratio 7 : 4 so that it becomes 5 : 2
(E)
6. Two numbers are in the ratio 3 : 5. If 5 is added to each term, they are in the ratio 2 : 3.
Find the numbers. (E)
7. A ratio is in the lowest form given to be 3:8. If the difference between the quantities is 25.
Find the quantities. (E)
8. Find the ratio between two numbers such that their sum is 40 and their difference is 8. (A)
2𝑥 2 +5𝑦 2
9. If x:y = 2:3. Then find . (A)
𝑥 2 +𝑦 2

10.If 10 persons can do a job in 30 days, then in how many days can 15 persons do the same
job? (E)
11.Find the numbers which when added to the terms of numerator and denominator of 25 : 37
will make it 5 : 6. (E)
12.An article is sold at 40% gain on the cost price. Find the ratio of the selling price and cost
price. (E)
13.If a:3:15 = 5:b:5 then find ‘a’ and ‘b’. (E)

14.If a :b =3:4 , b:c = 8: 15 find a: b:c (E)
15.X,Y and Z play cricket. The run scored by X andY are in the ratio 3:2 and that of Y and Z
are in the ratio 3:2 . Together they all score 342 runs. How many runs did each score? (A)

Three Marks Questions:

1. The monthly incomes of A and B are in the ratio 2:3 and their monthly expenditures are in
the ratio 3:5. If each save Rs 100 per month , find the monthly incomes of A and B.
(E)
2. X, Y and Z play cricket . The runs scored by X and Y are in the ratio 3:2. The ratio of the
runs scored by Y and Z are in the ratio 3:2. Together they all score 342 runs. How many
runs did each score? (E)
3. Three numbers are in the ratio 2:3:4. If the sum of their squares is 1856, find the numbers.
(E)
4. The monthly incomes of A and B are in the ratio 9:7 and those of B and C are in the ratio
3:2. If 10% of A’s income and 15% of the C’s income differ by Rs 18, Find the incomes
of A,B and C. (D)
5. If a:b = 2:3 and x:y = 4:5 then find (5ax+3by): (10ax+4by). (E)
6. Two taps can separately fill the tank in 12 min and 15 min respectively. The tank when
full can be emptied by a drain pipe in 20 min. When the tank was empty all the three pipes
were opened simultaneously, in what time will the tank be filled up?
(E)
7. 5Workers, each working 9 hours a day can finish a work in 30 days. How many workers
are required to finish eight times the work in 25 days each working 8 hours a day?
(D)
8. A jar contains two liquids A and B in the ratio 7:5. When 9 liters of the mixture is drawn
and the jar is filled with the same quantity of B, the ratio of A and B becomes 7:9. Find
the quantity of A in the jar initially. (A)

9. Three carpenters can earn Rs 360 in 6 days working 9 hours a day. How much will 8
carpenters can earn in 12 days working 6 hours a day? (E)
10. Rs 5625 is divided among A, B and C so that A receives one half as much as B and C
together receive and B receives one fourth of what A and C together receive . Find the
shares of A, B and C . (A)
11. Walking 4 kmph a student reaches his college 5 minutes late and if he
walks at 5kmph then he reaches the college 2.5 minutes early. What is the distance from
his house to the college? (E)
12.In a fort, there was ration for 560 soldiers that would last the soldiers for 70 days. After
20 days, 60 soldiers left the Fort. For how many days the remaining ration can support
the remaining soldiers? (A)
13.A railway train is 100 meters long and is running at the speed of 30kmph.In what time
will it pass through
i) a man standing on the platform and
ii) a bridge which is 200 meters long? (E)
14.Distribute Rs 8000 in the ratio 3:4:5 (E)

Chapter 8: BILL DISCOUNTING

Two Marks Questions:

1. Find Banker’s discount and True discount for a bill of Rs 415 due 9 months at 9% p.a. (E)
2. Find the present value of Rs 900 due 4 months hence at the rate of 15% p.a. Also find T.D
(E)
3. The Banker’s gain on a bill is 1/9th of the bankers discount, rate of interest is 10% find the
unexpired period the bill. (A)
4. Find the present worth and the discounted cash value of the bill of Rs 2320, due 2 Years
hence at 8% p.a. (E)

5. Banker’s discount and Banker’s gain on a sum of money due 3 months hence are Rs 154.50
and Rs 150 respectively. Find the sum of money and rate of interest. (A)

6. A Banker discounts a bill for a certain amount having 73 days to run before it matures at
15% p.a.The discounted value of the bill is Rs 970 What is the face value of the bill and
also find the Banker’s discount (D)
7. Banker’s discount and Banker’s gain on a bill due after sometime are Rs 1250 and Rs 50
Respectively. Find the face value of the bill. (E)
8. A banker pays Rs 2380 on a bill of Rs 2500, 73 days before the legally due date. Find the
rate of discount charged. (A)
9. The difference between BD and TD on a certain sum of money due 6 months is Rs 27 Find
the amount of the bill if the Rate of interest is 6% p.a. (E)
10.True discount on a bill was Rs100 and Banker’s gain is Rs 10 Find the Face value of the
bill. (E)
11.For Rs 512.50 due 6 months at 15% per annum find the true present value and the
discounted value of the bill. (E)
12.Find the present value of Rs 750 due 4 months hence at the rate of 15% p.a. Also find T.D
(E)

Three Marks Questions:

1. The difference between the Banker’s discount and true discount on a bill due after 6
months at 4% interest p.a. is Rs 20. Find the true discount, banker’s discount and the face
value of the bill. (A)
2. The bankers gain on a certain bill due 6 months hence is 10 the rate of interest being 10%
p.a. Find the face value of the bill and the true present value. (E)
3. A bill for Rs 14,600 drawn at 3 months after date was discounted on 11-11-99 for Rs 14320
if the discount rate is 20% p.a, on what date was the bill drawn? (D)

4. A Bill for Rs 3500 due for 3 months was drawn on 27th march 2012 and was discounted on
18th April 2012, at a rate of 7% p.a. Find the bankers discount and discounted value of the
bill. (A)
5. A bill of Rs 12,900 was drawn on 3rd of February 2004 at 6 months and discounted on 13th
march 2004 at 8% p.a. For what sum was the bill discount? And how much did the banker
gain in this transaction? (E)
6. The bankers gain on a bill is 1/15th of the banker’s discount, rate of interest being 10% p.a.
Find the unexpired period of the bill also find the true discount. (E)
7. The present value of a bill due sometime hence is Rs 1100 and the true discount is Rs
110.Find the bankers discount. (E)
8. A bill of Rs 1460 was drawn on 1stAugust for 6 months after date was discounted at 5% for
Rs 1451.On what date was the bill discounted? (D)
9. A bill of Rs 5000 drawn on 10/4/1998 at 12% p.a. for what sum was the bill discounted?
And how much was the bankers gain? (E)
10.A bill for Rs 12900 was drawn on 3rd February 2017 for 6 months and was discounted on
13th march 2017 at 8% p.a. Find the bankers present value and bankers gain? (E)
11.The difference between the Banker’s discount and true discount on a bill due after 6
months at 6% interest p.a. is Rs 27. Find the face value of the bill. (E)
12.The difference between the Banker’s discount and true discount on a bill due after 6
months at 6% interest p.a. is Rs 27. Find the true discount, Bankers discount and the
amount of the bill. (A)

Chapter 9: STOCK AND SHARES

Three Marks Questions:

1. A man invests equal sums of money in 4%, 5% and 6% stock, each stock being at par. If
the total income of the man is Rs. 3600 Find his total Investment? [E]
2. Ramesh has invested Rs 4300 partly in 4.5% stock at Rs 72 and partly in 5% stock at Rs95.
If the total income from both is Rs 250, find the investment in both the types of stock. [A]
3. Sanjana invests Rs 3240 in a stock at 108 and sells when the price falls to 104. How much
stock at 130 can Sanjana buy now? [D]
4. Prathik sells out Rs 6000 of 7.5% stock at 108 and re-invests and proceeds in 9% stock. If
Prathik’s income increases by 270. At what price did Prathik buy 9% stock? [D]
5. A person invests Rs 15000 partly in 3% stock at 75 and partly in 6% stock at 125. If the
income from both is 675. Find his investment in the 2 types of stocks. [A]
6. Rohan invested Rs 55000 partly in 8% stock at 80 and partly in 12% stock at 150 in such a
way that to get a written of 9% for his money. How much did Rohan invest in each? [A]
7. Sankeeth holds 2100 Rs of 3% stock. He sells at Rs 121 and invests then proceeds in 5%
stock. Thereby his income increases by Rs 14. Find the market price of 5% stock. [A]
8. Mr.Sandeep invests Rs 15000 cash partly in 3% stock at 75 and partly in 6% Debentures at
125 in such a way that to get a return on 4.5% for money. How much money does he need
to invest? How much does he invest in each? [D]
9. Rakshita invests 15000 in a company paying 7% per annum, when a share of value Rs 100
is selling for 150. What is her annual income and what percentage does she get on her
money? [D]
10.Which is a better investment 7% stock at 115 or 5% stock at 88. [E]
11.What is the market value of 9.5% stock when an investment of Rs.12400 produces an
income of Rs.1472.5? [A]

12.What is the quoted value of 12% stock if it earns an interest of 8%after deducting the
income tax of 8%? [A]
13.A person ivested 42000 partly in 5% stock at 125 and the remaining in 7.5% stock at 75. If
the income derived from two stocks is the same then find the investments in each type.
Also find the total income. [A]
14.Ayush buys Rs.10000 stock at 96 and sells when its price rises to Rs.102. find his gain. [A]
15.Raksha invets in shares which are selling at Rs.2020 per share. How much money is
required to purchase 10 shares if the brokerage is 0.5%? [D]
16.Parashara sold 25 Biocon shares when the market price was Rs.400 per share. He then
bought 10 Tata motors shares which were selling at Rs.800 per share. Brokerage for each
transaction was 0.25%. The balance amount he gave to his daughter for travelling. How
much did the daughter receive? [D]
17.Which is a better investment 7.5 % stock at 125 or 5% stock at 75. [E]
18.What is the market value of 12% stock when an investment of Rs.6900 produces an income
of Rs. 720? [A]
19.How much of 8% stock can be purchased for Rs.4800? find the income obtained. [E]
20.What income can be derived from an investment of Rs.10725 in 6.5% stock at 143? What
is the amount of stock purchased? [A]
21.What amount is realized by selling Rs.30000 stock at 20 discount? [D]

10. LEARNING CURVE
Five Marks Questions:

1. An engineering company has 80% learning effect and spends 1000 hours to produce 1 lot
of the product. Estimate the labour cost of producing 8 lots of the product, if the labour
cost is Rs. 40 per hour. [E]

2. A motor company limited has observed that a 90% learning effect applies to all labour
related costs. Whenever a new product is taken up for production, the anticipated
production is 320 units for the coming year. The production is done in lots of 10 units
each. Each lot requires 1000 hours at 𝑅𝑠. 15/hour. Calculate the total labour hours and
labour cost to manufacture 320 units. [A]
3. ABC Company required 1000 hours to produce first 30 engines. If the learning effect is
90%. Find the total labour cost at 𝑅𝑠. 20/hour to produce total of 120 engines.
[A]
4. An engineering company has 80% learning effect and spends 500 hours for the prototype.
Estimate the labour cost of producing 7 engines of new order, if the labour cost is 𝑅𝑠. 40
per hour. [A]
5. An aircraft manufacturer supplies aircraft engines to different airlines. They have just
completed an initial order for 30 engines involving a total of 6000 direct labour hours a5
Rs. 20 per hour. They have been asked to bid for a prospective contract for a supply of 90
engines. It is expected that there will be 80% learning effect. Estimate that labour cost for
the New Year. [E]

6. XYZ Company supplies water tankers to the government. The first water tanker takes
20,000 labour hours. The government auditors suggest that there should be 90% learning
effect rate. The management expects an order of 8 water tankers in the next year, what will
be the labour cost, the company will incur at the rate of Rs. 20 per hour.
[A]
7. A first sample batch of 50 units of product A took 80 hours to make. The company now
wishes to estimate the average time per unit will be, if the total output of product A is 200
units and 80% learning rate applies. [E]

8. A company requires 100 hours to produce the first 10 units at Rs. 15 per hour. The learning
curve effect is 80%. Find the total labour cost to produce a total of 160 units.
[E]

9. An engineering company has 80% learning effect and spends 1000 hours to produce 1 lot
of the product. Estimate the labour cost of producing 8 lots of the product, if the labour
cost is 𝑅𝑠. 100 per hour. [E]

11. LINEAR PROGRAMMING PROBLEMS
Five Marks Questions:

Solve the following LPP graphically:

1. Maximize 𝑧 = 60𝑥 + 15𝑦
Subject to the constraints 𝑥 + 𝑦 ≤ 50
3𝑥 + 𝑦 ≤ 90

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]

2. Maximize 𝑧 = 5𝑥 + 3𝑦
Subject to the constraints 3𝑥 + 5𝑦 ≤ 15
5𝑥 + 2𝑦 ≤ 10

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]

3. Maximize 𝑧 = 0.08𝑥 + 0.10𝑦
Subject to the constraints 3𝑥 + 5𝑦 ≤ 15
5𝑥 + 2𝑦 ≤ 10

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]

4. Maximize 𝑧 = 6𝑥 + 8𝑦
Subject to the constraints 4𝑥 + 2𝑦 ≤ 20
2𝑥 + 5𝑦 ≤ 24

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]
5. Maximize 𝑧 = 2.5𝑥 + 𝑦
Subject to the constraints 𝑥 + 3𝑦 ≤ 12
3𝑥 + 𝑦 ≤ 12

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]

6. Minimize 𝑧 = 50𝑥 + 75𝑦
Subject to the constraints 2𝑥 + 3𝑦 ≥ 6
3𝑥 + 4𝑦 ≥ 8

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]

7. Maximize 𝑧 = 2𝑥 + 3𝑦
Subject to the constraints 𝑥 + 𝑦 ≤ 400
2𝑥 + 𝑦 ≤ 600

and 𝑥 ≥ 0, 𝑦 ≥ 0 [A]

8. Maximize 𝑧 = 100𝑥 + 150𝑦
Subject to the constraints 𝑥 + 𝑦 ≤ 50
2𝑥 + 2𝑦 ≤ 85

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]
9. Minimize 𝑧 = 3𝑥 + 8𝑦
Subject to the constraints 𝑥 + 𝑦 ≥ 200
𝑥 ≤ 80

𝑦 ≥ 60

and 𝑥 ≥ 0, 𝑦 ≥ 0 [A]

10. Maximize 𝑧 = 10500𝑥 + 9000𝑦
Subject to the constraints 𝑥 + 𝑦 ≤ 50
2𝑥 + 𝑦 ≤ 80

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]
11. Maximize 𝑧 = 400𝑥 + 200𝑦
Subject to the constraints 5𝑥 + 2𝑦 ≥ 30
2𝑥 + 𝑦 ≤ 15

𝑥≤𝑦

and 𝑥 ≥ 0, 𝑦 ≥ 0 [D]

12. Maximize 𝑧 = 3𝑥 + 4𝑦
Subject to the constraints 4𝑥 + 2𝑦 ≤ 80
2𝑥 + 6𝑦 ≤ 180

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]
13. Maximize 𝑧 = 6𝑥 + 11𝑦
Subject to the constraints 2𝑥 + 𝑦 ≤ 104
𝑥 + 2𝑦 ≤ 76

and 𝑥 ≥ 0, 𝑦 ≥ 0 [A]
14. Maximize 𝑧 = 20𝑥 + 16𝑦
Subject to the constraints 6𝑥 + 2𝑦 ≥ 12
2𝑥 + 2𝑦 ≥ 8

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]

15. Maximize 𝑧 = 2000𝑥 + 3000𝑦
Subject to the constraints 400𝑥 + 600𝑦 ≤ 6000
400𝑥 + 200𝑦 ≤ 20000

and 𝑥 ≥ 0, 𝑦 ≥ 0 [A]
16. Maximize 𝑧 = 100𝑥 + 120𝑦
Subject to the constraints 2𝑥 + 3𝑦 ≤ 30
3𝑥 + 𝑦 ≤ 17

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]
17. Maximize 𝑧 = 3𝑥 + 10𝑦
Subject to the constraints 𝑥+𝑦 ≤4
2𝑥 + 𝑦 ≤ 6

and 𝑥 ≥ 0, 𝑦 ≥ 0 [E]
18. Maximize 𝑧 = 𝑥 + 𝑦
Subject to the constraints 2𝑥 + 𝑦 ≤ 50
𝑥 + 2𝑦 ≤ 40

and 𝑥 ≥ 0, 𝑦 ≥ 0 [A]

Chapter 12: SALES TAX AND VALUE ADDED TAX
Three Marks Questions:
1. The price of a washing machine inclusive of sales tax is 13,530. If the sales tax is 10%.
Find the basic price. [A]
2. A shopkeeper purchase an article for 7,000 and sell it to a customer for 8,200. If the VAT
rate is 6%. Find the VAT paid by the shopkeeper. [A]
3. Mr. Arya purchase an article for 3,100 and sell it to Mr. Aravind for 4,250. Mr. Aravind
in turn sells it to Mr. Anil for 5,000. If the VAT levied 10%. Find the VAT levied on
Arya and Aravind? [A]
4. Bharath bought a shirt for Rs. 336 including 12% sales tax and a neck tie for Rs. 110
including 10% sales tax. Find the printed price of shirt and neck tie together. [A]
5. A furniture dealers sold furniture for Rs. 21,000 and added 5% sales tax to the
quoted price. The customer agrees to buy it for Rs. 21,000 including sales tax. Find
the discount he received. [A]
6. A shopkeeper sells an item at the price of Rs. 810 including ST of 8%. What should
a customer pay for the same item, if the ST is reduced to 6%. [D]
7. Sanju goes to a shop to buy a Bicycle quoted at Rs. 2,000. The rate of sales tax is
12% on it. He asks the shopkeeper for a rebate on the price of the Bicycle to such an
extent that he has to pay Rs. 2,016 including sales tax. Find the rebate percentage on
the price of the bicycle. [D]
8. A shopkeeper announces a discount of 10% on a washing machine set. The marked
priced of the washing machine is Rs. 22,000. How much will a customer have to pay
for buying the washing machine set, if the rate of sales tax is 8%? [E]
9. A shopkeeper purchases an audio system for Rs. 3,000 and sells it off at a gain of
15%. He also charges a sales tax of 10% on the selling price. Calculate the amount
that the buyer will pay to the shopkeeper. [A]
10.The price of T.V set inclusive of sales tax of 9% is Rs.13,407. Find its marked price. If the
ST is increased to 13%, how much more does the customer Sneha pay for the T.V.?
[A]

11.When the rate of sales tax is decreased from 9% to 7%. For a Radio, Rahul has to pay Rs.632
less for it. What is the listed price of the radio? [A]
12.If the rate of sales tax is 5%, Sushma has to pay Rs. 7,140 for the steel cupboard. What
amount she has to pay, if the sales tax is increased by 2%. [A]
13. Chandana purchases an article for Rs. 5,400 which includes 10% rebate on the marked price
and 20% sales tax on the remaining price. Find the marked price of the article. [E]
14. Mr. Ramesh buys a tape recorder for Rs.10,260 including sales tax. If the listed price of the
tape recorder is Rs. 9,500. Find the rate of sales tax charged. [A]
15. Rohith goes to purchase a motorcycle which is priced at Rs. 35,640 including 10% as sales
tax. How was the actual rate of sales tax at the time of purchase is 7%. Find the extra profit
made by the shopkeeper, if he still charges the original list price. [D]
16. A shopkeeper bought a T.V at a discount of 30% of the listed price of Rs. 24,000. The
shopkeeper offer a discount of 10% on the listed price to the customer. If the VAT is 10%.
Find
i) The amount paid by the customer.
ii) The VAT to be paid by the shopkeeper. [A]
17. A shopkeeper buys a mobile set at a discount of 20% from the wholesaler, the printed price
of the mobile set being Rs. 1,600 and the rate of sales tax is 6%. The shopkeeper sells it to
the buyer at the printed price and charges tax at the same time. Find
i) The price at which the mobile set can be brought from the wholesaler.
ii) The VAT to be paid by the shopkeeper. [A]
18. ‘A’ is the manufacturer of electric iron. The cost price of each electric iron is Rs. 1,600. He
sells to ‘B’ and ‘B’ sells it to ‘C’ and ‘C’ sells it to ‘D’ the retailer. The tax rate is 12.5% and
the profit is Rs. 150 at each stage of the selling chain. Find
i) The total amount of VAT.
ii) The amount that the purchaser will have to pay. [A]

. Chapter 13: HEIGHTS AND DISTANCES

Four Marks Questions:

1. The angles of elevation of the top of a tower from the base and the top of a building are
60° and 45° . The building is 20 metres high. Find the height of the tower. (E)
2. A person standing on the bank of a river observes that the angle subtended by a tree on the
opposite bank is 60° . When he returns 40 metres from the bank, he finds the angle to
be 30° . Find the height of the tree and the breadth of the river. (A)
3. Two towers of height 14m and 25m stand on level ground. The angles of elevation of their
tops from a point on the line joining their feet are 45° and 60° respectively. Find the
distance between the towers. (E)
4. From the top of a house 32m high, the angle of elevation of the top of a tower is 45° and the
angle of depression of the foot of the tower is 30° .Find the height of the tower. (E)
5. From a point A due north of the tower, the elevation of the top of a tower is 60° . From a
point B due south, the elevation is45° .If AB= 100metres. Show that the height of the tower
is 50√3(√3 − 1) metres. (D)
6. A flagstaff stands upon the top of a building. At a distance of 20metres, the angles of the
top of the flag staff and building are 60° and 45° respectively. Find the height of the
flagstaff. (E)
7. The angles of elevation of the top of a tower from two points distant a and b (a<b) from its
foot and the same the straight line from it are 30° and 60° . Show that the height of the tower
is√𝑎𝑏. (A)
8. From a point on the line joining the feet of two poles of equal heights, the angles of
elevation of the tops of the poles are observed to be 30° and 60° . If the distance between the
poles is a, find
(i) The height of the poles
(ii) The position of the point of observation (E)

9. The angles of depressions of the two boats as observed from the mast head of a ship 50m
high are 45° and 30° .What is the distance between the boats if they lie on the same side of
the mast head in line with it. (E)
10.A person is at the top of a tower 75feet high. From there, he observes a vertical pole and
finds the angles of depression of the top and bottom of the pole which are 30° and
60° respectively. Find the height of the pole . (E)
11. An aeroplane when flying at a height of 2000m passes vertically above another plane at an
instant when their angles of elevation from the same point of observation are 60° and
45° respectively. Find the distance between the aeroplanes. (D)
12.The angles of elevation of the summit of a hill from the top and the bottom of a tower are
3ℎ
30° and60° respectively. If the height of the tower is h, show that the height of the hill is .
2

(A)
13.The shadow of a tower standing on a level plane is found to be 50 metres longer when the
Sun’s altitude is 30° than when it is60° . Find the height of the tower.
(A)
14.A person at the top of a hill observes that the angles of depression of two consecutive
kilometres stones on a road leading to the foot of the hill and in the same vertical plane
containing the position of the observer are 30° and60° . Find the height of the hill. (D)
15.The angles of elevation of the top of a tower from the base and the top of a building are
60° and 30° . The building is 20 metres high. Find the height of the tower. (E)
16.A flagstaff stands upon the top of a building. At a distance of 20metres, the angles of the
top of the flag staff and building are 60° and 30° respectively.
Find the height of the Flag staff. (E)

Chapter 14: COMPOUND ANGLES, MULTIPLE AND SUBMULTIPLE ANGLES AND
TRANSFORMATIOM FORMULAE

One Mark Questions:

Answer the following multiple choice questions

3 4
1. If sin A  , cos B  and A,B are acute angles, then the value of sin  A  B  is
5 5

3 1 3 1 3 1 3 1
a) b) c) d) [E]
2 2 2 2 2 2

2. The value of sin15 is [E]
a) 2  3 b) 2  3
3 1 3 1
c) d)
2 2 2 2

3. The value of cos 75 is [E]
1 2 3 1 c)
1 3
a) b) d)
2 2 2 2 2 2 2 2
1 1
4. If tan A  , tan B  ,then tan  A  B  is [A]
2 3

a) 0 b)
1 c) ∞ d) 1
7

1
5. If sin A  then sin 2A is [E]
2

√3
a) b) 0 c)1 d) 2
2

3
6. If cos A  then cos 2A is [E]
2
2 1
a) 2√3 b) c) d) 1
√3 2

7. The value of 3sin10  4sin 3 10 is [A]

a ) 10 b) 0 c) 1 d) ½

8. sin 4 A.cos 2 A expressed as sum of two trigonometric functions is [D]
1 1
a)  sin 6 A  sin 2 A b)  sin 6 A  cos 2 A
2 2
1 1
c)  cos 6 A  sin 2 A d)  cos 6 A  cos 2 A
2 2
9. cos65.cos15 expressed as sum of two trigonometric functions is [D]
1 1
a)  cos 70  cos 50  b)  cos80  cos 50 
2 2

1 1
c)  sin 70  sin 50  d)  sin 80  sin 50 
2 2

10. 2cos70.cos10 expressed as sum of two trigonometric functions is [D]
1 1 1 3
a)  cos 80   b)  cos80  
2 2 2 2 

1
c) cos80 
2

3
d) cos 80 
2

11. The value of 4 cos3 10  3cos10 is [A]
√3 1
b) 10 b) c) 30 d)
2 2

12.Match the following [A]

i. The value of cos3A if cos A 
4
is 1 3
5
a.
2 2
1 120
ii. If tan A  , tan 2A is b.
3 169

iii. If sin  
12
and  is acute, then 1 3
13
c.
2 2
sin 2 is
d.√3

iv. The value of cos105 is −44
e.
125

13.Express sin5  sin as product of two functions. [D]
14.Express cos 4 A.sin 2 A as sum or difference of two functions. [D]
15.Express sin80  sin 40 as product of two functions. [D]
16.Write sin12x  sin 4x as product of two trigonometric functions. [D]
17.Find the value of 𝑆𝑖𝑛80°𝐶𝑜𝑠10° + 𝐶𝑜𝑠80°𝑆𝑖𝑛10° [E]
3
18.If 𝐶𝑜𝑠 𝐴 = find 𝐶𝑜𝑠2𝐴. [E]
4
Five Marks Questions:
cos 7 x  cos 3 x  cos 5 x  cos x
1. Prove that  cot 2 x [A]
sin 7 x  sin 3 x  sin 5 x  sin x
sin 2 A  sin 5 A  sin A
2. Prove that  tan 2 A [A]
cos 2 A  cos 5 A  cos A
sin 5 A  sin 4 A  sin 2 A  sin A
3. Prove that  tan 3 A [A]
cos 5 A  cos 4 A  cos 2 A  cos A
sin 6 A  sin 2 A  2sin 4 A sin 4 A
4. Show that  [A]
sin 7 A  sin 3 A  2sin 5 A sin 5 A

5. If A  B  C  180 , prove that sin 2 A  sin 2B  sin 2C  4sin A.sin B.sin C [D]
6. Prove that cos 𝐴 + cos(120° + 𝐴) + cos (120° − 𝐴) = 0. [D]
7. Prove that tan 𝐴 . tan 3𝐴 . tan 4𝐴 = tan 4𝐴 − tan 3𝐴 − tan 𝐴 . [D]
1−cos 2𝐴+sin 2𝐴
8. Prove that = tan 𝐴 . [D]
1+cos 2𝐴+sin 2𝐴

9. If A+B+C = 𝜋, prove that sin 2𝐴 + sin 2𝐵 − sin 2𝐶 = 4 cos 𝐴 cos 𝐵 cos 𝐶. [D]
Chapter 15: CIRCLES
Six Mark Questions:

1. Show that the points (2, 0), (−1, 3), (−2, 0)and (1, −1) are concyclic. [A]
2. Show that the points (0, 0), (1, 1), (5, −5)and (6, −4) are concyclic. [A]
3. Show that the points (2, −4), (3, −1), (3, −3)and (0, 0) are concyclic [D]
4. Check whether the points (1, 0), (2, −7), (8, 1)and (9, −6) are concyclic or not. [A]
5. Show that the points (4, 8), (8, 6), (−1, 3)and (0, 0) are concyclic. [A]

Chapter 16: PARABOLA

One Mark questions:

1. The focus of the parabola x2 = 16y is (E)
a) (0, 4) b) (4,0) c) (16,0) d) (0,16)
2. The vertex of the parabola x2 = y is (E)
a) (0, 0) b) (1,0) c) (-1,0) d) (0,1)
3. The length of the latus rectum of the parabola y2 = 8x is (E)
a) 8 b) 4 c) 16 d) 2

4. The axis of the parabola x2 = 5y is (E)
a) y axis b) x axis c) origin d) y=0

5. The equation of the latus rectum of the parabola y2 = 4ax is (E)
a)x +a =0 b) x-a =0 c) y+a =0 d) y-a =0

6. If the length of the latus rectum of the parabola x2 = 4ky is 8, then the value of k is .. (A)

7. The axis of the parabola x2 = 16y is given by… (E)

8. The equation of the directrix of the parabola x2 = 8y is ….. (E)

9. Length of the latus rectum of the parabola y2 = 16x is….. (E)

Two Marks questions:

1. Find the equation of the parabola given that its vertex is (0,0) and focus is (4,0). (E)
2. Find the equation of the parabola given that its vertex is (0,0) and the directrix is y=-3. (E)
3. Find the equation of the parabola given that its focus is (0,1) and directrix is y= -1. (A)
4. Find the equation of the parabola whose focus is (0,-3) and its directrix is y= 3. (A)
5. Find the equation of the parabola whose focus is (0,6) and its vertex is (0,0). (E)
6. Find the equation of the parabola given that its vertex is (0,0) and focus is (-3, 0). (E)
7. Find the equation of the parabola given that its vertex is (0,0) and focus is (0,-5/2). (E)

8. Find the equation of the parabola given that the focus is (5/3, 0) and vertex is (0,0). (E)
9. Find the equation of the parabola given that the focus is (0,-3/5) and the
directrix is 5y = 3. (E)
10.Find the equation of the directrix, length of the latus rectum of the parabola y2= -6x. (E)
11.Find the vertex and focus of the parabola y2 = 12x. (E)
12.Find the equation of the latus rectum, length of the latus rectum of the
parabola x2= -24y. (E)
13.Find the equation of the axis and the length of the latus rectum of the parabola x2 = 15y. (E)
14. Find the focus, equation of latus rectum of the parabola y2 = 4x. (E)
15.Find the equation of the axis, equation of the directrix of parabola
x2 + 8y = 0. (E)
16.Find the axis and the length of latus rectum of the parabola x2= 16y. (E)
17.Find the end points of latus rectum of the parabola y2 = 18x. (E)
18.Find the value of ‘k’ if the length of the latus rectum x2 = -2ky is 8. (E)
19.Find the equation of the parabola whose vertex is (0,0), focus is (-4,0) and equation of the
directrix is x = 4. (E)
20. Find the equation of the parabola whose focus is (0,-4) and the vertex is (0,0). (E)
21.Find the equation of the parabola given that its vertex is (0,0) and the directrix is y=2. (E)
22.Find the focus and the equation of the directrix of the parabola
y2 = -12x. (E)
23. Find the equation of the parabola given that its vertex is (0,0) and passing through the
point (3,-4) and symmetric about the y axis (D)

24. Find the equation of the parabola given that its vertex is (0,0) and the axis along the x axis
and passing through the point P(2,3). (D)

23.Find the equation of the parabola whose focus is (0,6) and the vertex is (0,0). (E)
24.Find the equation of the parabola given that its vertex at the origin , axis is the y axis and
passing through the point (1,3). (D)

Chapter 17: LIMITS AND CONTINUITY

One Mark Questions:

𝑥 2 −4𝑥 2 𝑛
1. Evaluate: lim ( ). [E] 11.Evaluate: lim (1 + ) . [D]
𝑥→3 𝑥−2 𝑛→∞ 𝑛
sin 𝑎𝑥 1
2. Evaluate: lim , 𝑎, 𝑏 ≠ 0. [E] 12.Evaluate:lim(1 + 3𝑥)𝑥 . [D]
𝑥→0 𝑏𝑥 𝑥→0
sin 4𝑥 2
3. Evaluate: lim . [E] 𝑛+3 𝑛
𝑥→0 sin 2𝑥 13.Evaluate:lim ( ) . [D]
𝑛→0 3
sin 𝑥 0
4. Evaluate:lim . [D] 3𝑥 −2𝑥
𝑥→0 𝑥 14.Evaluate:lim ( ). [D]
𝑥→0 𝑥
sin 𝑥 2
5. Evaluate:lim . [A] 2𝑥 −1
𝑥→0 𝑥 15.Evaluate:lim ( ). [D]
𝑥→0 3𝑥
4𝑥+3
6. Evaluate: lim ( ). [A] 𝑒 −𝑥 −1
𝑥→4 𝑥−2 16.Evaluate:lim ( ). [D]
𝑥→0 𝑥
𝑎𝑥 2 +𝑏𝑥+𝑐
7. Evaluate: lim ( ). [A] 𝑥 2 −1
𝑥→1 𝑐𝑥 2 +𝑏𝑥+𝑎 17.Evaluate: lim ( ) [D]
𝑥→1 𝑥−1
8. Evaluate: lim(𝑥 2 + 6𝑥 + 4). [A]
𝑥→1
𝑥 3 +4
9. Evaluate: lim ( ). [D]
𝑥→1 1+𝑥

10.Evaluate: lim(√36 − 𝑥 2 ). [A]
𝑥→2

1

18.Evaluate: Show that the function 𝑓(𝑥) = { (1 + 3𝑥) 𝑥 ≠ 0 is continuous at x = 0. [D]
𝑥

𝑒3 𝑥=0
𝑘 + 𝑥, 𝑥 = 1
19.Evaluate: Find k for which 𝑓(𝑥) = { iscontinuous at x=1. [A]
4𝑥 + 3, 𝑥 ≠ 1
1

20.Evaluate: If the function 𝑓(𝑥) = {(1 + 2𝑥)𝑥 , 𝑥 ≠ 0is continuous at x=0, find k. [D]
𝑘, 𝑥=0
𝑒 2𝑥 −1
21.Evaluate:Find k for which𝑓(𝑥) = { 𝑥 ≠ 0is continuous at x=0. [D]
𝑥
𝑘 𝑥=0
Six Marks Question:

1. If n is a rational number and A is a non-zero real number then prove that
𝑥 𝑛 −𝑎𝑛
lim [ ] = 𝑛𝑎𝑛−1 . [A]
𝑥→𝑎 𝑥−𝑎

Chapter 18: DIFFERENTIAL CALCULUS

One Mark Questions:
𝑑𝑦
1. If y= sin(𝑡𝑎𝑛𝑥) is [E]
𝑑𝑥
a)Cos(tanx) b) cosx. Sec2x c) Cos(tanx) Sec2x d) - Cos(tanx) Sec2x
𝑑𝑦
2. If x-y=0 then is
𝑑𝑥
[A]
a) 1 b) 0 c) -1 d) x
3. If f(x)=x +2x+3 the f1(1) is
2
[A]
a) 6 b) 5 c) 8 d) 4
4. The derivative of√𝑥 is [E]
1 1 1
a) b) 2√𝑥 𝑐) d) √𝑥
√𝑥 2√𝑥 2
5. The derivative of sin2x is [E]
a) Cos2x b) 2sinx c) 2 cosx d) sin2x
𝑑𝑦
6. If y= log(sinx) then is [E]
𝑑𝑥
a ) cosecx b) cosx c ) cotx d) tanx
𝑑𝑦
7. If y= sin2x+ Cos2x then is [D]
𝑑𝑥
a ) sin2x b) sinx +cosx c) 1 d) 0
𝑑𝑦
8. If y=2x then is [A]
𝑑𝑥
a) 2x b) x 2x-1 c) 2x log2 d) 2x log2
4𝑥 2 −3𝑥 𝑑𝑦
9. If y= 𝑡ℎ𝑒𝑛 [A]
𝑥 𝑑𝑥
4𝑥 2 −3𝑥
a) 8x – 3 b) 4 c) d) 4x-3
𝑥2
10.The derivative of log𝑒 𝑒 w.r.t. x is [A]
𝑒 𝑒
a) 1 b) 0 c) 𝑒 d) 1/𝑒
𝑑𝑦
11. If 𝑦 = 𝑙𝑜𝑔𝑒 𝑒 then [E]
𝑑𝑥

a) -1 b) 0 c) e d) 1/e
𝑑𝑦 𝑑𝑦
12. If y=𝑥 𝑒 + 𝑒 𝑥 + 𝑒 𝑒 Find [E] 17.If y= log (𝑥 2 − 2) Find [E]
𝑑𝑥 𝑑𝑥
13.Differentiate the following w.r.t. x 18.Differentiate 𝑡𝑎𝑛4 𝑥 𝑤. 𝑟. 𝑡. 𝑥 [A]
𝑑𝑦
5𝑒 𝑥 − 𝑙𝑜𝑔𝑥 − 3√𝑥 [E] 19.If y = sec(secx) Find [E]
1 2 𝑑𝑦 𝑑𝑥
14.If y= (√𝑥 + ) Find [E] 𝑑𝑦
√𝑥 𝑑𝑥 20.If y =( 𝑎2 − 𝑥 2 )10 Find [A]
𝑑𝑦 𝑑𝑥
15.If y = (3𝑥 2 + 4𝑥 + 5)6 𝐹𝑖𝑛𝑑 [A] 21.If y = log (log (logx) Find
𝑑𝑦
[E]
𝑑𝑥
𝑑𝑦 𝑑𝑥
𝑥2
16.If y= 𝑒 Find [E] 22.If y = cos 𝑥 3 Find
𝑑𝑦
[E]
𝑑𝑥
𝑑𝑥

𝑑𝑦 𝑑𝑦
23.If y = √𝑐𝑜𝑡√𝑥 Find [E] 40.If y = 𝑥 9 ∙ 9𝑥 Find [A]
𝑑𝑥 𝑑𝑥
𝑑𝑦 𝑑𝑦
24.If y = log(sin √𝑥)) Find [E] 41.If y = 2𝑥 − 𝑥 2 − 3𝑙𝑜𝑔𝑥 Find [A]
𝑑𝑥 𝑑𝑥
𝑑𝑦 3𝑥 2 +2𝑥+5 𝑑𝑦
25.If y = 𝑠𝑖𝑛3 √𝑥 Find [E] 42. If y= Find [A]
𝑑𝑥 √𝑥 𝑑𝑥
𝑑𝑦 1 3 𝑑𝑦
26. If 𝑥 2 + 𝑦 2 = 𝑎2 𝐹𝑖𝑛𝑑 [E] 43.If 𝑦 = − 3/2 Find [A]
𝑑𝑥 𝑥 4/3 𝑥 𝑑𝑥
𝑑𝑦 4 2
27. If √𝑥 + √𝑦 = 3 𝐹𝑖𝑛𝑑
𝑑𝑥
[E] 44.If 𝑦 = 𝑥 2 + 2 − 𝑡𝑎𝑛𝑥 + 6𝑒
𝑥 3
2 𝑑𝑦 𝑑𝑦
28.If 𝑦 = 4𝑎𝑥 𝐹𝑖𝑛𝑑 [E] Find [E]
𝑑𝑥 𝑑𝑥
𝑑𝑦 𝑑𝑦
29.If 𝑥 2 − 𝑦 2 = 𝑎2 𝐹𝑖𝑛𝑑 [E] 45.If 𝑦 = √𝑡𝑎𝑛√𝑥 Find [A]
𝑑𝑥 𝑑𝑥
𝑑𝑦 𝑑𝑦
30.If 𝑥 2 = 4𝑎𝑦 𝐹𝑖𝑛𝑑 [E] 46.If 𝑦 = 𝑙𝑜𝑔(𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥) Find
𝑑𝑥 𝑑𝑥
2 2 𝑑𝑦
31.3𝑥 + 4𝑦 = 10 𝐹𝑖𝑛𝑑 [E] [A]
𝑑𝑥 𝑑𝑦
32.If y =
𝑥+1
𝐹𝑖𝑛𝑑
𝑑𝑦
[E] 47.If 𝑦 = 𝑡𝑎𝑛(𝑙𝑜𝑔𝑠𝑖𝑛𝑥) Find [A]
𝑑𝑥
𝑥 𝑑𝑥 𝑑𝑦
2 2 2
𝑑𝑦 48.If 𝑦 = 𝑠𝑖𝑛𝑥. 𝑆𝑖𝑛2𝑥 Find [A]
33.If 𝑥 + 𝑦 = 𝑎 𝐹𝑖𝑛𝑑
3 3 3 [A] 𝑑𝑥
𝑑𝑥 𝑑𝑦
2 𝑑𝑦 49.If 𝑦 = 𝑒 2𝑥 . 𝑆𝑖𝑛3𝑥 Find [A]
34.If y = Find [E] 𝑑𝑥
𝑥+1 𝑑𝑥 𝑑𝑦
𝑑𝑦 50.If 𝑦 = 𝑙𝑜𝑔(𝑙𝑜𝑔(𝑡𝑎𝑛𝑥)) Find [A]
35.If y = (x-a) (x-b) Find [A] 𝑑𝑥
𝑑𝑥 𝑑𝑦
𝑥−𝑎 𝑑𝑦 51.If 𝑦 = 7𝑆𝑖𝑛√𝑥 Find [A]
36.If y = Find [A] 𝑑𝑥
𝑥−𝑏 𝑑𝑥 𝑑𝑦
2 𝑑𝑦 52.If 𝑦 = [𝑙𝑜𝑔(𝑐𝑜𝑠𝑥)]2 Find [A]
37.If y = 𝑠𝑖𝑛 𝑥 Find [A] 𝑑𝑥
𝑑𝑥
6𝑙𝑜𝑔2 𝑥 𝑑𝑦
38.If y = 2 Find [A]
𝑑𝑥
𝑑𝑦
39.If y = sec x ∙ tan x Find [A]
𝑑𝑥

Two Marks Questions:

1. Differentiate 𝑥 𝑥 w.r.t. x [E]
𝑑𝑦 1
2. y= √𝑥 + √𝑥 + √𝑥 + ⋯ ∞ Prove that = [A]
𝑑𝑥 2𝑦−1

𝑑𝑦 𝑠𝑒𝑐 2 𝑥
3. y= √𝑡𝑎𝑛𝑥 + √𝑡𝑎𝑛𝑥 + √𝑡𝑎𝑛𝑥 + ⋯ ∞ Prove that = [A]
𝑑𝑥 2𝑦−1

𝑑𝑦 𝑐𝑜𝑠𝑥
4. y= √𝑠𝑖𝑛𝑥 + √𝑠𝑖𝑛𝑥 + √𝑠𝑖𝑛𝑥 + ⋯ ∞ Prove that = [A]
𝑑𝑥 2𝑦−1
5. Differentiate( 𝑙𝑜𝑔 𝑥 )𝑐𝑜𝑠𝑥 w.r.t x [A]
6. Differentiate 𝑥 𝑠𝑖𝑛𝑥 w.r.t x [A]
𝑑𝑦 𝑙𝑜𝑔𝑥
7. If 𝑥 𝑦 = 𝑒 𝑥−𝑦 Prove that = [A]
𝑑𝑥 (1+𝑙𝑜𝑔𝑥)2

𝑑𝑦
8. If x= a cos𝜃 , y=a sin𝜃 Find [E]
𝑑𝑥
𝑑𝑦
9. If x=𝑎𝑡 2 , y=2at Find [E]
𝑑𝑥
𝑎 𝑑𝑦 𝑦
10.If x = a𝜃 , y= Prove that + =0 [A]
𝜃 𝑑𝑥 𝑥
11.Differentiate 𝑠𝑖𝑛3 𝑥 w.r.t 𝑐𝑜𝑠 2 𝑥 [A]
1 𝑑𝑦
12.If x=log t , y= Find [E]
𝑡 𝑑𝑥
𝑑𝑦
13.If x = a sec𝜃 , y= b tan𝜃 Find [E]
𝑑𝑥
4 𝑑𝑦
14.If x=4t , y= Find [E]
𝑡 𝑑𝑥
15. Differentiate tan√𝑥 w.r.t √𝑥
1
16.Differentiate logx w.r.t [E]
𝑥
𝑑𝑦
17.If 𝑥 𝑦 = 𝑦 𝑥 Find [A]
𝑑𝑥
𝑥2 𝑦2 𝑑𝑦
18.If 2
+ 3 = 1. Find [A]
𝑎 𝑏 𝑑𝑥
𝑑𝑦
19.If 𝑥 3 + 𝑦 3 = 3𝑎𝑥𝑦 . Find [A]
𝑑𝑥
𝑑𝑦 1
20.If y=log (x + √1 + 𝑥 2 ) Prove that = [A]
𝑑𝑥 √1+𝑥 2
1−𝑐𝑜𝑠2𝑥 𝑑𝑦
21.If y = √ Prove that = 𝑠𝑒𝑐 2 𝑥 [A]
1+𝑐𝑜𝑠2𝑥 𝑑𝑥
22.If f(x) =𝑥 − 3𝑥 + 10 Find 𝑓 | (50)
2
[E]
23.If f(x)=𝑥 𝑛 and 𝑓 | (1) = 10. Find n [E]
1 𝑥 2 𝑑𝑦
24.If y= x + Show that − 𝑥𝑦 + 2 = 0 [A]
𝑥 𝑑𝑥
∞
. .. 𝑑𝑦 𝑦2
25.If y= 𝑥 𝑥 Prove that = [A]
𝑑𝑥 𝑥(1−𝑦𝑙𝑜𝑔𝑥)
26.Differentiate 𝑥 𝑥 w.r.t. x [E]
27.Differentiate 𝑥 √𝑥 w.r.t. x [A]
28.Differentiate (𝑠𝑖𝑛𝑥)𝑥 w.r.t. x [A]
29.Differentiate (𝑠𝑖𝑛𝑥)𝑡𝑎𝑛𝑥 w.r.t. x [A]
1 𝑑𝑦
30.x= log(1+t) , y= Find [E]
1+𝑡 𝑑𝑥

Three Marks questions:
𝑑𝑦 𝑦(𝑦−𝑥𝑙𝑜𝑔𝑥)
1. 𝑥 𝑦 = 𝑦 𝑥 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 = [A]
𝑑𝑥 𝑥(𝑥−𝑦𝑙𝑜𝑔 𝑥)
𝑑𝑦 −𝑥
2. If 𝑥 = 𝑒 𝑙𝑜𝑔𝑐𝑜𝑠4𝜃 𝑦 = 𝑒 𝑙𝑜𝑔𝑠𝑖𝑛4𝜃 show that = [A]
𝑑𝑥 𝑦
𝑑𝑦
3. If 𝑥 = 𝑎𝑐𝑜𝑠 4 𝜃 𝑦 = 𝑎𝑠𝑖𝑛4 𝜃 show that = −𝑡𝑎𝑛2 𝜃 [A]
𝑑𝑥

𝑑𝑦
4. If 𝑥 = 𝑒 𝑡 (𝑐𝑜𝑠𝑡 + 𝑠𝑖𝑛𝑡) 𝑦 = 𝑒 𝑡 (𝑐𝑜𝑠𝑡 − 𝑠𝑖𝑛𝑡) show that = −𝑡𝑎𝑛𝑡 [A]
𝑑𝑥
𝑑𝑦
5. If 𝑥 = 𝑎 log(𝑠𝑒𝑐𝜃) 𝑦 = 𝑎(𝑡𝑎𝑛𝜃 − 1) show that = −2𝐶𝑜𝑠𝑒𝑐2𝜃 [A]
𝑑𝑥
1
6. Differentiate 𝑙𝑜𝑔𝑥 𝑤. 𝑟. 𝑡. [A]
𝑥
7. Differentiate 𝑙𝑜𝑔𝑠𝑖𝑛𝑥 𝑤. 𝑟. 𝑡. √𝑐𝑜𝑠𝑥 [A]
8. Differentiate 𝑡𝑎𝑛√𝑥 𝑤. 𝑟. 𝑡. √𝑥 [A]
𝑡 𝑑𝑦
9. If 𝑥 = 𝑎 [𝑐𝑜𝑠𝑡 + 𝑙𝑜𝑔𝑡𝑎𝑛 ] 𝑦 = 𝑎 𝑠𝑖𝑛𝑡 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 = 𝑡𝑎𝑛𝑡 [D]
2 𝑑𝑥
10.Differentiate 𝑡𝑎𝑛2 𝑥 𝑤. 𝑟. 𝑡. 𝑐𝑜𝑠 2 𝑥. [A]
Five Marks Questions:

1. If 𝑦 = 𝑎𝐶𝑜𝑠(𝑙𝑜𝑔𝑥) + 𝑏𝑠𝑖𝑛(𝑙𝑜𝑔𝑥) 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑥 2 𝑦2 + 𝑥𝑦1 + 𝑦 = 0 [A]
2. If 𝑦 = log(𝑥 − √𝑥 2 + 1). Show that (𝑥 2 + 1)𝑦2 + 𝑥𝑦1 = 0 [A]
3. If y=𝑥 + √𝑥 2 − 1. Show that (𝑥 2 − 1)𝑦2 + 𝑥𝑦1 − 𝑦 = 0 [A]
4. If y=(𝑥 + √𝑥 2 + 1)𝑚 .S how that (𝑥 2 + 1)𝑦2 + 𝑥𝑦1 − 𝑚2 𝑦 = 0 [A]
5. If y=sin(logx) . Show that (𝑥 2 𝑦2 + 𝑥𝑦1 + 𝑦 = 0) [A]
𝑑2𝑦
6. If y=a cos m x + b sin m x. Show that + 𝑚2 𝑦 = 0 [A]
𝑑𝑥 2
7. If y= (𝑎2 + 𝑥 2 )6 . Show that (𝑥 2 + 𝑎2 )𝑦2 − 10𝑥𝑦1 − 12𝑦 = 0 [A]
𝑑2𝑦 −24
8. If xy + 6y = 2x. Show that = [D]
𝑑𝑥 2 (𝑥+6) 3
9. If y= 𝑒 𝑥 𝑙𝑜𝑔𝑥 Show that x𝑦2 − (2𝑥 − 1)𝑦1 + (𝑥 − 1)𝑦 = 0 [A]
10.If y=log (x +√𝑥 2 + 1). Show that (𝑥 2 + 1)𝑦2 + 𝑥𝑦1 = 0 [A]
𝑛
11.𝐼𝑓 𝑦 = (𝑥 + √𝑎2 + 𝑥 2 ) Prove that (𝑎2 + 𝑥 2 )𝑦2 + 𝑥𝑦1 − 𝑛2 𝑦 = 0 [A]

19 PPLICATION OF DERIVATIVES
Two Marks Questions:

1. If 𝑆 = 𝑡 3 − 6𝑡 2 + 9𝑡 + 8. Find the initial velocity & acceleration. [E]
2. A particle moves a distance 𝑆 = 6𝑡 2 − 𝑡 3 + 5 then find the velocity and acceleration.
[E]
3. The distance S travelled by a particle in time t seconds is given by
𝑆 = 𝑡 3 − 6𝑡 2 + 15𝑡 + 2, find when the acceleration is zero. [E]

4. If the displacement S at any time t is given by 𝑆 = √1 − 𝑡. Show that the velocity is
inversely proportional to the displacement. [A]
5. If the displacement S at any time t is given by 𝑆 = 2𝑡 3 − 5𝑡 2 + 4𝑡 − 3. Find the time,
when acceleration is 14ft/sec2. [A]
6. If the displacement S at any time t is given by 𝑆 = 2𝑡 3 − 5𝑡 2 + 4. Find the velocity and
displacement at that time. [E]
7. When brakes are applied to a moving car, the car travels a distance S feet in t seconds given
by 𝑆 = 20𝑡 − 40𝑡 2 . When and where the car stops? [E]
8. A particle moves according to the laws 𝑆 = 𝑡 3 + 𝑎𝑡 2 + 𝑏𝑡. Find a and b, if initial velocity
is 5units and when 𝑡 = 1 sec, it is moving with a velocity which is 4 times its initial
velocity. [D]
9. A particle shot vertically upwards rises ‘s’ feet in ‘t’ sec, where 𝑆 = 40𝑡 − 16𝑡 2 . Find the
greatest height attained by the particle. [A]
10. The displacement S of a particle at time t is given by 𝑆 = 4𝑡 3 − 6𝑡 2 + 𝑡 − 7. Find the
velocity and acceleration, when 𝑡 = 2 𝑠𝑒𝑐. [E]
11. If 𝑆 = 5𝑡 2 + 4𝑡 − 8. Find the initial velocity and acceleration. [E]
12. A stone thrown vertically upward rises ‘s’ feet in ‘t’ sec, where 𝑆 = 80𝑡 − 16𝑡 2 . What
will be the velocity and acceleration after 2 secs. [E]
13. A body is thrown vertically upwards, its distance s feet in t sec is given by
𝑆 = 5 + 12𝑡 − 𝑡 2 . Find the greatest height reached by the body. [A]
14. If 𝑆 = 𝑎𝑡 3 + 𝑏𝑡, find ‘a’ and ‘b’ given that when 𝑡 = 3𝑠𝑒𝑐, velocity is 0 and acceleration
is 14 units. [A]
15. When brakes are applied to a moving car, the car travels a distance S feet in t seconds
given by 𝑆 = 8𝑡 − 6𝑡 2 . When and where the car stops? [A]
16. If the cost function of a firm is given by 𝐶(𝑥) = 𝑥 3 − 3𝑥 + 7. Find the average cost and
marginal cost, when the output is 𝑥 = 6 units. [E]
17. The total cost function is given by 𝑞 3 − 3𝑞 2 + 15𝑞 + 27. Find the average cost and
marginal cost. [E]

q3
18. If total cost function is given by 𝐶 = 9𝑞 − 3q2 + . Find the level of output at which
3

average cost is minimized. [A]
19. The total cost of a commodity is given by 𝐶 = 𝑥 2 − 7𝑥 + 2, where 𝑥 is the number of
units and price/unit is Rs. 5.00. find the profit function. [E]
20. Find the average cost and marginal cost, If total cost function of an article is given by
𝐶 = 5x 2 + 2x + 3. [E]
21. The total cost of a commodity is given by 𝐶 = −𝑥 2 + 5𝑥 + 7 and price/unit is 12.00rs.
Find the profit function. [E]
22. For the demand function 2𝑥 − 5𝑦 = 7. Find the total revenue, marginal revenue and
average revenue. [A]
Q3
23. The total cost C for output Q is given by 𝐶 = 300𝑄 − 10Q2 + . Find the level of output
3

at which marginal cost is minimized. [A]
24. If 𝑅 = 250𝑥 + 45𝑥 2 − 𝑥 3 . What will be the marginal revenue, if 𝑥 = 25𝑢𝑛𝑖𝑡𝑠 and what
is average revenue, when 𝑥 = 10𝑢𝑛𝑖𝑡𝑠. [A]
25. The sum of 2 natural numbers is 48. Find them, if their product is maximum. [E]
26. The product of 2 numbers is 64. Find the numbers, when their sum is minimum. [A]
27. The total revenue function is given by 𝑅 = 400𝑥 − 2𝑥 2 and the total cost function is
given by 𝐶 = 2𝑥 2 + 40𝑥 + 4000. Find the marginal revenue and marginal cost function.
[E]
28. If the total cost 𝐶(𝑥) = 𝑥 2 + 2𝑥 + 1 then find the marginal cost and average cost.
[E]
𝑑𝑦
29. If 𝑥 = 𝑎𝑡 2 , 𝑦 = 2𝑎𝑡 then find [A]
𝑑𝑥

30. The total revenue function is given by 𝑅 = 400𝑥 − 2𝑥 2 . Find the marginal revenue and
average revenue function. [E]

Three Marks Questions:

1. The sides of an equilateral triangle is increasing at the rate of √3𝑐𝑚/𝑠𝑒𝑐. Find the rate at
which its area is increasing, when its side is 200cms. [A]
2. The surface area of a spherical bubble is increasing at the rate of 0.6𝑐𝑚2 /𝑠𝑒𝑐. Find the
rate at which its volume is increasing, when its radius is 3cm. [A]
3. A circular patch of oil spreads on water, the area growing at the rate of 16𝑐𝑚2 /𝑚𝑖𝑛.
How fast are the radius and the circumference increasing, when the diameter is 12cms?
[D]
8
4. The radius of a circular plate increases at the rate of 𝑐𝑚/𝑠𝑒𝑐. Find the rate of change
3𝜋

in the area, when the diameter is 12cm. Also find the rate of increase of circumference of
the circle after 3secs. [D]
5. The volume of a sphere is increasing at the rate of 4𝜋 𝑐. 𝑐/𝑠𝑒𝑐. Find the rate at which
area of its surface increases, when the radius is 10cm. [A]
6. A spherical balloon is being inflated so that its volume is increasing at the rate of
30 𝑐. 𝑐/𝑚𝑖𝑛. How fast its surface area increasing, when its volume is 36𝜋 𝑐. 𝑐. [E]
7. A square plate is expanding uniformly, the side is increasing at the rate of 5𝑐𝑚/𝑠𝑒𝑐.
What is the rate at which the area and its perimeter is increasing, when the side is 20cm
long. [A]
8. The sides of an equilateral triangle are increasing at the rate of 3 𝑐𝑚/𝑠𝑒𝑐. How fast is its
area increasing, when the side is 10 cm. [E]
9. The edge of a variable cube is increasing at the rate of 6 𝑐𝑚/𝑚𝑖𝑛. How fast is the volume
and its surface area increasing when the edge is 10 cm long. [A]
10. A ladder 17 feet long leaning against a smooth vertical wall. If the lower end is moving
at the rate of 12 𝑓𝑡/𝑚𝑖𝑛. Find the rate at which the upper end is moving, when the lower
end is 8𝑓𝑡 from the wall. [D]
11. The radius of a sphere is increasing at the rate of 0.5 𝑚𝑡/𝑠𝑒𝑐. Find the rate of increase of
its Surface area and volume after 3 sec. [A]

12. The surface area of a spherical bubble is increasing at the rate of 0.8 𝑐𝑚²/𝑠𝑒𝑐. Find at
what rate is its volume increasing when 𝑟 = 2.5𝑐𝑚. [E]
13. The surface area of a spherical bubble is increasing at the rate of 0.6 𝑐𝑚²/𝑠𝑒𝑐. Find at
what rate is its volume increasing when 𝑟 = 3𝑐𝑚. [E]
14. A spherical balloon is being inflated at the rate of 35𝑐𝑐/𝑠𝑒𝑐. Find the rate at which the
surface area of the balloon increases, when its diameter is 14 cm. [E]
2
15. The radius of circular plate is increasing at the rate of 𝑐𝑚/𝑠𝑒𝑐. Find the rate of change
3𝜋

of its area, when the radius is 6 cm. [E]
16. Water is being poured at the rate of 30𝑚𝑡 3 /𝑚𝑖𝑛 into a cylindical vessel whose base is a
circle of radius 3mts. Find the rate at which the level of water is rising. [A]
17.A man 6 ft tall is moving directly away from a lamp post of height 10 ft above the
ground. If he is moving at the rate of 3 ft/sec. Find the rate at which the length of his
shadow is increasing and also tip of shadow is moving. [D]
18. The volume of a spherical ball is increasing at the rate of 4𝜋 𝑐. 𝑐/𝑠𝑒𝑐. Find the rate of
increase of the radius of the ball, when the volume is 288𝜋 𝑐. 𝑐. [A]
19. A drop of ink spreads over a blotting paper so that the circumference of the blot is
4𝜋 𝑐𝑚 and it changes at 3𝑐𝑚/𝑠𝑒𝑐. Find the rate of increase of its radius and also find
rate of increase of its area. [A]
20. A circular plate of metal is heated so that its radius increases at the rate of 0. 1 𝑚𝑚/𝑚𝑖𝑛.
At what rate is the plates area increasing, when the radius is 25 𝑐𝑚 (1𝑐𝑚 = 10𝑚).
[A]
2
21. A circular patch of oil spreads on water, the area growing at the rate of 16𝑐𝑚 ⁄𝑚𝑖𝑛. How
fast are the radius and the circumference increasing when the diameter is 12cms?
[A]
250
22. Find the minimum value of 𝑥 2 + [D]
𝑥

23. Find maximum and minimum value of the function
𝑓(𝑥) = 3𝑥 3 − 9𝑥 2 − 27𝑥 + 30 [A]

24. Find the maximum and minimum value of 𝑓(𝑥) = 𝑥 5 − 5𝑥 4 + 5𝑥 3 − 1 [D]
25. Show that 𝑥 3 − 6𝑥 2 + 12𝑥 − 3 has neither a maximum nor minimum at 𝑥 = 2. [A]
26.Divide the number 40 into two parts such that their product is maximum. [D]
27. The product of two natural number is 144, find the numbers if their sum is minimum.
[E]
28.Divide 64 into two parts such that the sum of the cubes of two parts in minimum.
29.Find the maximum and minimum of the function 𝑓(𝑥) = 𝑥 3 − 12𝑥 2 + 36𝑥 − 4. [A]
30. Find the maximum and minimum value of the function 𝑓(𝑥) = 2𝑥 3 − 3𝑥. [E]
31.Find the maximum and minimum value of the function
𝑓(𝑥) = 𝑥 3 − 6𝑥 2 + 9𝑥 + 15 (0 ≤ 𝑥 ≤ 6). [A]
32.Find the maximum and minimum value of the function
𝑓(𝑥) = 𝑥 4 − 62𝑥 2 + 120𝑥 + 9 [D]
33. Find the maximum and minimum value of the function
𝑓(𝑥) = 2𝑥 3 − 3𝑥 2 + 312𝑥 + 12 [A]
34.Find the maximum and minimum value of the function
𝑓(𝑥) = 2𝑥 3 − 3𝑥 2 + 36𝑥 + 10 [A]
35.Find the maximum and minimum value of the function 𝑓(𝑥) = 9𝑥 3 + 12𝑥 + 2. [A]

Chapter 20: INDEFINITE INTEGRALS

One Mark Questions:

1. The value of  e3x dx is _______ [E]
a) e3 x e3 x e3 x
c) c d)
3 3
b) 3e3 x  c
x 4  3x 2  5 x
2.  dx is equal to ______ [A]
x2
x3 b) 3x 2  3x  log x  c x2
a)  3x  5log x  c  3  log x  c
3 c) x3  3x  5log x  c d) 2
3.   xe  e x  log e a  dx is equal to _____ [A]
a) e  xe1   e x  c b)
x e 1
 ex 1  c
e 1
x e1 d) xe  e x  1  c
c)  e x  x log e a  c
e 1
1
4.  dx is equal to _____ [A]
7x  8
a) log 7 x  8  c log 7 x  8 log 7 x  8 7x  8
b) c c) c d) log c
7 8 7

5. The value of  e5 x 7 dx is ______ [E]
a) e5 x  c b) e5 x  7  c c) 5e5 x 7  c d) 7e5 x 7  c
6.  tan xdx is _____ [A]
a) log(sin x)+c b) log(cot x)+c c) log(sec x)+c d) log(sec2 x)+c
7.  e34 x dx is _______ [A]
a) 3e3-4x +c 𝑒 3−4𝑥 𝑒 3−4𝑥
c) +𝑐 d) +𝑐
b) -4e3-4x +c 4 −4

8. The value of  sec2  x  5dx is [A]
a) tan(x-5)+c b) sec(x-5).tan(x-5)+c
c) tan2(x-5)+c d) none of these
1
9. The value of  dx is [A]
10 x  3
1 1 3 log 10 x  3
a) c b) c c) c d) c
10 x  3 10 x  3
2
10 x 10
9 cos x
10.Integral of with respect to x is [A]
5sin 2 x

9 9 9sin x 9sin x
a) cos ecx  c b) cos ecx  c c) c d) c
5 5 5cos 2 x 5cos 2 x

2x
11.The value of  dx is [E]
1  x2
a) 1+x2+c b) 2x+c c) log(1+x2)+c d) log 2x+c
x
e
12.  x dx is equal to [A]
e 1
a) log(ex+1)+c b) log ex + c c) ex + c d) ex +1 + c
1
13.The value of  2 dx is [E]
x
1 2 −2 −1
a) + 𝑐 b) + 𝑐 c) 2
+𝑐 d) +𝑐
𝑥 𝑥 𝑥 𝑥
2
14.The integral of 4 cos ec x with respect to x is [E]
a) sec2 x +c b) 4 sec2 x +c c) – cot x +c d) – 4 cot x +c
9
15.Find the integral of w.r.t x. [E]
sin 2 x

16.Evaluate   4sec2 x  dx [E]
3
17.Evaluate  dx [E]
x
1
18.Find  dx [E]
5e  x

19.Find   7 x  10.9x  dx [A]

20.Evaluate   tan 2 x dx [E]
1
21.Evaluate ∫ 𝑑𝑥 [E]
𝑥

Two Marks Questions:
Evaluate the following

1. 
x5  5 x 2  7 x
dx [E] 5.   1  cos 2x  dx [A]
x
2
6. 
 1  sin 2x  dx [A]
2.   x   dx
1
[E]
 x

7.  sec x  sec x  tan x  dx
3

3.   x   dx
1 [E]
[E]
 x
cos 2 x
4.   1  sin 2x  dx [A] 8. 
1  sin x
dx [A]

9.  x  x  dx ex 1
1
[A] 19.  x dx [E]
 x  e x

10.   sin 2 x  dx [A] 20. 
1
dx [E]
x  2 log x  5 
11.   sin3 x  dx [A]
1  tan x
21.  dx [A]
12.   cos2 x  dx [A] 1  tan x
tan x
13.   cos3 x  dx [A] 22. 
2  log  sec x 
dx [A]

14.   tan 2 x  dx [E]
23.  e x  e  1dx
x
[E]
15.  tan 2  3  2x  dx [E]
1
24.   log x  dx
3
[A]
16.  cos ec  3  5x  .cot 3  5x  dx [E] x

25.   2 x  3  x 2  3x  5  2 dx
3
3x  2 [E]
17.  2 dx [E]
3x  4 x  5
26.  
log x  dx [E]
4x  3
18.  2 dx [E]
2 x  3x  5

Three Marks Questions:
Evaluate the following
1.   x.sin x dx [E] 10. 
3
dx [E]
 x  1 x  2 
2.   x2 .log x dx [A]
x2
11.  dx [E]
3.   x.log x dx [A]  2 x  1 x  3
4.   x.e x dx [A] 12. 
1
dx [E]
x  x  1 x  2 
5.   x 2 .cos x dx [A]
13.   6 x  5  3x 2  5 x  4  3 dx
5
[E]
6.   x.e dx
2x
[A]
14.  cos ec 2 x  1  cot x  dx [E]
7.   x.sec2 x dx [E]
1  ex
3x  2 15.  dx [D]
8. 
 2 x  3 3x  1
dx [E] x  e  x 5

2
16.   log x  dx
2
3 [A]
9.  2 dx [E] x
x  5x  6

17.  cos2 x  sin x  dx [A] 25. 
2x
dx [A]
2x  3

18.  x 3x 2  5 dx  [A]
26. 
2x  5
dx [A]
3x  4
1
19.  [D]
27.   1  sin 2x  dx
dx
xx [A]
1
20.  dx [A] 28. 
x
x  2 log x  5  dx [A]
x 5

21. 
3x 2
dx [A] 29.  cot 2  5x  3 dx [A]
1  x2
4𝑥+5
2x  5 30.∫ (𝑥−1)(𝑥+2) [E]
22.  2 dx [A]
x  5x  3
1
31.∫ 𝑑𝑥 [D]
23.   cos 7 x.cos 6 x  dx [D] 𝑥(𝑥+2)

24.   sin 2 x.cos3x  dx [D]

Chapter 21: DEFINITE INTEGRALS AND ITS APPLICATIONS TO AREAS

One Mark Questions:
Evaluate the following
1 1 
1.  x 2 dx 5.  e x dx
2
[E] [E]
0 0
8.  sin xdx
0
[E]
 2
6.  7x 2 dx
2 2
1
  sin x  dx
2. [E] [E] 9.  dx [E]
0
1
1
2x  3

2 2
3.   x  dx [E] 7.  cos xdx [E]
1 0

1
4.  e2 x dx [E]
0

Two Marks Questions:

 2

  sec x  dx   x  e  dx
4 x
1. Evaluate 2
[E] 2. Evaluate [E]
1

4


ex  1
1
4. Find the value of 
2
[E]
  sin x  cos x  dx
3. Find [E] dx
0
ex
0

2 2
5. Evaluate  x.e x dx [A] 6. Evaluate   log x  dx [E]
1 1


8. Evaluate   x  x  dx
4
2
[E]
7. Evaluate   sin 2x  dx
0
[E]
0

9. Find the area bounded by x-axis, the curve y  x 2 and ordinates x  0 and x  1 . [A]
10.Find the area bounded by the curve x 2  8 y , y-axis and the lines y  3 and y  6 . [A]
11.Find the area enclosed by the curve y  x 2 , x-axis and the lines x  1 and x  3 . [A]
12.Find the area enclosed by the curve x  2 y 2 , y-axis and the lines y  2 and y  4 . [A]
13.Find the area enclosed by the curve 𝑦 = 𝑥 2 + 2𝑥 between the ordinates
𝑥 = 0 𝑎𝑛𝑑 𝑥 = 2 [E]

Three Marks Questions:
Evaluate the following

  x3
1 3
1.   6 x  1 3x  x  5 dx 2
[E] 6.  dx [A]
0 0
x2

 10 x  3  5 x  3x  7  dx
2x  3
1 1
2. 2
[E] 7.  dx [A]
1 0
3x  5
 3
1
8. 
2

 x  sin x  dx dx [E]
3.
0
[D]
2  x  1 x  2 

2
4.  x  cos x  dx
0
[D]

2x  5
2

5.  2 dx [A]
1
x  5x  3

6. If the MC of a firm is f ( x)  10  6 x  6 x 2  x  output  , find the TC given fixed cost is
₹125. [E]
7. If the MC of a product is 2 x 2  4 x , find TC of producing 16 units, x is the output. [E]
8. If MC function is 3x 2  x  5 , x=output, then find TC, AC, TVC and average variable cost,
given that its fixed cost is Rs.25. [E]
x
9. If MR is f ( x)  20  , find the TR and AR obtained from an output of 30 units. [E]
20
10.If MR  10 x  x 2 , find the TR. Also find its demand function. [E]
11.Find the area between the curves y 2  4 x and x 2  4 y . [A]
12.Find the area bounded by the curve y 2  5 x and the line y  x . [A]
13.Find the area enclosed by the parabola y 2  4 x and the line x  y  0 .[A]

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Study Materials
Notes

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

Class 9 Notes
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Periodic Table
Class 11 Notes
Writing Skills / Formats

Maps of India / World Class 12 Notes

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