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DPP No. 1
DPP Year : 2022
Topic : Matrix

1. If 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛 is a matrix of rank 𝑟 and 𝐵 is a square submatrix of order 𝑟 + 1, then

a) 𝐵 is invertible
b) 𝐵 is not invertible
c) 𝐵may or may not be invertible
d) None of these

1
2. If 𝐴 is square matrix, 𝐴’, is its transpose, then (𝐴 − 𝐴′ )is
2
a) A symmetric matrix b) A skew-symmetric matrix
c) A unit matrix d) An elementary matrix

1 −2
3. Inverse of the matrix 𝐴 = [ ]is
3 4
1 1 −2 1 4 2 4 2 1 4 −2
a) 10 [ ] b) 10 [ ] c) [ ] d) 10 [ ]
3 4 −3 1 −3 1 −3 1

4. Let 𝐴 be a matrix of rank 𝑟.Then,

a) rank (𝐴𝑇 ) = 𝑟 b) rank (𝐴𝑇 ) < 𝑟 c) rank (𝐴𝑇 ) > 𝑟 d) None of these

3 −3 4
5. The adjoint matrix of [2 −3 4] is
0 −1 1
4 8 3 1 −1 0 11 9 3 1 −2 1
a) [2 1 6] b) [−2 3 −4] c) [ 1 2 8] d) [−1 3 3]
0 2 1 −2 3 −3 6 9 1 −2 3 −3

6. If a matrix 𝐴 is such that 3 𝐴3 + 2 𝐴2 + 5 𝐴 + 𝐼 = 0, then 𝐴−1 is equal to

a) −(3 𝐴2 + 2 𝐴 + 5) b) 3 𝐴2 + 2 𝐴 + 5 c) 3 𝐴2 − 2 𝐴 − 5 d) None of these

7. Let 𝐴 = [𝑎𝑖𝑗 ]𝑛×𝑛 be a square matrix, and let 𝑐𝑖𝑗 be cofactor of 𝑎𝑖𝑗 in 𝐴.If𝐶 = [𝑐𝑖𝑗 ], then
a) |𝐶| = |𝐴| b) |𝐶| = |𝐴|𝑛−1 c) |𝐶| = |𝐴|𝑛−2 d) None of these

8. The system of equations 𝑥 + 𝑦 + 𝑧 = 0, 2𝑥 + 3𝑦 + 𝑧 = 0 and 𝑥 = 2𝑦 = 0 has

a) A unique solution; 𝑥 = 0, 𝑦 = 0, 𝑧 = 0 b) Infinite solutions
c) No solutions d) Finite number of non-zero solutions

1 2 3 2
9. If 2𝑋 − [ ]=[ ], then 𝑋 is equal to
7 4 0 −2
2 2 1 2 2 2
a) [ ] b) [ 7 2] c) [ 7 1] d) None of these
7 4 2 2

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1 2
10. Let 𝐴 = [ ] and 𝐴−1 = 𝑥𝐴 + 𝑦𝐼, then the values of 𝑥 and 𝑦 are
−5 1
1 2 1 2 1 2 1 2
a) 𝑥 = − 11 , 𝑦 = 11 b) 𝑥 = − 11 , 𝑦 = − 11 c) 𝑥 = 11 , 𝑦 = 11 d) 𝑥 = 11 , 𝑦 = − 11

11. Let 𝐴 and 𝐵 be two symmetric matrices of same order. Then, the matrix 𝐴𝐵 − 𝐵𝐴 is
a) A symmetric matrix b) A skew-symmetric matrix
c) A null matrix d) The identity matrix

1 𝑥 −3 1 1 0
12. If 𝐴 = [ 2 ] 𝑎, 𝐵 = [ ] and adj 𝐴 + 𝐵 = [ ], then the values of 𝑥 and 𝑦 are respectively
𝑥 4𝑦 1 0 0 1
a) (1, 1) b) (−1, 1) c) (1, 0) d) None of these

13. Let 𝑝 is a non-singular matrix such that 1 + 𝑝 + 𝑝2 +. . . +𝑝𝑛 = 𝑂 (𝑂 denotes the null matrix), then 𝑝−1 is
a) 𝑝𝑛 b) −𝑝𝑛 c) −(1 + 𝑝+. . . +𝑝𝑛 ) d) None of these

𝑥 5 10 −5 5
1
14. If [𝑦] = [−5 −2 13 ] [0], then the value of 𝑥 + 𝑦 + 𝓏 is
40
𝓏 10 −4 6 5
a) 3 b) 0 c) 2 d) 1

0 1
15. The matrix [ ] is the matrix reflection in the line
1 0
a) 𝑥 = 1 b) 𝑥 + 𝑦 = 1 c) 𝑦 = 1 d) 𝑥 = 𝑦

1 − tan 𝜃 1 tan 𝜃 −1 𝑎 − 𝑏
16. If [ ][ ] =[ ] , then
tan 𝜃 1 − tan 𝜃 1 𝑏 𝑎
a) 𝑎 = 1, 𝑏 = 1 b) 𝑎 =sin 2𝜃, 𝑏 = cos 2𝜃
c) 𝑎 =cos2𝜃, 𝑏 = sin 2𝜃 d) None of the above

−1 −2 −2
17. If 𝐴 = [ 2 1 −2], then adj 𝐴 is equal to
2 −2 1
a) 𝐴 b) 𝐴′ c) 3𝐴 d) 3𝐴′

18. Let the homogeneous system of linear equations 𝑝𝑥 + 𝑦 + 𝑧 = 0, 𝑥 + 𝑞𝑦 + 𝑧 = 0, and 𝑥 + 𝑦 + 𝑟𝑧 =

1 1 1
0, where 𝑝, 𝑞, 𝑟 ≠ 1, have a non-zero solution, then the value of 1−𝑝 + 1−𝑞 + 1−𝑟is
a) -1 b) 0 c) 2 d) 1
θ
1 tan 2
19. If 𝐴 = [ θ
]and 𝐴𝐵 = 𝐼, then 𝐵 is equal to
− tan 2 1
θ θ θ
a) cos 2 2 ∙ 𝐴 b) cos 2 2 ∙ 𝐴𝑇 c) cos2 θ ∙ 𝐼 d) sin2 2 ∙ 𝐴

20. The values of 𝑥, 𝑦, 𝓏 in order, if the system of equations 3𝑥 + 𝑦 + 2𝓏 = 3, 2𝑥 − 3𝑦 − 𝓏 = −3, 𝑥 + 2𝑦 + 𝓏 = 4

has unique solution, are
a) 2, 1, 5 b) 1, 1, 1 c) 1, −2, −1 d) 1, 2, −1

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B B B A B D B B C B

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B A A A D C D D B D

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DPP No. 2
DPP Year : 2022
Topic : Matrix

1. Matrix A is such that 𝐴2 = 2𝐴 − 𝐼, where 𝐼 is the indentity matrix, then for 𝑛 ≥ 2, 𝐴𝑛 is equal to
a) 𝑛𝐴 − (𝑛 − 1)𝐼 b) 𝑛𝐴 − 𝐼 c) 2𝑛−1 𝐴 − (𝑛 − 1)𝐼 d) 2𝑛−1 𝐴 − 𝐼

𝑟 𝑟−1
2. Matrix 𝑀𝑟 is defined as 𝑀𝑟 = [ ] , 𝑟 ∈ 𝑁 value of det(𝑀1 ) + det(𝑀2 ) + det(𝑀3 ) +. . . + det(𝑀2007 )
𝑟−1 𝑟
is
a) 2007 b) 2008 c) 20082 d) 20072

3. The number of solutions of the system of equations 𝑥2 − 𝑥3 = 1, −𝑥1 + 2𝑥3 = −2, 𝑥1 − 2, 𝑥1 − 2𝑥2 = 3 is
a) Zero b) One c) Two d) Infinite

4. If 𝐴 = [𝑎𝑖𝑗 ] is a scalar matrix of order 𝑛 × 𝑛 such that 𝑎𝑖𝑖 = 𝑘 for all 𝑖, then trace of 𝐴 is equal to
a) 𝑛𝑘 b) 𝑛 + 𝑘 c) 𝑛/𝑘 d) None of these

5. If 𝐷 = diag[𝑑1 , 𝑑2 , 𝑑3 , … , 𝑑𝑛 ], where 𝑑𝑖 ≠ 0∀ 𝑖 = 1, 2, … , 𝑛 then 𝐷 −1 is equal to

a) 𝑂 b) 𝐼𝑛
c) diag [𝑑1 , 𝑑2 , … , 𝑑𝑛 ]
−1 −1 −1
d) None of the above

1 𝑎 1
6. If 𝐴 = [ ], then lim 𝑛 𝐴𝑛 is
0 1 n→∞
0 𝑎 0 0 0 1
a) [ ] b) [ ] c) [ ] d) None of these
0 0 0 0 0 0

7. The system of equations 2𝑥 + 𝑦 − 5 = 0, 𝑥 − 2𝑦 + 1 = 0,2𝑥 − 14𝑦 − 𝑎 = 0, is consistent.Then, 𝑎 is equal to

a) 1 b) 2 c) 5 d) None of these

8. The system of equation

𝑎𝑥 + 𝑦 + 𝑧 = 𝛼 − 1
𝑥 + 𝛼𝑦 + 𝑧 = 𝛼 − 1
𝑥 + 𝑦 + 𝛼𝑧 = 𝛼 − 1
Has no solution, if 𝛼 is
a) 1 b) Not -2 c) Either-2 or1 d) -2

9. A matrix 𝐴 = |𝑎𝑖𝑗 | is an upper triangular matrix, if

a) It is a square matrix and 𝑎𝑖𝑗 = 0, 𝑖 < 𝑗
b) It is a square matrix and 𝑎𝑖𝑗 = 0, 𝑖 > 𝑗
c) It is not a square matrix and 𝑎𝑖𝑗 = 0, 𝑖 > 𝑗
d) It is not a square matrix and 𝑎𝑖𝑗 = 0, 𝑖 < 𝑗

𝑥 1
10. If A=[ ]and 𝐴2 is the identity matrix, then 𝑥 is equal to
1 0
a) -1 b) 0 c) 1 d) 2

0 3
11. 𝐴 = [ ]and 𝐴−1 = 𝜆 (adj 𝐴), then 𝜆 equal to
2 0
1 1 1 1
a) − b) c)− d)
6 3 3 6

12. If 𝐴 = [𝑎𝑖𝑗 ] is a 4 × 4 matrix and 𝐶𝑖𝑗 is the cofactor of the element 𝑎𝑖𝑗 in |𝐴|, then the expression 𝑎11 𝐶11 +
𝑎12 𝐶12 + 𝑎13 𝐶13 + 𝑎14 𝐶14 is equal to
a) 0 b) −1 c) 1 d) |𝐴|

13. For what value of 𝜆,the system of equations 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑦 + 3𝑧 = 10, 𝑥 + 2𝑦 + 𝜆𝑧 = 10 is

consistent?
a) 1 b) 2 c) -1 d) 3

1 1
14. If 𝐴 = [ ] , then 𝐴100is equal to
1 1
a) 2100 𝐴 b) 299 𝐴 c) 100 𝐴 d) 299𝐴

cos 2θ − sin 2θ
15. Inverse of the matrix [ ] is
sin 2θ cos 2θ
cos 2θ − sin 2θ cos 2θ sin 2θ
a) [ ] b) [ ]
sin 2θ cos 2θ sin 2θ −cos 2θ
cos 2θ sin 2θ cos 2θ sin 2θ
c) [ ] d) [ ]
sin 2θ cos 2θ −sin 2θ cos 2θ

16. Which of the following is correct?

a) Determinant is square matrix
b) Determinant is a number associated to a matrix
c) Determinant is a number associated to a square matrix
d) None of these

1 0 0 1 cos 𝜃 sin 𝜃
17. If 𝐼 = [ ],𝐽 = [ ] and 𝐵 = [ ], then 𝐵 equals
0 1 −1 0 − sin 𝜃 cos 𝜃
a) 𝐼 cos 𝜃 + 𝐽 sin 𝜃 b) 𝐼 sin 𝜃 + 𝐽 cos 𝜃 c) 𝐼 cos 𝜃 − 𝐽 sin 𝜃 d) −𝐼 cos 𝜃 + 𝐽 sin 𝜃

1 2 3 8
18. What must be the matrix𝑋 if 2𝑋 + [ ]=[ ]?
3 4 7 2
1 3 1 −3 2 6 2 −6
a) [ ] b) [ ]c) [ ]d) [ ]
2 −1 2 −1 4 −2 4 −2

19. 𝐴 and 𝐵 be 3 × 3 matrices. Then, 𝐴𝐵 = 𝑂 implies

a) 𝐴 = 𝑂 and 𝐵 = 𝑂
b) |𝐴| = 𝑂 and |𝐵| = 𝑂

c) Either |𝐴| = 𝑂 or |𝐵| = 𝑂

d) 𝐴 = 𝑂or𝐵 = 𝑂

𝑥 3 1 −1 −2
𝑦
20. Let 𝑋 = [ ] , 𝐷 = [ 5 ]and 𝐴 = [2 1 1 ], if 𝑋 = 𝐴−1 𝐷, then 𝑋 is equal to
𝓏 11 4 −1 −2
8 8 8
1 3 −3 3
a) [0] b) [−1] c) [ 1 ] d) [ 1 ]
2 3 3
0 0 −1

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A D A A C A D D B B

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A D D B D C A A C B

DPP No. 3
DPP Year : 2022
Topic : Matrix

1. If 𝐴 and 𝐵 are matrics such that 𝐴𝐵 and 𝐴 + 𝐵 both are defined, then
a) 𝐴and𝐵 can be any two matrices
b) 𝐴and𝐵 are square matrices not necessarily of the same order
c) 𝐴, 𝐵are square matrices of the same order
d) Number of columns of 𝐴 is same as the number of rows of 𝐵

2. Let 𝑎, 𝑏, 𝑐 be any real numbers. Suppose that there are real numbers 𝑥, 𝑦, 𝑧 not all zero such that 𝑥 = 𝑐𝑦 +
𝑏𝑧, 𝑦 = 𝑎𝑧 + 𝑐𝑥, and 𝑧 = 𝑏𝑥 + 𝑎𝑦 have non-zero solution. Then, 𝑎2 + 𝑏 2 + 𝑐 2 + 2 𝑎𝑏𝑐 is equal to
a) 1 b) 2 c) -1 d) 0

3. If 𝐼𝑛 is the identity matrix of order 𝑛, then rank of 𝐼𝑛 is

a) 1 b) 𝑛 c) 0 d) None of these

8 −6 2
4. If the matrix 𝐴 = [−6 7 −4] is singular, then 𝜆 is equal to
2 −4 𝜆
a) 3 b) 4 c) 2 d) 5

1 2
5. If 𝐴 = [ ], then 𝐼 + 𝐴 + 𝐴2 + 𝐴3 + ⋯ ∞ equals to
3 4
1 1 1
1 0 −1 −2 ½ −3 −
a) [ ] b) [ ] c) [ 1 ] d) [ 1 4 3
]
0 1 −3 −4 −2 0 0
2

6. If 𝐴 is a non-singular square matrix of order 𝑛, then the rank of 𝐴 is

a) Equal to 𝑛 b) Less than 𝑛 c) Greater than 𝑛 d) None of these

1 −2 3 6
7. If A=[ ] and 𝑓(𝑡) = 𝑡 2 − 3𝑡 + 7, then 𝑓(𝐴) + [ ]is equal to
4 5 −12 − 9
1 0 0 0 0 1 1 1
a) [ ] b) [ ] c) [ ] d) [ ]
0 1 0 0 1 0 0 0

8. The system of linear equations

𝑥+𝑦+𝑧 = 2
2𝑥+𝑦−𝑧 =3
3 𝑥 + 2 𝑦 + 𝑘𝑧 = 4 has a unique solution if
a) 𝑘 ≠ 0 b) −1 < 𝑘 < 1 c) −2 < 𝑘 < 2 d) 𝑘 = 0

9. The number of solutions of the system of equations

2𝑥 + 𝑦 − 𝑧 = 7, 𝑥 − 3𝑦 + 2𝑧 = 1, 𝑥 + 4𝑦 − 3𝑧 = 5is
a) 0 b) 1 c) 2 d) 3
3 −4
10. If 𝑋 = [ ], the value of 𝑋 𝑛 is equal to
1 −1
3 𝑛 −4 𝑛 2+𝑛 5−𝑛 3𝑛 (−4)𝑛
a) [ ] b) [ ] c) [ 𝑛 ] d) None of these
𝑛 −𝑛 𝑛 −𝑛 1 (−1)𝑛

11. If 𝐼3 is the identity matrix of order 3, then (𝐼3 )−1 =

a) 0 b) 3 𝐼3 c) 𝐼3 d) Not necessarily exists

12. If 𝐴 = [𝑎𝑖𝑗 ] is a square matrix of order 𝑛 × 𝑛 and 𝑘 is a scalar, then |𝑘𝐴| =

a) 𝑘 𝑛 |𝐴| b) 𝑘|𝐴| c) 𝑘 𝑛−1 |𝐴| d) None of these

1 0 0
13. If 𝐴 = [0 1 0 ], then 𝐴2 is equal to
𝑎 𝑏 −1
a) Null matrix b) Unit matrix c) −𝐴 d) 𝐴

α 0 1 0
14. If 𝐴 = [ ]and 𝐵 = [ ], then value of α for which 𝐴2 = 𝐵 is
1 1 5 1
a) 1 b) −1 c) 4 d) No real values

4 0 0
15. If 𝐴 is a square matrix such that 𝐴 (adj 𝐴) = [0 4 0], then |adj 𝐴| =
0 0 4
a) 4 b) 16 c) 64 d) 256

𝜔 0
16. If 𝜔 is a complex cube root of unity and 𝐴 = [ ], then 𝐴50 is
0 𝜔
a) 𝜔2 𝐴 b) 𝜔𝐴 c) 𝐴 d) 0
1 2 𝑥 1 −2 𝑦
17. If 𝐴 = [0 1 0] and 𝐵 = [0 1 0] and 𝐴𝐵 = 𝐼3 , then 𝑥 + 𝑦 equals
0 0 1 0 0 1
a) 0 b) −1 c) 2 d) None of these

cos θ sin θ
18. The adjoint of the matrix [ ]is
− sin θ cos θ
cos θ − sin θ sin θ cos θ cos θ sin θ −sin θ cos θ
a) [ ] b) [ ] c) [ ] d) [ ]
sin θ cos θ cos θ sin θ − sin θ cos θ cos θ sin θ

0 1 2
19. The inverse matrix of 𝐴 = [1 2 3]is
3 1 1

1 11
− 22 1 5
2 −4 2 1 2 3 1 −1 −1
2 1 1
a) [−4 3 − 1] b) [1 − 6 3] c) 2
[3 2 1] d) 2
[−8 6 −2]
5 31
− 22 1 2 −1 4 2 3 5 −3 1
2

cos θ − sin θ 0
20. If 𝑓(𝜃) = [ sin θ cos θ 0], then{𝑓(𝜃)−1 } is equal to
0 0 1
a) 𝑓(−𝜃) b) 𝑓(𝜃)−1 c) 𝑓(2𝜃) d) None of these

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B A B A C A B A A D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C A B D B A A A A A

DPP No. 4
DPP Year : 2022
Topic : Matrix

1. If the three linear equations

𝑥 + 4𝑎𝑦 + 𝑎𝑧 = 0
𝑥 + 3𝑏𝑦 + 𝑏𝑧 = 0
𝑥 + 2𝑐𝑦 + 𝑐𝑧 = 0
Have a non-trivial solution, where 𝑎 ≠ 0, 𝑏 ≠ 0, 𝑐 ≠ 0, then 𝑎𝑏 + 𝑏𝑐 is equal to
a) 2𝑎𝑐 b) −𝑎𝑐 c) 𝑎𝑐 d) −2𝑎𝑐

2. If 𝐴 and 𝐵 are two matrices such that rank of 𝐴 = 𝑚 and rank of 𝐵 = 𝑛, then
a) rank (𝐴𝐵) = 𝑚𝑛
b) rank (𝐴𝐵) ≥ rank (𝐴)
c) rank (𝐴𝐵) ≥ 𝑟𝑎𝑛𝑘(B)
d) rank (𝐴𝐵) ≤ min(rank 𝐴, rank 𝐵)

3. If 𝐴 is a non-zero column matrix of order 𝑚 × 1 and 𝐵 is a non-zero row matrix of order 1 × 𝑛, then rank of
𝐴𝐵 equats
a) 1 b) 2 c) 3 d) 4

2 1 −3 2 1 0
4. If [ ]𝐴[ ]=[ ], then 𝐴 is equal to
3 2 5 −3 0 1
1 1 1 1 1 0 0 1
a) − [ ] b) [ ] c) [ ] d) [ ]
1 0 0 1 1 1 1 1

5. If 𝐴2 − 𝐴 + 𝐼 = 0, then the inverse of 𝐴 is

a) 𝐼 − 𝐴 b) 𝐴 − 𝐼 c) 𝐴 d) 𝐴 + 𝐼

6. If 𝐵 is an invertible matrix and 𝐴 is a matrix, then

a) rank(𝐵𝐴) = rank(𝐴) b) rank(𝐵𝐴) ≥ rank(𝐵)
c) rank(𝐵𝐴) > rank(𝐴) d) rank(𝐵𝐴) > rank(𝐵)

4 2
7. If 𝐴 = [ ] , |adj 𝐴|is equal to
3 4
a) 6 b) 16 c) 10 d) None of these

cos 𝜃 sin 𝜃 sin 𝜃 − cos 𝜃

8. cos 𝜃 [ ] + sin 𝜃 [ ] is equal to
− sin 𝜃 cos 𝜃 cos 𝜃 sin 𝜃
0 0 1 0 0 1 1 0
a) [ ] b) [ ] c) [ ] d) [ ]
0 0 0 0 1 0 0 1

9. Let 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛 be a matrix such that 𝑎𝑖𝑗 = 1 for all 𝑖, 𝑗. Then,

a) rank (𝐴𝑇 ) > 1 b) rank (𝐴) = 1 c) rank (𝐴) = 𝑚 d) rank (𝐴) = 𝑛

10. Let 𝐴 be a square matrix all of whose entries are integers. Then, which one of the following is true?
a) If det (𝐴)=±1, then 𝐴−1 need not exist
b) If det (𝐴)=±1, then𝐴−1 exists but all its entries are not necessarily integers
c) If det (𝐴)≠ ±1, then 𝐴−1 exists and all its entries are non − integers
d) If det (𝐴) =±1,then 𝐴−1 exists and all its entries are integers

1 0 −𝑘
11. Matrix 𝐴 = [2 1 3 ] is invertible for
𝑘 0 1
a) 𝑘 = 1 b) 𝑘 = −1 c) 𝑘 = ±1 d) None of these

1 − tan 𝜃 1 tan 𝜃 −1 𝑎 −𝑏
12. If [ ][ ] =[ ], then
tan 𝜃 1 − tan 𝜃 1 𝑏 𝑎
a) 𝑎 = 1, 𝑏 = 1 b) 𝑎 = cos 2 𝜃, 𝑏 = sin 2 𝜃
c) 𝑎 = sin 2 𝜃, 𝑏 = cos 2 𝜃 d) None of these

13. If 𝑥 2 + 𝑦 2 + 𝑧 2 ≠ 0, 𝑥 = 𝑐𝑦 + 𝑏𝑧, 𝑦 = 𝑎𝑧 + 𝑐𝑥 and 𝑧 = 𝑏𝑥 + 𝑎𝑦, then 𝑎2 + 𝑏 2 + 𝑐 2 + 2𝑎𝑏𝑐 =

a) 2 b) 𝑎 + 𝑏 + 𝑐 c) 1 d) 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎

1 −3
14. If 𝐴 = [ ]and𝐴2 − 4𝐴 = 10𝐼 = 𝐴 then 𝑘 is equal to
2 𝑘
a) 0 b) -4 c) 4 and not 1 d) 1 or 4

15. Matrix 𝐴 such that 𝐴2 = 2𝐴 − 𝐼, where 𝐼 is the identity matrix. Then, for 𝑛 ≥ 2, 𝐴𝑛 is equal to
a) 𝑛𝐴 − (𝑛 − 1)𝐼 b) 𝑛𝐴 − 𝐼
c) 2 𝐴 − (𝑛 − 1)𝐼
𝑛−1
d) 2𝑛−1 𝐴 − 𝐼

1 3 1 1
16. The matrix 𝐴 satisfying the equation [ ]𝐴 = [ ]is
0 1 0 −1
1 4 1 −4 1 4
a) [ ] b) [ ] c) [ ] d) None of these
−1 0 1 0 0 −1

17. If 𝐴 is an orthogonal matrix, then 𝐴−1 equals

a) 𝐴 b) 𝐴𝑇 c) 𝐴2 d) None of these

1 2 3
18. By elementary transformation method, the inverse of [2 3 4] is
3 4 6
−2 0 1 2 0 −1 1 2 3
a) [ 0 3 −2] b) [ 0 −3 2 ] c) [2 3 4] d) None of these
1 −2 1 −1 2 −1 3 4 6

1 2 3 8
19. What must be the matrix 𝑋, if 2𝑋 + [ ]=[ ]?
3 4 7 2
1 3 1 −3 2 6 2 −6
a) [ ] b) [ ] c) [ ] d) [ ]
2 −1 2 −1 4 −2 4 −2
4 𝑥+2
20. If 𝐴 = [ ] is symmetric, then 𝑥 =
2𝑥 − 3 𝑥 + 1
a) 3 b) 5 c) 2 d) 4

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A D C C A A C D B D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C B C C A C B A A B

DPP No. 5
DPP Year : 2022
Topic : Matrix

3 3 3
1. If A=[3 3 3] , 𝐴4 is equal to
3 3 3
a) 27𝐴 b) 81𝐴 c) 243𝐴 d) 729𝐴
1 𝜔2 𝜔
2. If 𝜔 is a complex cube root of unity, then the matrix 𝐴 = [𝜔2 𝜔 1 ] is a
𝜔 1 𝜔2
a) Singular matrix b) Non-symmetric matrix
c) Skew-symmetric matrix d) None of these

3. The values of 𝜆 and 𝜇 for which of the system of equations 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 = 2𝑦 + 3𝑧 = 10 and 𝑥 + 2𝑦 +

𝜆𝑧 = 𝜇 have infinite number of solutions, are
a) 𝜆 = 3, 𝜇 = 10 b) 𝜆 = 3, 𝜇 ≠ 10
c) 𝜆 ≠ 3, 𝜇 = 10 d) 𝜆 ≠ 3, 𝜇 ≠ 10

4. If 𝐴 and 𝐵 are square matrices of the same order such that (𝐴 + 𝐵)(𝐴 − 𝐵)=𝐴2 − 𝐵2 ,
than(𝐴𝐵𝐴−1 )2 is equal to
a) 𝐵2 b) I c) 𝐴2 𝐵2 d) 𝐴2

5. If 𝐴 is a skew-symmetric matrix, then trace of 𝐴 is

a) 1 b) −1 c) 0 d) None of these

6. A square matrix 𝑃satisfies 𝑃2 = 𝐼 − 𝑃, where 𝐼 is the identity matrix. If 𝑃𝑛 = 5𝐼 − 8𝑃, then 𝑛 is equal to
a) 4 b) 5 c) 6 d) 7

7. Let 𝐴 and 𝐵 are two square matrices such that 𝐴𝐵 = 𝐴 and 𝐵𝐴 = 𝐵, then𝐴2 equals to
a) 𝐵 b) 𝐴 c)𝐼 d) 𝑂

8. 𝐴 and 𝐵 are two square matrices of same order and 𝐴′ denotes the transpose of 𝐴, then
a) (𝐴𝐵) = 𝐵′ 𝐴′ b) (𝐴𝐵)′ = 𝐴′ 𝐵′
c) 𝐴𝐵 = 0 ⇒ |𝐴| = 0or|𝐵| = 𝑂 d) 𝐴𝐵 = 0 ⇒ 𝐴 = 0or𝐵 = 0

1 2 −3
9. The element in the first row and third column of the inverse of the matrix [0 1 2 ] is
0 0 1
a)−2 b) 0 c) 1 d) 7

cos 𝑥 sin 𝑥 0
10. If 𝐴 = [− sin 𝑥 cos 𝑥 0] = 𝑓(𝑥), then 𝐴−1 is equal to
0 0 1
a) 𝑓(−𝑥) b) 𝑓(𝑥) c) −𝑓(𝑥) d) −𝑓(−𝑥)
0 0 1
11. If 𝐴 = [0 1 0],then 𝐴−1 is
1 0 0
a) −𝐴 b) 𝐴 c) 1 d) None of these

1 3
12. If 𝐴 = [ ] and 𝐴2 − 𝑘𝐴 − 5 𝐼2 = 𝑂, then the value of 𝑘 is
3 4
a) 3 b) 5 c) 7 d) −7

13. Consider the following statements :

1. There can exist two matrices 𝐴, 𝐵 of order 2 × 2 such that 𝐴𝐵 − 𝐵𝐴 = 𝐼2
2. Positive odd integral power of a skew-symmetric matrix is symmetric
a) only (1) b) Only (2) c) Both of these d) None of these

1 1 1 𝑥 0 𝑥
14. If [1 −2 −2] [𝑦] = [3], then [𝑦] is equal to
1 3 1 𝓏 4 𝓏
0 1 5 1
a) [1] b) [ 2 ] c) [−2] d) [−2]
1 −3 1 3

15. The number of non-trivial solutions of the system 𝑥 − 𝑦 + 𝑧 = 0, 𝑥 + 2𝑦 − 𝑧 = 0, 2𝑥 + 𝑦 + 3𝑧 = 0 is

a) 0 b) 1 c) 2 d) 3

1−1 𝑥
16. If [ 1 𝑥 1 ]has no inverse, then the real value of 𝑥 is
𝑥 −1 1
a) 2 b) 3 c) 0 d) 1

2+𝑥 3 4
17. If [ 1 −1 2 ] is a singular matrix, then 𝑥 is
𝑥 1 −5
13 25 5 25
a) 25 b) − 13 c) 13 d) 13

2 3 1 4
18. The rank of the matrix 𝐴 = [0 1 2 −1] is
0 −2 −4 2
a) 2 b) 3 c) 1 d) Indeterminate

𝑎 𝑏 𝛼 𝛽
19. If 𝐴 = [ ] and 𝐴2 = [ ], then
𝑏 𝑎 𝛽 𝛼
a) 𝛼 = 𝑎 + 𝑏 , 𝛽 = 𝑎𝑏
2 2

b) 𝛼 = 𝑎2 + 𝑏 2 , 𝛽 = 2 𝑎𝑏
c) 𝛼 = 𝑎2 + 𝑏 2 , 𝛽 = 𝑎2 − 𝑏 2
d) 𝛼 = 2 𝑎𝑏, = 𝑎2 + 𝑏 2

20. If 𝐴 = [𝑎𝑖𝑗 ] is a scalar matrix, then trace of 𝐴 is

a) ∑𝑖 ∑𝑗 𝑎𝑖𝑗 b) ∑𝑖 𝑎𝑖𝑗 c) ∑𝑗 𝑎𝑖𝑗 d) ∑𝑖 𝑎𝑖𝑖

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. D A A A C C B A D A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B B D B A D B A B D

DPP No. 1
DPP Year : 2022
Topic : Indefinite Integration

1+𝑥 4
1. ∫ (1−𝑥4 )3/2 𝑑𝑥 is equal to
1 1
(a) 1 +𝐶 (b) √1 +𝐶
√𝑥 2 − 2 −𝑥 2
𝑥 𝑥2
1
(c) √1 +𝐶 (d) None of these
+𝑥 2
𝑥2

1
2. ∫ 2 𝑑(𝑥 2 + 1) is equal to
√𝑥 +2
(a) 2√𝑥 2 + 2 + 𝐶 (b) √𝑥 2 + 2 + 𝐶
1
(c) (𝑥 2 +𝐶 (d)None of these
+2)3/2

3. Integration of 𝑓(𝑥) = 1 + 𝑥 2 with respect to 𝑥 2 , is

3
2 (1+𝑥 2 )2 2
(a) +𝐶 (b) (1 + 𝑥 2 )3/2 + 𝐶
3 𝑥 3
2𝑥
(c) 3 (1 + 𝑥 2 )3/2 +𝐶 (d) None of these

√cot 𝑥
5. If ∫ sin 𝑥 cos 𝑥 𝑑𝑥 = 𝑃√cot 𝑥 + 𝑄, then P equals
(a) 1 (b) 2
(c) -1 (d) -2

3𝑥−4
6. If 𝑓 (3𝑥+4) = 𝑥 + 2 then ∫ 𝑓(𝑥)𝑑𝑥 is equal to
3𝑥−4 8 2
(a)𝑒 𝑥+2 log 𝑒 |3𝑥+4| (b)− 3 log 𝑒 |1 − 𝑥| + 3 + 𝐶
8 𝑥
(c) 3 log 𝑒 |1 − 𝑥| + 3 + 𝐶 (d)None of these

7. ∫ 𝑓(𝑥)(1 + log 𝑒 𝑥)𝑑𝑥 is equal to

𝑥
(a)𝑥 𝑥 log 𝑒 𝑥 + 𝐶 (b) 𝑒 𝑥 + 𝐶
(c) 𝑥 𝑥 + 𝐶 (d) None of these

1
8. ∫ 1 4 1/2
𝑑𝑥 is
𝑥 5 (1+𝑥 5 )

5
(a) √1 + 𝑥 4/5 + 𝐶 (b) 2 √1 + 𝑥 4/5 + 𝐶
4
(c) 𝑥 5 √1 + 𝑥 4/5 + 𝐶 (d) None of these

5
𝑥2
9. ∫ 𝑑𝑥 is
√1+𝑥 7
2 1 𝑥 7 +1
(a) log|𝑥 7/2 + √1 + 𝑥 7 | + 𝐶 (b) log | |+ 𝐶
7 2 𝑥 7 −1
(c)2√1 + 𝑥 7 + 𝐶 (d)None of these

7𝑥
10. ∫ 77 . 7𝑥 𝑑𝑥 is equal to
7𝑥 7𝑥
77 77
(a) (𝑙𝑜𝑔 3 +𝑐 (b) (log 2 +𝑐
𝑒 7) 𝑒 7)
𝑥
77
(c)7 . (log 𝑒 7)3 + 𝑐 (d)None of these

𝑙𝑜𝑔𝑒 (𝑥+√𝑥 2 +1)

12. The value of ∫ 𝑑𝑥 is
√𝑥 2 +1
2
(a)2 log 𝑒 (𝑥 + √𝑥 2 + 1) + 𝐶 (b) {2 log 𝑒 (𝑥 + √𝑥 2 + 1)} + 𝐶
(c) log 𝑒 (𝑥 + √𝑥 2 + 1) + 𝐶 (d) None of these

√1+𝑥
13. The value of ∫ 𝑥
𝑑𝑥 is
√1+𝑥−1
(a) 2√1 + 𝑥 + log | 1+𝑥+1| + 𝐶 (b) 2√1 + 𝑥 + 𝐶
√
√1+𝑥−1 √1+𝑥−1
(c) 𝑙𝑜𝑔𝑒 | 1+𝑥+1| + 𝐶 (d) +𝐶
√ √1+𝑥+1

1+cos 4𝑥
14. If ∫ cot 𝑥−tan 𝑥 𝑑𝑥 = 𝐴 cos 4𝑥 + 𝐵, then the values of A and B are
1 1
(a) 𝐴 = 8 , 𝐵 ∈ 𝑅 (b) 𝐴 = − 8 , 𝐵 ∈ 𝑅
1
(c) 𝐴 = 4 , 𝐵 ∈ 𝑅 (d)None of these

𝑥7
15. The value of ∫ (1−𝑥2 )5 𝑑𝑥 is
𝑥8 1 𝑥8
(a) (1−𝑥2 )4 + 𝐶 (b) 8 (1−𝑥 2 )4
+𝐶

1 𝑥4
(c) 8 (1−𝑥 2 )4
+𝐶 (d)None of these

1
16. If ∫ 𝑓(𝑥) sin 𝑥 cos 𝑥 𝑑𝑥 = log|𝑓(𝑥)| + 𝐶, then 𝑓(𝑥) =
2(𝑎 2 −𝑏2 )
1 1
(a) 𝑎2 sin2 𝑥+𝑏2 sin2 𝑥 (b) 𝑎2 sin2 𝑥−𝑏2 cos2 𝑥
1 1
(c) 𝑎2 cos2 𝑥+𝑏2 sin2 𝑥 (d) 𝑎2 cos2 𝑥−𝑏2 sin2 𝑥

1
17. The value of ∫ 𝑥 𝑛 (1+𝑥𝑛 )1/𝑛 𝑑𝑥, (𝑛 ∈ 𝑁), is
1 1
1 1 1−𝑛 1 1 1−𝑛
(a) {1 + 𝑛} +𝐶 (b) {1 − 𝑛} +𝐶
1−𝑛 𝑥 1+𝑛 𝑥
1 1
1 1 1− 1 1 1−
(c) {1 − 𝑛} 𝑛 +𝐶 (d) {1 + 𝑛} 𝑛 +𝐶
1−𝑛 𝑥 1+𝑛 𝑥

cos 𝑥−sin 𝑥 sin 𝑥+cos 𝑥

18. If ∫ 𝑑𝑥 = sin−1 ( )+𝐶 then a =
√8−sin 2𝑥 𝑎
(a) 2 (b) 3
(c) 4 (c) None of these

1 1
19. The value of ∫(3𝑥 2 tan 𝑥 − 𝑥 sec 2 𝑥)𝑑𝑥, is
1 1
(a) 𝑥 3 tan + 𝐶 (b) 𝑥 2 tan + 𝐶
𝑥 2
1
(c) 𝑥 tan + 𝑐 (d) None of these
𝑥

1
20. If ∫ 𝑥 log (1 + 𝑥) 𝑑𝑥 = 𝑓(𝑥). log 𝑒 (𝑥 + 1) + 𝑔(𝑥) log 2 𝑥 2 + 𝐿𝑥 + 𝐶, then
𝑥2
(a) 𝑓(𝑥) = 2
(b) 𝑔(𝑥) = 𝑙𝑜𝑔𝑒 𝑥
1
(c) L=1 (d) 𝐿 =
2

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B A B A D B C B A A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. D B A B B A A B A D

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DPP No. 2
DPP Year : 2022
Topic : Indefinite Integration

2 2
𝑒 (𝑥 +4 ln 𝑥) −𝑥 3 3𝑥
1. ∫ 𝑑𝑥 is equal to
𝑥−1
3 ln 𝑥 ln 𝑥 2
𝑒 −𝑒 2 (𝑥−1)𝑥𝑒 𝑥
(𝑎) (
2𝑥
) 𝑒𝑥 + 𝐶 (b) 2
+𝐶
(𝑥 2 −1) 𝑥 2
(c) 2𝑥
𝑒 +𝐶 (d)None of these

2
𝑥 sin 𝑥 2 𝑒 sec 𝑥
2. The value of the integral ∫ 𝑑𝑥 is
cos2 𝑥 2
1 2 1 2
(a)2 𝑒 sec 𝑥 + 𝐶 (b) 2 𝑒 sin 𝑥 + 𝐶
1 2 𝑥2
(c) 2 sin 𝑥 2 𝑒 cos +𝐶 (d)None of these

1
3. ∫ 𝑑𝑥 is equal
(𝑥−1)√𝑥 2 −1
𝑥−1 𝑥−1
(a) − √ + 𝐶 (b) √𝑥+1 + 𝐶
𝑥+1

𝑥+1 𝑥+1
(c)√ +𝐶 (d) √𝑥−1 + 𝐶
𝑥−1

4. ∫ √𝑥 − 3{sin−1(ln 𝑥) + cos −1 (ln 𝑥)}𝑑𝑥 is equal to

𝜋
(a) (𝑥 − 3)3/2 + 𝐶 (b)0
3
(c) does not exist (d)None of these

1−𝑥 7
5. If ∫ 𝑑𝑥 = 𝑎 ln|𝑥| + 𝑏 ln|𝑥 7 + 1| + 𝐶, then
𝑥(1+𝑥 7 )
2 2
(a)𝑎 = 1, 𝑏 = 7 (b)𝑎 = −1, 𝑏 = 7
2 2
(c) 𝑎 = 1, 𝑏 = − 7 (d) 𝑎 = −1, 𝑏 = − 7

sin3 𝑥
6. ∫ (cos4 𝑥+3 cos2 𝑥+1) tan−1 (sec 𝑥+cos 𝑥) 𝑑𝑥 =
(a)tan−1 (sec 𝑥 + cos 𝑥) + 𝐶 (b)𝑙𝑜𝑔𝑒 |tan−1 (sec 𝑥 + cos 𝑥)| + 𝐶
1
(c)(sec 𝑥+cos 𝑥)2 + 𝐶 (d)None of these

1/5
(𝑥−𝑥 5 )
7. ∫ 𝑥6
𝑑𝑥 is equal to
5 1 6/5 5 1 6/5
(a) ( − 1) +𝐶 (b) 24 (1 − 𝑥 4 ) +𝐶
24 𝑥 4
5 1 6/5
(c) − ( − 1) +𝐶 (d)None of these
24 𝑥 4

tan 𝑥
8. ∫ 𝑑𝑥 is equal to
√sin4 𝑥+cos4 𝑥
1
(a) log 𝑒 (tan2 𝑥 + √1 + tan4 𝑥) + 𝐶 (b) 2 log 𝑒 (tan2 𝑥 + √1 + tan4 𝑥) + 𝐶
1
(c) log 𝑒 (tan2 𝑥 + √1 + tan4 𝑥) + 𝐶 (d)None of these
4

9 5
cos3 𝑥
9. ∫ √sin11 𝑥 𝑑𝑥 = −2 {𝐴 tan−2 𝑥 + 𝐵 tan−2 𝑥} + 𝐶 then
1 1 1 1
(a) 𝐴 = 9 , 𝐵 = − 5 (b) 𝐴 = 9 , 𝐵 = 5
1 1
(c) 𝐴 = − 9 , 𝐵 = 5 (d)None of these

𝑓(𝑥)𝑔′ (𝑥)−𝑔(𝑥)𝑓′(𝑥)
10. ∫ 𝑓(𝑥).𝑔(𝑥)
{log 𝑔(𝑥) − log 𝑓(𝑥)}𝑑𝑥 is equal to
𝑔(𝑥) 1 𝑔(𝑥) 2
(a) 𝑙𝑜𝑔𝑒 { }+𝐶 (b) 2 𝑙𝑜𝑔𝑒 {𝑓(𝑥)} + 𝐶
𝑓(𝑥)
𝑔(𝑥) 𝑔(𝑥)
(c) 𝑙𝑜𝑔𝑒 𝑓(𝑥) + 𝐶 (d) None of these
𝑓(𝑥)

𝑓(𝑥)𝑔′ (𝑥)+𝑓′(𝑥)𝑔(𝑥)
11. ∫ {log 𝑓(𝑥) + log 𝑔(𝑥)}𝑑𝑥 is equal to
𝑓(𝑥)𝑔(𝑥)
1
(a) 𝑓(𝑥)𝑔(𝑥) log{𝑓(𝑥)𝑔(𝑥)} + 𝐶 (b) 2 [log{𝑓(𝑥)𝑔(𝑥)}]2 + 𝐶
(c) [log{𝑓(𝑥)𝑔(𝑥)}]2 + 𝐶 (d) log{𝑓(𝑥)𝑔(𝑥)} + 𝐶

12. ∫(𝑥 𝑥 )𝑥 (2𝑥 𝑙𝑜𝑔𝑒 𝑥 + 𝑥)𝑑𝑥 is equal to

𝑥
(a) 𝑥 (𝑥 ) +C (b) (𝑥 𝑥 )𝑥 + 𝐶
(c) 𝑥 𝑥 . log 𝑒 𝑥 + 𝐶 (d) None of these

3
13. The equation of a curve passing through the point (0,1) be given by 𝑦 = ∫ 𝑥 2 . 𝑒 𝑥 𝑑𝑥. If the equation of the
curve be written in the form 𝑥 = 𝑓(𝑦), then 𝑓(𝑦) =
3
(a) √log 𝑒 (3𝑦 − 2) (b) √log 𝑒 (3𝑦 − 2)
(c) 3√log 𝑒 (2 − 3𝑦) (d) None of these

1
14. ∫ sin4 𝑥+cos4 𝑥 𝑑𝑥 =
1 2𝑥 1 1+cos 2𝑥
(a) tan−1 (tan 2) + 𝐶 (b) tan−1 ( )+𝐶
√2 √ √2 √2
1 tan 𝑥+cot 𝑥 √𝑡𝑎𝑛 𝑥+√𝑐𝑜𝑡𝑥
(c) tan−1 ( )+ 𝐶 (d) √2 tan−1 ( )+ 𝐶
√2 √2 √2

sec 𝑥
15. ∫ 𝑑𝑥 is equal to
√sin(2𝑥+𝛼)+sin 𝛼
(a)√2 sec 𝛼(tan 𝑥 + tan 𝛼) + 𝐶 (b) √2 sec 𝛼(tan 𝑥 − tan 𝛼) + 𝐶
(c) √2 sec 𝛼(tan 𝛼 − tan 𝑥) + 𝐶 (d)None of these

16. Let ∫ 𝑒 𝑥 {𝑓(𝑥) − 𝑓 ′ (𝑥)}𝑑𝑥 = 𝜙(𝑥). Then, ∫ 𝑒 𝑥 𝑓(𝑥)𝑑𝑥 is equal to

(a) 𝜙(𝑥) + 𝑒 𝑥 𝑓(𝑥) (b) 𝜙(𝑥) − 𝑒 𝑥 𝑓(𝑥)
1 1
(c) 2 {𝜙(𝑥) + 𝑒 𝑥 𝑓(𝑥)} (d) 2 {𝜙(𝑥) + 𝑒 𝑥 𝑓′(𝑥)}

1 𝑥4
17. If ∫ 𝑥+𝑥 5 𝑑𝑥 = 𝑓(𝑥) + 𝐶, then the value of ∫ 𝑥+𝑥 5 𝑑𝑥 is
(a) log 𝑥 − 𝑓(𝑥) + 𝐶 (b) 𝑓(𝑥) + log 𝑥 + 𝐶
(c) 𝑓(𝑥) − log 𝑥 + 𝐶 (d) None of these

18. If ∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑥), then ∫ 𝑥 3 𝑓(𝑥 2 )𝑑𝑥 is equal to

1 1
(a)2 [𝑥 2 {𝐹(𝑥)}2 − ∫{𝐹(𝑥)}2 𝑑𝑥] (b) 2 [𝑥 2 𝐹(𝑥)2 − ∫ 𝐹(𝑥 2 )𝑑(𝑥 2 )]
1 1
(c) 2 [𝑥 2 𝐹(𝑥) − 2 ∫{𝐹(𝑥 2 )}𝑑𝑥] (d)None of these

19. If n is odd positive integer, then ∫|𝑥 𝑛 | 𝑑𝑥 is equal to

𝑥 𝑛 +1 𝑥 𝑛 +1
(a)| 𝑛+1 | + 𝐶 (b) 𝑛+1 + 𝐶
|𝑥 𝑛 |
(c)𝑛+1 + 𝐶 (d)None of these

𝑒 2𝑥
20. If ∫ 𝑒 𝑎𝑥 cos 𝑏𝑥 𝑑𝑥 = 29
𝑓(𝑥) + 𝐶, then 𝑓 ′ (𝑥) =
(a) 29𝑓(𝑥) (b) −29𝑓(𝑥)
(c) 25𝑓(𝑥) (d) −25𝑓(𝑥)

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. D A D C C B C B B B

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B B B A A C A B C D

DPP No. 1
DPP Year : 2022
Topic : Limit

𝜋
−𝜃
1. lim
𝜋
2
is equal to
𝜃→ cot 𝜃
2
a) 0 b) -1 c) 1 d) ∞

𝑎 𝑥 −𝑏𝑥
2. lim is equal to
𝑥→0 𝑒 𝑥 −1
𝑎 𝑏
a) log 𝑒 ( ) b) log 𝑒 ( ) c) log 𝑒 (𝑎𝑏) d) log 𝑒 (𝑎 + 𝑏)
𝑏 𝑎

2𝑥−1
3. lim is equal to
𝑥→−∞ √𝑥 2 +2𝑥+1
a) 2 b) -2 c) 1 d) -1

√1+√2+𝑥−√3
4. lim = 𝑥−2
is equal to
𝑥→2
1 1
a) b) c) 8√3 d) √3
8√3 √3

1 1
5. lim { 3 − } is equal to
𝑥→0 𝑥 √8+𝑥 2𝑥
1 −4 −16 −1
a) 12 b) 3
c) 3
d) 48

𝑎 𝑥 −𝑏𝑥
6. The value of lim , is
𝑥→0 𝑥
𝑎 𝑏
a) log (𝑏 ) b) log (𝑎) c) log(𝑎𝑏) d) − log(𝑎𝑏)

∑𝑛 𝑟
𝑟=1 𝑥 −𝑛
7. lim 𝑥−1
is equal to
𝑥→1
𝑛 𝑛(𝑛+1)
a) 𝑥 b) 2
c) 1 d) 0

8. The value of lim (log 5 5𝑥)log𝑥 5 is

𝑥→1
a) 1 b) 𝑒 c) −1 d) None of these

𝑥
𝑒 tan 𝑥−𝑒
9. lim =
𝑥→0 tan 𝑥−𝑥
a) 1 b) 𝑒 c) 𝑒 − 1 d) 0

𝑥
𝑥 2 −2𝑥+1
10. The value of lim (𝑥 2 −4𝑥+2) , is
𝑥→∞
a) 𝑒 2 b) 𝑒 −2 c) 𝑒 6 d) None of these

log(𝑥+𝑎)−log 𝑎 log 𝑥−1

11. If lim 𝑥
+ 𝑘 lim 𝑥−𝑒
= 1, then the value of 𝑘 is
𝑥→0 𝑥→𝑒
1 1
a) 1 − b) 𝑒(1 − 𝑎) c) 𝑒 (1 − ) d) 𝑒(1 + 𝑎)
𝑎 𝑎

sin 𝑥
12. The value of lim 𝑥
, is
𝑥→∞
a) 1 b) 0 c) −1 d) None of these

13. lim 𝑥 log sin 𝑥 is equal to

𝑥→0
a) 0 b) ∞
c) 1 d) Cannot be determined

𝑑 1−cos 𝑥
14. lim 𝑑𝑥 ∫ 𝑥2
𝑑𝑥 is equal to
𝑥→0
a) 1 b) 0 c) 1/2 d) None of these

1 𝑎 2 𝑎 2𝑡
15. lim {∫𝑦 𝑒 sin 𝑡 𝑑𝑡 − ∫𝑥+𝑦 𝑒 sin 𝑑𝑡} is equal to (where 𝑎 is a constant)
𝑥
𝑥→0
2𝑦 2𝑦
a) 𝑒 sin b) sin 2𝑦 𝑒 sin c) 0 d) None of these

2𝑓(𝑥)−3𝑓(2𝑥)+𝑓(4𝑥)
16. Let 𝑓′′(𝑥) be continuous at 𝑥 = 0and 𝑓 ′′ (0) = 4. Then lim 𝑥2
is equal to
𝑥→0
a) 11 b) 2 c) 12 d) None of these

[(𝑎−𝑛)𝑛𝑥−tan 𝑥] sin 𝑛𝑥
17. If lim 𝑥2
= 0, where 𝑛 is non-zero real number, then 𝑎 is equal to
𝑥→0
𝑛+1 1
a) 0 b) c) 𝑛 d) 𝑛 +
𝑛 𝑛

𝑥(1+𝑎 cos 𝑥)−𝑏 sin 𝑥

18. The values of 𝑎 and 𝑏 such that lim 𝑥3
= 0, are
𝑥→0
5 3 5 3 5 3
a) 2 , 2 b) 2 , − 2 c) − 2 , − 2 d) None of these

𝑥
𝑥 2 −2𝑥+1
19. The value of lim (𝑥 2 −4𝑥+2) is
𝑥→∞
a) 𝑒 2 b) 𝑒 −2 c) 𝑒 6 d) None of these

(1−cos 2𝑥)
20. The value of lim is
𝑥→0 𝑥2
a) Does not exist b) Infinite c) 0 d) 2

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C A B A D A B B A A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C B A C A C D C A D

DPP No. 3
DPP Year : 2022
Topic : Limit

2𝑥 2 −4𝑓′ (𝑥)
1. If 𝑓 ′ (2) = 2, 𝑓 ′′ (2) = 1, then lim 𝑥−2
, is
𝑥→2
a) 4 b) 0 c) 2 d) ∞

sec2 𝑥
∫2 𝑓(𝑡)𝑑𝑡
2. lim equals
𝑥→𝜋/4 𝑥 2 −𝜋2 /16
8 2 2 1
a) 𝜋 𝑓(2) b) 𝜋 𝑓(2) c) 𝜋 𝑓 (2) d) 4𝑓(2)

𝑓(𝑎)𝑔(𝑥)−𝑓(𝑥)𝑔(𝑎)
3. If 𝑓(𝑎) = 2, 𝑓 ′ (𝑎) = 1, 𝑔(𝑎) = 3, 𝑔′ (𝑎) = −1, then lim 𝑥−𝑎
is equal to
𝑥→𝑎
a) 6 b) 1 c) -1 d) -5

𝑥
𝑥 2 +5𝑥+3
4. If 𝑓(𝑥) = ( ) then lim 𝑓(𝑥) is equal to
𝑥 2 +𝑥+2 𝑥→∞
a) 𝑒 4 b) 𝑒 3 c) 𝑒 2 d) 24

sin−1 𝑥−𝑥
5. lim is equal to
𝑥→0 𝑥 3 cos 𝑥
a) 1/2 b) 1/3 c) 1/6 d) 1/12

1 sin 𝑥
6. For 𝑥 > 0, lim ((sin 𝑥)1/𝑥 + ( ) ) is
𝑥→0 𝑥
a) 0 b) -1 c) 1 d) 2

𝑥+1
3𝑥−4 3
7. The value of lim ( ) is equal to
𝑥→∞ 3𝑥+2
a) 𝑒 −1/3 b) 𝑒 −2/3 c) 𝑒 −1 d) 𝑒 −2

1 2𝑥
8. lim sin−1 (1+𝑥2 ) is equal to
𝑥→0 𝑥
a) -2 b) 0 c) 2 d) ∞

2 sin2 3𝑥
9. lim 𝑥2
is equal to
𝑥→0
a) 0 b) 1 c) 18 d) 36

𝑎 𝑥 +𝑎 −𝑥 −2
10. lim 𝑥2
is equal to
𝑥→0
a)(log 2 𝑎) b) log 𝑎 c) 0 d) None of these

1, when 𝑥 is rational
11. Let 𝑓(𝑥) = { , then lim 𝑓(𝑥) is
0, when 𝑥 irrational 𝑥→0
1
a) 0 b) 1 c) 2 d) None of these

𝑥−3 𝑥
12. For 𝑥 ∈ 𝑅 lim (𝑥+2) is equal to
𝑥→∞
a) 𝑒 b) 𝑒 −1 c) 𝑒 −5 d) 𝑒 5

𝜋 𝜋
13. The value of lim 𝑥 cos (4 𝑥) sin (4 𝑥), is
𝑥→∞
𝜋 𝜋
a) b) c) 1 d) None of these
2 4

𝑓(𝑥)
14. The derivative of function 𝑓(𝑥) istan4 𝑥. If 𝑓(𝑥) = 0, then lim is equal to
𝑥→0 𝑥
a) 1 b) 0 c) -1 d) None of these

(1⁄2){𝑔(𝑥) + (𝑥)}sin(𝑥), 𝑥 ≥ 1
15. Let 𝑓(𝑥) = {
sin 𝑥/𝑥 , 𝑥 < 1
1, if 𝑥 > 0
Where 𝑔(𝑥) = {−1, if 𝑥 < 0
0, if 𝑥 = 0
Then, lim 𝑓(𝑥) is equal to
𝑥→1
a) 0 b) 2 c) sin 1 d) None of these

𝑥 3 +1
16. If lim [ 2 − (𝑎𝑥 + 𝑏)] = 2, then
𝑥→∞ 𝑥 +1
a) 𝑎 = 1and𝑏 = 1 b) 𝑎 = 1 and 𝑏 = −1 c) 𝑎 = 1 and 𝑏 = −2 d) 𝑎 = 1 and 𝑏 = 2

17. If 𝑓: 𝑅 → 𝑅 is defined by
𝑥−2
2
, if 𝑥 ∈ 𝑅 − {1, 2}
𝑓(𝑥) = {𝑥 − 3𝑥 + 2
2, if 𝑥=1
1, if 𝑥=2
𝑓(𝑥)−𝑓(2)
Then lim 𝑥−2
is equal to
𝑥→2
1
a) 0 b) -1 c) 1 d) − 2

𝑓(𝑥)
1 ∫3 2𝑡 3 𝑑𝑡
18. Let 𝑓: 𝑅 → 𝑅 be a differentiable function such that 𝑓(3) = 3, 𝑓 ′ (3) = , Then, the value of lim is
2 𝑥→3 𝑥−3
a) 25 b) 26 c) 27 d) None of these

19. Let 𝑓(𝑎) = 𝑔(𝑎) = 𝑘 and their 𝑛th derivatives 𝑓 𝑛 (𝑎), 𝑔𝑛 (𝑎) exist and are not equal for some𝑛.
Further if
𝑓(𝑎)𝑔(𝑥)−𝑓(𝑎)−𝑔(𝑎)𝑓(𝑥)+𝑔(𝑎)
lim = 4, then the value of 𝑘 is equal to
𝑥→𝑎 𝑔(𝑥)−𝑓(𝑥)

a) 4 b) 2 c) 1 d) 0
sin 𝑥
20. The value of lim , is
𝑥→0 √𝑥 2
a) 1 b) −1 c) 0 d) None of these

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A A D A C C B C C A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B C B B C C B C A D

DPP No. 1
DPP Year : 2022
Topic : Limit

1+sin 𝑥−cos 𝑥+log(1−𝑥)

1. The value of lim 𝑥3
, is
𝑥→0
a) 1/2 b) −1/2 c) 0 d) 1

1+tan 𝑥 cosec 𝑥
2. lim { 1+sin 𝑥 } is equal to
𝑥→0
1
a) 𝑒 b) 1 c) 𝑒 d) 𝑒 2

2 sin2 𝑥+sin 𝑥−1

3. lim𝜋 2 sin2 𝑥−3 sin 𝑥+1 is equal to
𝑥→
6
a) 3 b) -3 c) 6 d) 0

1/𝑥 2
1+5𝑥 2
4. The value of lim (1+3𝑥2 ) is
𝑥→0
1 1
a) 𝑒 2 b) 𝑒 c) 𝑒 d) 𝑒 2

𝑥, 𝑥 < 0
5. If 𝑓(𝑥) = { 1, 𝑥 = 0 , then lim 𝑓(𝑥) is
𝑥→0
𝑥2, 𝑥 > 0
a) 0 b) 1 c) 2 d) Does not exist

6. If 𝑥 is a real number in [0, 1], then the value of lim lim [1 + cos 2𝑚 (𝑛! 𝜋 𝑥)] is given by
𝑚→∞ 𝑛→∞
a) 2 or 1 according as 𝑥 is rational or irrational
b) 1 or 2 according as 𝑥 is rational or irrational
c) 1 for all 𝑥
d) 2 or 1 for all 𝑥

7. lim (1 + cos 𝜋 𝑥) cot 2 𝜋 is equal to

𝑥→1
a) 1 b) −1 c) 1/2 d) −1/2

(𝑒 𝑘𝑥 −1) sin 𝑘𝑥
8. If lim 𝑥2
= 4, then 𝑘 is equal to
𝑥→0
a) 2 b) -2 c) ±2 d) ±4

log(1+𝑥 3 )
9. lim is equal to
𝑥→0 sin3 𝑥
a) 0 b) 1 c) 3 d) None of these
cos 𝑥
10. If 𝑙1 = lim+(𝑥 + [𝑥]), 𝑙2 = lim−(2𝑥 − [𝑥])and 𝑙3 = lim , then
𝑥→2 𝑥→2 𝑥→𝜋/2 (𝑥−𝜋/2)

a) 𝑙1 < 𝑙2 < 𝑙3 b) 𝑙2 < 𝑙3 < 𝑙1 c) 𝑙3 < 𝑙2 < 𝑙1 d) 𝑙1 < 𝑙3 < 𝑙2

sin(1+[𝑥])
, 𝑓𝑜𝑟 [𝑥] ≠ 0
11. If 𝑓(𝑥) = { [𝑥]
0, 𝑓𝑜𝑟 [𝑥] = 0
Where [𝑥] denotes the greatest integer not exceeding 𝑥, then lim− 𝑓(𝑥) is equal to
𝑥→0
a) -1 b) 0 c) 1 d) 2

(1−𝑐𝑜𝑠 2𝑥) sin 5𝑥

12. lim equals
𝑥→0 𝑥 2 sin 3𝑥
a) 10/3 b) 3/10 c) 6/5 d) 5/6

sin(𝑒 𝑥−2 −1)

13. If 𝑓(𝑥) = log(𝑥−1)
, then lim 𝑓(𝑥) is given by
𝑥→2
a) −2 b) −1 c) 0 d) 1

𝑆𝑛+1 −𝑆𝑛
14. If 𝑆𝑛 = ∑𝑛𝑘=1 𝑎𝑘 and lim 𝑎𝑛 = 𝑎, then lim is equal to
𝑛→∞ 𝑛→∞ √∑𝑛 𝑘
𝑘=1

a) 0 b) 𝑎 c) √2 𝑎 d) 2𝑎
sin √𝑥+ℎ−sin √𝑥
15. lim is equal to
ℎ→0 ℎ
a) cos √𝑥 b) 1/(2 sin √𝑥) c) (cos √𝑥)/2√𝑥 d) sin √𝑥

16. The value of lim𝜋(sin 𝑥)tan 𝑥 is

𝑥→
2
a) 1 b) 0 c) 𝑒 d) None of these

17. The value of

𝑥 𝑥 𝑥 𝑥
lim cos (2) cos (4) cos (8) … cos (2𝑛 )is
𝑛→∞
𝑥 𝑥 (sin 𝑥) (cos 𝑥)
a) sin 𝑥 b) cos 𝑥 c) 𝑥
d) 𝑥

sin(𝑒 𝑥−1 −1)

18. The value of the limit lim log 𝑥
is
𝑥→1
1
a) 0 b) 𝑒 c) 𝑒 d) 1

tan 𝜋 𝑥 1 𝑥
19. If 𝑙 = lim + lim (1 + ) , then which one of the following is not correct?
𝑥→−2 𝑥+2 𝑥→∞ 𝑥2
a) 𝑙 > 3
b) 𝑙 > 4
c) 𝑙 < 4
d) 𝑙is a transcendental number

√1−cos 2(𝑥−1)
20. lim 𝑥−1
𝑥→1
a) Exists and is equals √2
b) Exists and is equals −√2
c) Does not exist because 𝑥 − 1 → 0
d) Does not exist because left hand limit is not equal to right hand limit

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B B B A A A C C B C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B A D A C A C D C D

DPP No. 4
DPP Year : 2023
Topic : Limit

(𝑎+ℎ)2 sin(𝑎+ℎ)−𝑎 2 sin 𝑎

1. The value of lim ℎ
, is
ℎ→0
a) 2𝑎 sin 𝑎 + 𝑎 cos 𝑎
2
b) 2𝑎 sin 𝑎 − 𝑎2 cos 𝑎 c) 2𝑎 cos 𝑎 + 𝑎2 sin 𝑎 d) None of these

2𝑓(𝑥)−3𝑓(2𝑥)+𝑓(4𝑥)
2. If 𝑓(𝑥) is differentiable function and 𝑓 ′′ (0) = 𝑎, then lim 𝑥2
is equal to
𝑥→0
a) 3𝑎 b) 2𝑎 c) 5𝑎 d) 4𝑎

𝑒 𝑥 +log(1+𝑥)−(1−𝑥)−2
3. The value of lim is equal to
𝑥→0 𝑥2
a) 0 b) -3 c) -1 d) Infinity

4. If for some real number 𝑘

lim 𝑘𝑥 cosec(𝑥) = lim 𝑥 cosec (𝑘𝑥), then the possible values of 𝑘 are
𝑥→0 𝑥→0
a) 1, -1 b) 0, 1 c) 1, 2 d) 0, 𝜋

|𝑥|
5. The value of lim is
𝑥→0 𝑥
a) 1 b) −1 c) 0 d) None of these

𝜋
6. lim 𝑥 2 sin 𝑥 , us
𝑥→0
a) 1 b) 0 c) Non-existent d) ∞

𝑥 2 +𝑏𝑥+4
7. The value of lim ( ) is
𝑥→∞ 𝑥 2 +𝑎𝑥+5
𝑏 4
a) 𝑎 b) 0 c) 1 d) 5

2 sin 𝑥−sin 2𝑥
8. If 𝑓(𝑥) is the integral function of the function 𝑥3
,𝑥 ≠ 0, then lim 𝑓 ′ (𝑥) is equal to
→
a) 0 b) 1 c) -1 d) None of these

𝑥+1
3𝑥−4 ( 3 )
9. The value of lim (3𝑥+4) , is
𝑥→∞
a) 𝑒 −2/3 b) 𝑒
−1/3
c) 𝑒 −2 d) 𝑒 −1

(1+𝑥)8 −1
10. lim (1+𝑥)2 −1 is equal to
𝑥→0
a) 8 b) 6 c) 4 d) 2

2√2−(cos 𝑥+sin 𝑥)3

11. The value of lim , is
𝑥→𝜋/4 1−sin 2𝑥
3 √2 1
a) b) c) d) √2
√2 3 √2

12. lim (3𝑥 + √9𝑥 2 − 𝑥)equals

𝑥→−∞
a) 1/3 b) 1/6 c) −1/6 d) −1/3

𝑒 5𝑥 −𝑒 4𝑥
13. lim is equal to
𝑥→0 𝑥
a) 1 b) 2 c) 4 d) 5

𝑎−√𝑎 2−𝑥 2 −𝑥 2 /4
14. Let 𝐿 = lim , 𝑎 > 0. If 𝐿 is finite, then
𝑥→0 𝑥4
1 1 1 1
a) 𝑎 = 2, 𝐿 = b) 𝑎 = 1, 𝐿 = c) 𝑎 = 3, 𝐿 = d) 𝑎 = 1, 𝐿 =
64 64 32 32

15. lim 𝑥 log 𝑒 (sin 𝑥) is equal to

𝑥→0
a) -1 b) log 𝑒 1 c) 1 d) None of these

16. The value of lim+ 𝑥 𝑚 (log 𝑥)𝑛 , 𝑚, 𝑛. 𝑁 is

𝑥→0
a) 0 b) 𝑚/𝑛 c) 𝑚𝑛 d) 𝑛/𝑚

√1+𝑥 4 −(1+𝑥 2 )
17. The value of lim is equal to
𝑥→∞ 𝑥2
a) 0 b) -1 c) 2 d) None of these

18. lim (cosec 𝑥)1/ log 𝑥 is equal to

𝑥→0
a) 0 b) 1 c) 1/𝑒 d) None of these

2𝑥−𝑓(𝑥)
19. If 𝑓(1) = 2and 𝑓 ′ (1) = 1, then value of lim 𝑥−1
is
𝑥→1
a) -1 b) 0 c) 1 d) 2

1−cos(1−cos 𝑥)
20. The value of lim 𝑥4
is
𝑥→0
1 1 1 1
a) 2 b) 4 c) 6 d) 8

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A A B A D B C B A C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A B A A B A A C C D

DPP No. 5
DPP Year : 2023
Topic : Limit

1. The value of lim 𝑥 3/2 (√𝑥 3 + 1 − √𝑥 3 − 1), is

𝑥→∞
a) 1 b) −1 c) 0 d) None of these

𝑎 𝑥 −𝑥 𝑎
2. If lim 𝑥 𝑥 −𝑎𝑎 = −1, then 𝑎 equal to
𝑥→𝑎
a) 1 b) 0 c) 𝑒 d) (1/𝑒)

𝑥 3 +1
3. If lim {𝑥 2 +1 − (𝑎𝑥 + 𝑏)} = 2, then
𝑥→0
a) 𝑎 = 1, 𝑏 = 1 b) 𝑎 = 1, 𝑏 = 2 c) 𝑎 = 1, 𝑏 = −2 d) None of these

2
𝑒 𝑥 −cos 𝑥
4. lim 𝑥2
is equal to
𝑥→0
1 3
a) 0 b) c) 1 d)
2 2

12 22 𝑛2
5. lim ( + +. . . + ) is equal to
𝑛→∞ 1−𝑛3 1−𝑛3 1−𝑛3
1 1 1 1
a) b) − c) d) −
3 3 6 6

sin 2𝑥
6. lim𝜋 is equal to
𝑥→ sin 𝑥
6
1 1
a) √3 b) c) 2 d) 2
√3

1−cos(1−cos 𝑥))
7. The value of lim 𝑥4
, is
𝑥→0
1 1 1
a) 8 b) 2 c) 4 d) None of these

|𝑎𝑥 2 +𝑏𝑥+𝑐|
8. Let 𝛼 and 𝛽 be the roots of the equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 1 < 𝛼 < 𝛽. If lim = 1, then
𝑥→𝑚 𝑎𝑥 2 +𝑏𝑥+𝑐
a) 𝑎 < 0and𝛼 < 𝑚 < 𝛽 b) 𝑎 > 0 and 𝑚 > 1 c) 𝑎 > 0 and 𝑚 < 1 d) All the above

𝑥 𝑚 −1
9. lim 𝑥 𝑛 −1 is equal to
𝑥→1
𝑛 𝑚 2𝑚 2𝑛
a) 𝑚 b) 𝑛 c) 𝑛
d) 𝑚

1−cos(𝑎𝑥 2 +𝑏𝑥+𝑐)
10. Let 𝛼 and 𝛽 be the distinct roots of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, then lim (𝑥−𝛼)2
is equal to
𝑥→𝛼
1 𝑎2 𝑎2
a) (𝛼 − 𝛽)2 b) − (𝛼 − 𝛽)2 c) 0 d) (𝛼 − 𝛽)2
2 2 2

2𝑥 −1
11. lim [ ] is equal to
𝑥→0 √1+𝑥−1
a) log 𝑒 2 b) log 𝑒 √2 c) log 𝑒 4 d) 2

𝑥
12. lim ( )is equal to
𝑥→0 √1+𝑥−√1−𝑥
a) 0 b) 1 c) 2 d) -1

log(3+𝑥)−log(3−𝑥)
13. If lim 𝑥
= 𝑘, the value of 𝑘 is
𝑥→0
a) 0 b) -1/3 c) 2/3 d) -2/3

1 𝑐+𝑑𝑥
14. If 𝑎, 𝑏, 𝑐, 𝑑 are positive, then lim (1 + 𝑎+𝑏𝑥) =
𝑥→∞
a) 𝑒 𝑑/𝑏 b) 𝑒 𝑐/𝑎
c) 𝑒 (𝑐+𝑑)/𝑎+𝑏 d) 𝑒

𝑥2
∫0 sec2 𝑡 𝑑𝑡
15. The value of lim ( 𝑥 sin 𝑥
) is
𝑥→0

a) 3 b) 2 c) 1 d) 0

sin 𝑥
16. lim 𝑥
is equal to
𝑥→∞
a) ∞ b) 1 c) 0 d) Does not exist

𝑥 cos 𝑥−log(1+𝑥)
17. lim 𝑥2
equals
𝑥→0
a) 1/2 b) 0 c) 1 d) −1/2

log(𝑟+𝑛)−log 𝑛 1
18. Given that lim ∑𝑛𝑟=1 𝑛
= 2 (log 2 − 2) ,
𝑛→∞
1
lim [(𝑛 + 1)𝑘 (𝑛 + 2)𝑘 … (𝑛 + 𝑛)𝑘 ]1/𝑛 , is
𝑛→∞ 𝑛𝑘
4𝑘 𝑘 4 4 𝑘 𝑒 𝑘
a) 𝑒
b) √𝑒 c) (𝑒) d) (4)

sin |𝑥|
19. lim 𝑥
is equal to
𝑥→0
a) 1 b) 0 c) positive infinity d) does not exist

1
𝑥 2 sin( )−𝑥
20. The value of lim { 𝑥
1−|𝑥|
}, is
𝑥→∞

a) 0 b) 1 c) −1 d) None of these

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A A C D D A A D B D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C B C A C C A C D A

DPP No. 1
DPP Year : 2023
Topic : Continuity & Differentiability

1. Let [𝑥] denotes the greatest integer less than or equal to 𝑥 and 𝑓(𝑥) = [tan2 𝑥]. Then,
a) lim 𝑓(𝑥) does not exist
𝑥→0
b) 𝑓(𝑥) is continuous at 𝑥 = 0
c) 𝑓(𝑥) is not differentiable at 𝑥 = 0
d) 𝑓 ′ (0) = 1

(−𝑒 𝑥 +2𝑥 )
2. The value of 𝑓(0) so that 𝑥
may be continuous at 𝑥 = 0 is
1
a) log ( ) b) 0 c) 4 d) −1 + log 2
2

3. Let 𝑓(𝑥) be an even function. Then 𝑓′(𝑥)

a) Is an even function b) Is an odd function c) May be even or odd d) None of these

[cos 𝜋 𝑥], 𝑥 < 1

4. If 𝑓(𝑥) = { , then 𝑓(𝑥) is
|𝑥 − 2|, 2 > 𝑥 ≥ 1
a) Discontinuous and non-differentiable at 𝑥 = −1 and 𝑥 = 1
b) Continuous and differentiable at 𝑥 = 0
c) Discontinuous at 𝑥 = 1/2
d) Continuous but not differentiable at 𝑥 = 2

|𝑥+2|
, 𝑥 ≠ −2
5. If 𝑓(𝑥) = {tan−1 (𝑥+2) , then 𝑓(𝑥) is
2, 𝑥 = −2
a) Continuous at 𝑥 = −2
b) Not continuous 𝑥 = −2
c) Differentiable at 𝑥 = −2
d) Continuous but not derivable at 𝑥 = −2

6. If 𝑓(𝑥) = | log |𝑥| |, then

a) 𝑓(𝑥) is continuous and differentiable for all 𝑥 in its domain
b) 𝑓(𝑥) is continuous for all 𝑥 in its domain but not differentiable at 𝑥 = ±1
c) 𝑓(𝑥) is neither continuous nor differentiable at 𝑥 = ±1
d) None of the above

𝑥𝑓(𝑎)−𝑎𝑓(𝑥)
7. If 𝑓 ′ (𝑎) = 2 and 𝑓(𝑎) = 4, then lim 𝑥−𝑎
equals
𝑥→𝑎
a) 2𝑎 − 4 b) 4 − 2𝑎 c) 2𝑎 + 4 d) None of these

8. If 𝑓(𝑥) = 𝑥(√𝑥 + √𝑥 + 1), then

a) 𝑓(𝑥) is continuous but not differentiable at 𝑥 = 0 b) 𝑓(𝑥) is differentiable at 𝑥 = 0

c) 𝑓(𝑥) is not differentiable at 𝑥 = 0 d) None of the above

𝑎𝑥 2 + 𝑏, 𝑏 ≠ 0, 𝑥 ≤ 1
9. If 𝑓(𝑥) = { , then, 𝑓(𝑥) is continuous and differentiable at 𝑥 = 1, if
𝑥 2 𝑏 + 𝑎𝑥 + 𝑐, 𝑥 > 1
a) 𝑐 = 0, 𝑎 = 2𝑏 b) 𝑎 = 𝑏, 𝑐 ∈ 𝑅 c) 𝑎 = 𝑏 , 𝑐 = 0 d) 𝑎 = 𝑏, 𝑐 ≠ 0

|𝑥 − 3|, 𝑥 ≥ 1
10. For the function 𝑓(𝑥) = {𝑥 2 3𝑥 13 which one of the following is incorrect?
4
− 2
+ 4
, 𝑥<1
a) Continuous at 𝑥 = 1 b) Derivable at 𝑥 = 1
c) Continuous at 𝑥 = 3 d) Derivable at 𝑥 = 3

11. If 𝑓: 𝑅 → 𝑅 is defined by
2 sin 𝑥 − sin 2𝑥
𝑓(𝑥) = { 2𝑥 cos 𝑥 , if 𝑥 ≠ 0,
𝑎, if 𝑥 = 0
Then the value of 𝑎 so that 𝑓 is continuous at 0 is
a) 2 b) 1 c) -1 d) 0

12. 𝑓(𝑥) = 𝑥 + |𝑥| is continuous for

a) 𝑥 ∈ (−∞, ∞) b) 𝑥 ∈ (−∞, ∞) − {0} c) Only 𝑥 > 0 d) No value of 𝑥

13. If the function

𝑎 𝜋
{1 + |sin 𝑥|}| sin 𝑥| , − <𝑥<0
6
𝑓(𝑥) = 𝑏, 𝑥=0
tan 2𝑥 𝜋
{ 𝑒 tan 3𝑥 , 0<𝑥<
6
Is continuous at 𝑥 = 0
2 2
a) 𝑎 = log 𝑒 𝑏 , 𝑏 = 3 b) 𝑏 = log 𝑒 𝑎 , 𝑎 = 3
c) 𝑎 = log 𝑒 𝑏 , 𝑏 = 2 d) None of these

𝑥2 𝑥2 𝑥2
14. If 𝑓(𝑥) = 𝑥 2 + + (1+𝑥 2 )2 + ⋯ + (1+𝑥2 )𝑛 + ⋯, then at 𝑥 = 0, 𝑓(𝑥)
1+𝑥 2
a) Has no limit
b) Is discontinuous
c) Is continuous but not differentiable
d) Is differentiable

1, ∀ 𝑥<0
15. Let𝑓(𝑥) = { , then what is the value of 𝑓 ′ (𝑥) at 𝑥 = 0?
1 + sin 𝑥, ∀ 0 ≤ 𝑥 ≤ 𝜋/2
a) 1 b) −1 c) ∞ d) Does not exist

16. The function 𝑓(𝑥) = 𝑥 − |𝑥 − 𝑥 2 | is

a) Continuous at 𝑥 = 1 b) Discontinuous at 𝑥 = 1
c) Not defined at 𝑥 = 1 d) None of the above

17. If 𝑓(𝑥 + 𝑦 + 𝑧) = 𝑓(𝑥). 𝑓(𝑦). 𝑓(𝑧) for all 𝑥, 𝑦, 𝑧 and 𝑓(2) = 4, 𝑓 ′ (0) = 3, then 𝑓′(2) equals
a) 12 b) 9 c) 16 d) 6

18. If 𝑓(𝑥) = | log 𝑒 |𝑥| |, then 𝑓′(𝑥) equals

1
a) |𝑥| , 𝑥 ≠ 0
1 −1
b) 𝑥 for |𝑥| > 1 and𝑥
for |𝑥| < 1
−1 1
c) 𝑥 for |𝑥| > 1 and 𝑥 for |𝑥| < 1
1 1
d) for |𝑥| > 0 and − for 𝑥 < 0
𝑥 𝑥

1−cos 𝑥
, for 𝑥 ≠ 0
19. If the function 𝑓(𝑥) = { 𝑥2 is continuous at 𝑥 = 0, then the value of 𝑘 is
𝑘, for 𝑥 = 0
1
a) 1 b) 0 c) 2 d) -1

20. Function 𝑓(𝑥) = |𝑥 − 1| + |𝑥 − 2|, 𝑥 ∈ 𝑅 is

a) Differentiable everywhere in 𝑅
b) Except 𝑥 = 1 and 𝑥 = 2 differentiable everywhere in 𝑅
c) Not continuous at 𝑥 = 1 and 𝑥 = 2
d) Increasing in 𝑅

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B D B C B B B C A D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. D A A B D A A B C B

DPP No. 2
DPP Year : 2023
Topic : Continuity & Differentiability

2
1. The set of points where the function 𝑓(𝑥) = √1 − 𝑒 −𝑥 is differentiable is
a) (−∞, ∞) b) (−∞, 0) ∪ (0, ∞) c) (−1, ∞) d) None of these

1
2. If 𝑓(𝑥) = 𝑥 sin (𝑥) , 𝑥 ≠ 0, then the value of function at 𝑥 = 0, so that the function is continuous at 𝑥 = 0
is
a) 1 b) −1 c) 0 d) Indeterminate

2−(256−7 𝑥)1/8
3. The value of 𝑓(0) so that the function 𝑓(𝑥) = (5𝑥+32)1/5 −2
(𝑥 ≠ 0) is continuous everywhere, is given
by
a) −1 b) 1 c) 26 d) None of these

4. The derivative of 𝑓(𝑥) = |𝑥|3 at 𝑥 = 0 is

a) −1 b) 0 c) Does not exist d) None of these

(4𝑥 −1)3
𝑥 𝑥2
,𝑥 ≠ 0
5. If 𝑓(𝑥) = { sin( ) log(1+ )
𝑎 3 is continuous function at 𝑥 = 0, then the value of 𝑎 is equal to
3
9(log 4) , 𝑥 = 0
a) 3 b) 1 c) 2 d) 0

6. 𝑓(𝑥) = |[𝑥] + 𝑥| in −1 < 𝑥 ≤ 2 is

a) Continuous at 𝑥 = 0
b) Discontinuous at 𝑥 = 1
c) Not differentiable at 𝑥 = 2, 0
d) All the above

7. Let𝑓(𝑥) = [𝑥 3 − 𝑥], where [𝑥]the greatest integer function is. Then the number of points in the interval
(1, 2), where function is discontinuous is
a) 4 b) 5 c) 6 d) 7

8. If 𝑦 = cos−1 cos (|𝑥| − 𝑓(𝑥)), where

1, 𝑖𝑓 𝑥 > 0
5𝜋
𝑓(𝑥) = {−1, 𝑖𝑓 𝑥 < 0. Then, (𝑑𝑦/𝑑𝑥) 𝑥 = is equal to
4
0, 𝑖𝑓 𝑥 = 0
a) -1 b) 1
c) 0 d) Cannot be determined

9. Let 𝑓(𝑥 + 𝑦) = 𝑓(𝑥) + 𝑓(𝑦) and 𝑓(𝑥) = 𝑥 2 𝑔(𝑥) for all 𝑥, 𝑦 ∈ 𝑅, where 𝑔(𝑥) is continuous function.
Then, 𝑓′(𝑥) is equal to
a) 𝑔′(𝑥) b) 𝑔(0) c) 𝑔(0) + 𝑔′(𝑥) d) 0

𝑥, 𝑥 ∈ 𝑄
10. Let a function 𝑓(𝑥) be defined by 𝑓(𝑥) = { Then, 𝑓(𝑥) is
0, 𝑥 ∈ 𝑅 − 𝑄
a) Everywhere continuous
b) Nowhere continuous
c) Continuous only at some points
d) Discontinuous only at some points

1 − 2𝑥 + 3𝑥 2 − 4𝑥 3 + ⋯ to ∞, 𝑥 ≠ −1
11. The function 𝑓(𝑥) = { is
1, 𝑥 = −1
a) Continuous and derivable at 𝑥 = −1
b) Neither continuous nor derivable at 𝑥 = −1
c) Continuous but not derivable at 𝑥 = −1
d) None of these

2𝑎 − 𝑥 in – 𝑎 < 𝑥 < 𝑎
12. 𝑓(𝑥) = { . Then, which of the following is true?
3𝑥 − 2𝑎 in 𝑎 ≤ 𝑥
a) 𝑓(𝑥) is discontinuous at 𝑥 = 𝑎 b) 𝑓(𝑥) is not differentiable at 𝑥 = 𝑎
c) 𝑓(𝑥) is differentiable at 𝑥 ≥ 𝑎 d) 𝑓(𝑥) is continuous at all 𝑥 < 𝑎

13. Let 𝑓(𝑥 + 𝑦) = 𝑓(𝑥)𝑓(𝑦) and 𝑓(𝑥) = 1 + (sin 2 𝑥)𝑔(𝑥) where 𝑔(𝑥) is continuous. Then, 𝑓′(𝑥) equals
a) 𝑓(𝑥)𝑔(0) b) 2𝑓(𝑥)𝑔(0) c) 2𝑔(0) d) None of these

14. If 𝑓(𝑥) = [𝑥 sin 𝜋 𝑥], then which of the following is incorrect?

a) 𝑓(𝑥) is continuous at 𝑥 = 0
b) 𝑓(𝑥) is continuous in (−1, 0)
c) 𝑓(𝑥) is differentiable at 𝑥 = 1
d) 𝑓(𝑥) is differentiable in (−1, 1)

1, 𝑥 < 0
15. If 𝑓(𝑥) = { 𝜋 then derivative of 𝑓(𝑥) at 𝑥 = 0
1 + sin 𝑥 , 0 ≤ 𝑥 ≤ 2
a) Is equal to 1 b) Is equal to 0 c) Is equal to −1 d) Does not exist

16. If the derivative of the function 𝑓(𝑥) is everywhere continuous and is given by
𝑏𝑥 2 + 𝑎𝑥 + 4; 𝑥 ≥ −1
𝑓(𝑥) = { , then
𝑎𝑥 2 + 𝑏; 𝑥 < −1
a) 𝑎 = 2, 𝑏 = −3 b) 𝑎 = 3, 𝑏 = 2 c) 𝑎 = −2, 𝑏 = −3 d) 𝑎 = −3, 𝑏 = −2

𝑥 log cos 𝑥
, 𝑥≠0
17. If 𝑓(𝑥) = {log(1+𝑥2 ) , then
0, 𝑥 = 0
a) 𝑓(𝑥) is not continuous at 𝑥 = 0
b) 𝑓(𝑥) is not continuous and differentiable at 𝑥 = 0
c) 𝑓(𝑥) is not continuous at 𝑥 = 0 but not differentiable at 𝑥 = 0
d) None of these

𝐴𝑥 − 𝐵, 𝑥 ≤ 1
18. If the function 𝑓(𝑥) = { 3𝑥, 1 < 𝑥 < 2 be continuous at 𝑥 = 1 and discontinuous at 𝑥 = 2, then
𝐵 𝑥 2 − 𝐴, 𝑥 ≥ 2
a) 𝐴 = 3 + 𝐵, 𝐵 ≠ 3 b) 𝐴 = 3 + 𝐵, 𝐵 = 3 c) 𝐴 = 3 + 𝐵 d) None of these

|𝑥 − 4|, for 𝑥 ≥ 1
19. If 𝑓(𝑥) = { , then
(𝑥 3
/2) − 𝑥 2 + 3𝑥 + (1/2), for 𝑥 < 1
a) 𝑓(𝑥) is continuous at 𝑥 = 1 and 𝑥 = 4
b) 𝑓(𝑥) is differentiable at 𝑥 = 4
c) 𝑓(𝑥) is continuous and differentiable at 𝑥 = 1
d) 𝑓(𝑥) is not continuous at 𝑥 = 1

20. The function 𝑓(𝑥) = 𝑎[𝑥 + 1] + 𝑏[𝑥 − 1], where [𝑥] is the greatest integer function, is continuous at
𝑥 = 1, is
a) 𝑎 + 𝑏 = 0 b) 𝑎 = 𝑏 c) 2𝑎 − 𝑏 = 0 d) None of these

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B C D B A D C B D B

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B B B C D C B A A A

DPP No. 01
DPP Year : 2023
Topic : Quadratic Equation

1. If 𝑝, 𝑞, 𝑟 are positive and are in AP, then roots of the quadratic equation 𝑝𝑥 2 + 𝑞𝑥 + 𝑟 = 0 are
complex for
𝑟 𝑝
a) | − 7| ≥ 4√3 b) | − 7| < 4√3 c) All 𝑝 and 𝑟 d) No 𝑝 and 𝑟
𝑝 𝑟
2. 1 1 1
If the roots of the equation 𝑥+𝑝 + 𝑥+𝑞 = 𝑟 , (𝑥 ≠ −𝑝, 𝑥 ≠ −𝑞, 𝑟 ≠ 0) are equal in magnitude but
opposite in sign, then 𝑝 + 𝑞 is equal to
1
a) 𝑟 b) 2𝑟 c) 𝑟 2 d)
𝑟
3. The solution set of the inequation |2𝑥 − 3| < |𝑥 + 2|, is
a) (−∞, 1/3) b) (1/3, 5) c) (5, ∞) d) (−∞, 1/3) ∪ (5, ∞)
4. 2
In writing an equation of the form 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0; the coefficient of 𝑥 is written incorrectly and
roots are found to be equal. Again in writing the same equation the constant term is written incorrectly
and it is found that one root is equal to those of the previous wrong equation while the other is
double of it. If 𝛼 and 𝛽 be the roots of correct equation, then (𝛼 − 𝛽)2 is equal to
a) 5 b) 5 𝛼 𝛽 c) −4 𝛼 𝛽 d) −4
5. If 𝑥 is complex, the expression
𝑥 2 +34𝑥−71
takes all which lie in the interval (𝑎, 𝑏) where
𝑥 2 +2𝑥−7
a) 𝑎 = −1, 𝑏 = 1 b) 𝑎 = 1, 𝑏 = −1 c) 𝑎 = 5, 𝑏 = 9 d) 𝑎 = 9, 𝑏 = 5
6. 2
Let 𝑎, 𝑏, 𝑐 be real, if 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 has two real roots 𝛼 and 𝛽, where 𝛼 < −2 and 𝛽 > 2, then
2𝑏 𝑐 2𝑏 𝑐 2𝑏 𝑐 2𝑏 𝑐
a) 4 − + <0 b) 4 + − <0 c) 4 − + =0 d) 4 + + =0
𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎
7. Two students while solving a quadratic equation in 𝑥, one copied the constant term incorrectly
and got the roots 3 and 2. The other copied the constant term coefficient of 𝑥 2 correctly as
−6 and 1 respectively the correct roots are
a) 3, −2 b) −3, 2 c) −6, −1 d) 6, −1
8. 𝐸1 : 𝑎 + 𝑏 + 𝑐 = 0, if l is a root of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, 𝐸2 : 𝑏 2 − 𝑎2 = 2𝑎𝑐, if sin 𝜃, cos 𝜃 are the roots
of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
Which of the following is true?
a) 𝐸1 is true, 𝐸2 is true b) 𝐸1 is true, 𝐸2 is false
c) 𝐸1 is false 𝐸2 is true d) 𝐸1 is false, 𝐸2 is false
9. If 𝜔 =
−1+√3𝑖
, then (3 + 𝜔 + 3𝜔2 )4 is
2
a) 16 b) -16 c) 16𝜔 d) 16𝜔2
10. The least value of |𝑎| for which tan θ and cot θ are roots of the equation 𝑥 + 𝑎𝑥 + 1 = 0, is
2

a) 2 b) 1 c) 1/2 d) 0
11. If 1, 2, 3 and 4 are the roots of the equation 𝑥 + 𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥 + 𝑑 = 0, then 𝑎 + 2𝑏 + 𝑐 is
4 3 2

equal to
a) -25 b) 0 c) 10 d) 24

12. The number of integral solutions of 2(𝑥 + 2) > 𝑥 2 + 1, is

a) 2 b) 3 c) 4 d) 5
13. If one root of the equation (𝑎 − 𝑏)𝑥 + 𝑎𝑥 + 1 = 0 be double the other and if 𝑎 ∈ 𝑅, then the greatest
2

value of 𝑏 is
a) 9/8 b) 7/8 c) 8/9 d) 8/7

14. If (𝑥 − 1)3 is a factor of 𝑥 4 + 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 − 1, then the other factor is

a) 𝑥 − 3 b) 𝑥 + 1 c) 𝑥 + 2 d) 𝑥 − 1
15. If 𝑝(𝑥) = 𝑎𝑥 + 𝑏𝑥 + 𝑐 and 𝑄(𝑥) = −𝑎𝑥 + 𝑑𝑥 + 𝑐, where 𝑎𝑐 ≠ 0, then 𝑃(𝑥)𝑄(𝑥) = 0 has at least
2 2

a) Four real roots b) Two real roots

c) Four imaginary roots d) None of these
16. If 𝑥 + 2𝑎𝑥 + 𝑏 ≥ 𝑐, ∀𝑥 ∈ 𝑅, then
2

a) 𝑎 − 𝑐 ≥ 𝑎2 b) 𝑐 − 𝑎 ≥ 𝑏 2 c) 𝑎 − 𝑏 ≥ 𝑐 2 d) None of these
17. If the sum of the roots of the equation (𝑎 + 1)𝑥 + (2𝑎 + 3)𝑥 + (3𝑎 + 4) = 0 is −1, then the product of
2

the roots is
a) 0 b) 1 c) 2 d) 3
18. The roots of the equation 2𝑥+2 33𝑥/(𝑥−1) = 9 are given by
2 log3
a) 1 − log 2 3, 2 b) log 2 ( ) , 1 c) 2, −2 d) −2, 1 −
3 log2
19. If 𝑎 + 𝑏 + 𝑐 = 0 and 𝑎 ≠ 𝑐 then the roots of the equation (𝑏 + 𝑐 − 𝑎)𝑥 + (𝑐 + 𝑎 − 𝑏)𝑥 + (𝑎 + 𝑏 − 𝑐) =
2

0, are
a) Real and unequal
b) Real and equal
c) Imaginary
d) None of these
20. If 𝛼, 𝛽 are the roots of the equation 𝑥 2 + √𝛼 𝑥 + 𝛽 = 0, then the values of 𝛼 and 𝛽 are
a) 𝛼 = 1, 𝛽 = −1 b) 𝛼 = 1, 𝛽 = −2 c) 𝛼 = 2, 𝛽 = 1 d) 𝛼 = 2, 𝛽 = −2

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B B B B C A D A C A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C B A B B A C D A B

DPP No. 01
DPP Year : 2023
Topic : Tangent & Normal

1. The equation of the tangent to the curve 𝑥 = 𝑡 cos 𝑡 , 𝑦 = 𝑡 sin 𝑡 at the origin is
a) 𝑥 = 0 b) 𝑦 = 0 c) 𝑥 + 𝑦 = 0 d) 𝑥 − 𝑦 = 0
2. The tangents to the curve 𝑥 = 𝑎(𝜃 − sin 𝜃), 𝑦 = 𝑎(1 + cos 𝜃) at the points 𝜃 = (2𝑘 + 1)𝜋, 𝑘 ∈ 𝑍 are
parallel to:
a) 𝑦 = 𝑥 b) 𝑦 = −𝑥 c) 𝑦 = 0 d) 𝑥 = 0
3. 5 3
The normal to the curve 5𝑥 − 10𝑥 + 𝑥 + 2𝑦 + 6 = 0 at 𝑃(0, −3) meets the curve again at the point
a) (−1, 1), (1, 5) b) (1, −1), (−1, −5) c) (−1, −5), (−1, 1) d) (−1, 5), (1, −1)
4. The normal to the curve represented parametrically by 𝑥 = 𝑎(cos 𝜃 + 𝜃 sin 𝜃) and
𝑦 = 𝑎(sin 𝜃 − 𝜃 cos 𝜃) at any point 𝜃, is such that it
a) Makes a constant angle with 𝑥-axis
b) Is at a constant distance from the origin
c) Passes through the origin
d) Satisfies all the three conditions
5. 3𝑥 2 + 12 𝑥 − 1, −1 ≤ 𝑥 ≤ 2
If 𝑓(𝑥) = { , then
37 − 𝑥, 2 < 𝑥 ≤ 3
a) 𝑓(𝑥) is increasing in [−1, 2]
b) 𝑓(𝑥) is continuous in [−1, 3]
c) 𝑓(𝑥) is maximum at 𝑥 = 2
d) All the above
6. The equation(s) of the tangent(s) to the curve 𝑦 = 𝑥 4 from the point (2, 0) not on
the curve is given by
4098
a) 𝑦 =
81
b) 𝑦 − 1 = 5(𝑥 − 1)
4096 2048 8
c) 𝑦 − = (𝑥 − )
81 27 3
32 80 2
d) 𝑦 − = (𝑥 − )
243 81 3
7. The point on the curve √𝑥 + √𝑦 = √𝑎 at which the normal is parallel to the 𝑥-axis, is
a) (0, 0) b) (0, 𝑎) c) (𝑎, 0) d) (𝑎, 𝑎)
8. If 𝑚 denotes the slope of the normal to the curve 𝑦 = −3 log(9 + 𝑥 2 ) at the point 𝑥 ≠ 0, then,
a) 𝑚 ∈ [−1, 1] b) 𝑚 ∈ 𝑅 − (−1, 1) c) 𝑚 ∈ 𝑅 − [−1, 1] d) 𝑚 ∈ (−1, 1)
9. 𝑥 𝑦 𝑥 𝑛 𝑦 𝑛
The line 𝑎 + 𝑏 = 2 touches the curve (𝑎) + (𝑏 ) = 2 at the point (𝑎, 𝑏) for
a) 𝑛 = 2 only b) 𝑛 = −3 only c) Any 𝑛 ∈ 𝑅 d) None of these
10. The shortest distance between the line 𝑦 − 𝑥 = 1 and the curve 𝑥 = 𝑦 2 is

3√2 2√3 3√2

a) b) c) d) √3
8 8 5 4
11. The length of the subtangent at any point (𝑥1 , 𝑦1 )on the curve 𝑦 = 𝑎 𝑥 , (𝑎 > 0) is
1
a) 2 log 𝑎 b) c) log 𝑎 d) 𝑎2𝑥1 log 𝑎
log 𝑎
12. The tangent to the curve 𝑦 = 2𝑥 2 − 𝑥 + 1 is parallel to the line 𝑦 = 3𝑥 + 9 at the point
a) (3, 9) b) (2, -1) c) (2, 1) d) (1, 2)
13. 2 3
The point 𝑃 of the curve 𝑦 = 2𝑥 such that the tangent at 𝑃 is perpendicular to the
line 4𝑥 − 3𝑦 + 2 = 0 is given by
a) (2, 4) b) (1, √2) c) (1/2, −1/2) d) (1/8, −1/16)
14. 𝑡 𝑡
If the parametric equation of a curve given by 𝑥 = 𝑒 cost, 𝑦 = 𝑒 sin 𝑡, then the tangent
to the curve at the point 𝑡 = 𝜋/4 makes with axis of 𝑥 the angle
a) 0 b) 𝜋/4 c) 𝜋/3 d) 𝜋/2
15. All points on the curve 𝑦 2 = 4𝑎 (𝑥 + 𝑎 sin 𝑥 ) at which the tangents are parallel to the axis
𝑎
of 𝑥 lie on a
a) Circle b) Parabola c) Line d) None of these
16. The point of the curve 𝑦 = 2(𝑥 − 3) at which the normal is parallel to line 𝑦 − 2𝑥 + 1=0
2

1 3
a) (5, 2) b) (− , −2) c) (5, −2) d) ( , 2)
2 2
17. The abscissa of the point on the curve
𝑦 = 𝑎(𝑒 𝑥/𝑎 + 𝑒 −𝑥/𝑎 )
Where the tangent is parallel to the x-axis, is
a) 0 b) 𝑎 c) 2𝑎 d) −2𝑎
18. If the subnormal at any point on 𝑦 = 𝑎 𝑥 is of constant length, then the value of 𝑛, is
1−𝑛 𝑛

a) 1 b) 1/2 c) 2 d) −2
19. The normal to the curve 𝑥 = 𝑎(1 + cos 𝜃), 𝑦 = 𝑎 sin 𝜃 at 𝜃 always passes through the fixed point
a) (𝑎, 0) b) (0, 𝑎) c) (0, 0) d) (𝑎, 𝑎)
20. If tangent to the curve 𝑥 = 𝑎𝑡 , 𝑦 = 2𝑎𝑡 is perpendicular to 𝑥-axis, then its point of contact is
2

a) (𝑎, 𝑎) b) (0, 𝑎) c) (0, 0) d) (𝑎, 0)

21. If 𝑦 = 4𝑥 − 5 is tangent to the curve 𝑦 = 𝑝𝑥 + 𝑞 at (2, 3) then (𝑝, 𝑞) is
2 3

a) (2, 7) b) (−2, 7) c) (−2, −7) d) (2, −7)

22. If 𝑦 = 4𝑥 − 5 is a tangent to the curve 𝑦 = 𝑝𝑥 + 𝑞 at (2, 3), then
2 3

a) 𝑝 = 2, 𝑞 = −7 b) 𝑝 = −2, 𝑞 = 7 c) 𝑝 = −2, 𝑞 = −7 d) 𝑝 = 2, 𝑞 = 7
23. The tangent to the curve 𝑦 = 2𝑥 − 𝑥 + 1 at a point 𝑃 is parallel to 𝑦 = 3𝑥 + 4, then the coordinates of
2

𝑃are
a) (2, 1) b) (1, 2) c) (−1, 2) d) (2, −1)
24. The point on the curve √𝑥 + √𝑦 = √𝑎, the normal at which is parallel to the 𝑥-axis, is
a) (0, 0) b) (0, 𝑎) c) (𝑎, 0) d) (𝑎, 𝑎)
25. The equation of tangent to the curve 𝑥 2 − 𝑦2 = 1, which is parallel to 𝑦 = 𝑥, is
3 2
a) 𝑦 = 𝑥 ± 1 b) 𝑦 = 𝑥 − 1/2 c) 𝑦 = 𝑥 + 1/2 d) 𝑦 = 1 − 𝑥

26. If the curves 𝑥 2 + 𝑦2 = 1 and 𝑥 2 − 𝑦2 = 1 cut each other orthogonally, then

𝑎2 𝑏2 𝑙2 𝑚2
a) 𝑎2 + 𝑏 2 = 𝑙 2 + 𝑚2 b) 𝑎2 − 𝑏 2 = 𝑙 2 − 𝑚2 c) 𝑎2 − 𝑏 2 = 𝑙 2 + 𝑚2 d) 𝑎2 + 𝑏 2 = 𝑙 2 − 𝑚2
27. The normal at point (1,1) of the curve 𝑦 2 = 𝑥 3 is parallel to the line
a) 3𝑥 − 𝑦 − 2 = 0 b) 2𝑥 + 3𝑦 − 7 = 0 c) 2𝑥 − 3𝑦 + 1 = 0 d) 2𝑦 − 3𝑥 + 1 = 0
28. 2𝑥 2
The distance between the origin and the normal to the curve 𝑦 = 𝑒 + 𝑥 at 𝑥 = 0 is
a) 2 2 2 1
b) c) d)
√3 √5 2
29. 2
The length of subnormal of parabola 𝑦 = 4𝑎𝑥 at any point is equal to
𝑎
a) √2𝑎 b) 2√2𝑎 c) d) 2𝑎
√2
30. If tangent to the curve 𝑥 = 𝑎𝑡 2 , 𝑦 = 2𝑎𝑡 is perpandicular to 𝑥-axis, then its point of contact is
a) (𝑎, 𝑎) b) (0, 𝑎) c) (0, 0) d) (𝑎, 0)

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B C B B D C B B C A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B D D D B C A B A C

Ques. 21 22 23 24 25 26 27 28 29 30

Ans. D A B B A C B C D C

DPP No. 01
DPP Year : 2023
Topic : Minima & Maxima

1. The maximum value of the function 𝑓(𝑥) given by 𝑓(𝑥) = 𝑥(𝑥 − 1)2 , 0 < 𝑥 < 2, is

a) 0 b) 4/27 c) −4 d) 1/4

2. If 𝜃 is the semi vertical angle of a cone of maximum volume and given slant height, then tan 𝜃is given by

a) 2 b) 1 c) √2 d) √3

𝑥 𝑥
3. The maximum value of 𝑓(𝑥) = 3 cos2 𝑥 + 4 sin2 𝑥 + cos 2 + sin 2, is

a) 4 b) 3 + √2 c) 4 + √2 d) 2 + √2

4. If 𝑎2 𝑥 4 + 𝑏 2 𝑦 4 = 𝑐 6 , then maximum value of 𝑥𝑦 is

𝑐2
a)
√𝑎𝑏
𝑐3
b)
𝑎𝑏
𝑐3
c)
√2𝑎𝑏
𝑐3
d)
2𝑎𝑏
5. The maximum value of (1/𝑥)𝑥 , is

a) 𝑒 b) 𝑒 𝑒 c) 𝑒 1/𝑒 d) (1/𝑒)1/𝑒

6. If 𝑓(𝑥) = 2𝑥 3 − 21𝑥 2 + 36𝑥 − 30, then which one of the following is correct

a) 𝑓(𝑥)has minimum 𝑎𝑡 𝑥 = 1 b) 𝑓(𝑥)has maximum 𝑎𝑡 𝑥 = 6

c) 𝑓(𝑥)has maximum 𝑎𝑡 𝑥 = 1 d) 𝑓(𝑥)has maxima or minima

7. 𝑏
If 𝑎𝑥 2 + 𝑥 ≥ 𝑐 for all positive 𝑥, where 𝑎, 𝑏, > 0, then

a) 27𝑎𝑏 2 ≥ 4𝑐 3 b) 27𝑎𝑏 2 < 4𝑐 3 c) 4𝑎𝑏 2 ≥ 27𝑐 3 d) None of these

8. The function 𝑓(𝑥) = |𝑝𝑥 − 𝑞| + 𝑟|𝑥|, 𝑥 ∈ (−∞, ∞), where 𝑝 > 0, 𝑞 > 0, 𝑟 > 0 assume its minimum value
only at one point if

a) 𝑝 ≠ 𝑞 b) 𝑟 ≠ 𝑞 c) 𝑟 ≠ 𝑝 d) 𝑝 = 𝑞 = 𝑟

9. Let 𝑓(𝑥) = ∫0
𝑥 cos 𝑡 𝜋
𝑑𝑡. Then, at 𝑥 = (2𝑛 + 1) , 𝑓(𝑥) has
𝑡 2

a) Maxima when 𝑛 = −2, −4, −6, … and minima when 𝑛 = −1, −3, −5, …
b) Maxima when 𝑛 = −1, −3, −5, … and minima when 𝑛 = 1, 3, 5, …
c) Minima when 𝑛 = 0, 2, 4, … and maxima when 𝑛 = 1, 3, 5, …
d) None of these

10. If 𝑓(𝑥) = 1
, then its maximum value is
4𝑥 2 +2𝑥+1

a) 4/3 b) 2/3 c) 1 d) 3/4

11. The minimum value of 2𝑥 + 3𝑦, when 𝑥𝑦 = 6, is

a) 9 b) 12 c) 8 d) 6

12. Suppose the cubic 𝑥 3 − 𝑝𝑥 + 𝑞 has three distinct real roots where 𝑝 > 0 and 𝑞 > 0. Then, which
one of the following holds?
𝑝 𝑝 𝑝 𝑝
a) The cubic has maxima at both 3 𝑎𝑛𝑑 − 3 b) The cubic has minima at 3 and maxima at − 3
The cubic has minima 𝑝 𝑝
c) 𝑝 𝑝 d) The cubic has minima at both 3 𝑎𝑛𝑑 − 3
at− and maxima at
3 3

13. The shortest distance between the line 𝑦 − 𝑥 = 1 and the curve 𝑥 = 𝑦 2 is

3√2 2√3 3√2 √3

a) b) c) d)
8 8 5 4

14. A cubic 𝑓(𝑥) vanishes at 𝑥 = −2 and has relative minimum/maximum at 𝑥 = −1 and 𝑥 = 1

3
1 14
such that ∫−1 𝑓(𝑥)𝑑𝑥 = 3
. Then, 𝑓(𝑥) is
a) 𝑥 3 + 𝑥 2 − 𝑥 b) 𝑥 3 + 𝑥 2 − 𝑥 + 1 c) 𝑥 3 + 𝑥 2 − 𝑥 + 2 d) 𝑥 3 + 𝑥 2 − 𝑥 − 2

15. The different between the greatest and least values of the function 𝑓(𝑥) = cos 𝑥 1 cos 2𝑥 −
2
1
3
cos 3 𝑥 𝑖𝑠
2 8 3 9
a) b) c) d)
3 7 8 4

16. If 𝑓(𝑥) = sin6 𝑥 + cos 6 𝑥, then which one of the following is false?

1 1
a) 𝑓(𝑥) ≤ 1 b) 𝑓(𝑥) ≤ 2 c) 𝑓(𝑥) > d) 𝑓(𝑥) ≤
4 8

17. The set {𝑥 3 − 12𝑥: −3 ≤ 𝑥 ≤ 3} is equal to

a) {𝑥: − 16 ≤ 𝑥 ≤ 16} b) {𝑥: − 12 ≤ 𝑥 ≤ 12} c) {𝑥: − 9 ≤ 𝑥 ≤ 9} d) {𝑥: 0 ≤ 𝑥 ≤ 10}

18. If 𝑥𝑦 = 𝑎2 and 𝑆 = 𝑏 2 𝑥 + 𝑐 2 𝑦 where 𝑎, 𝑏 and 𝑐 are constants, then the minimum value of 𝑆 is
a) 𝑎𝑏𝑐 b) √𝑎 𝑏𝑐 c) 2𝑎𝑏𝑐 d) None of these

19. The maximum value of 𝑥𝑦 subject to 𝑥 + 𝑦 = 8, is

a) 8 b) 16 c) 20 d) 24

20. The point of inflexion for the curve 𝑦 = 𝑥 5/2 is

a) (1, 1) b) (0, 0) c) (1, 0) d) (0, 1)

Answer Key

Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B C C C C C A C B A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B B A C D D A C B B

DPP No. 1
DPP Year : 2023
Topic : Permutations & Combinations

1. Let 𝐴 = {𝑥1 , 𝑥2, 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 },

𝐵 = {𝑦1 , 𝑦2 , 𝑦3 , 𝑦4 , 𝑦5 , 𝑦6 }.Then the number of one –one mapping from 𝐴 to 𝐵 such that
𝑓(𝑥𝑖 ) ≠ 𝑦𝑣 𝑖 = 1, 2, 3, 4, 5, 6 is
a) 720 b) 265 c) 360 d) 145

2. A man invites a party to (𝑚 + 𝑛) friends to dinner and places 𝑚 at one round table and 𝑛 at another.
The number of ways of arranging the guests is
(𝑚+𝑛) ! (𝑚+𝑛) !
a) 𝑚 !𝑛 !
b) (𝑚−1)!(𝑛−1) ! c) (𝑚 − 1)! (𝑛 − 1) ! d) None of these

3. The number of ways in which seven persons can be arranged at a round table, if two particular persons
may not sit together is
a) 480 b) 120 c) 80 d) None of these

4. If 2𝑛+1 𝑃𝑛−1 : 2𝑛−1

𝑃𝑛 : 3: 5, then the value of 𝑛 is equal to
a) 4 b) 3 c) 2 d) 1

5. The number of ways in which a committee can be formed of 5 members from 6 men and 4 women if the
committee has at least one woman, is
a) 186 b) 246 c) 252 d) 244

6. In how many ways can 5 books be selected out of 10 books, if two specific books are never selected?
a) 56 b) 65 c) 58 d) None of these

7. The number of parallelograms that can be formed from a set of four parallel lines intersecting another
set of three parallel lines, is
a) 6 b) 18 c) 12 d) 9

8. There is a set of 𝑚 parallel lines intersecting a set of another 𝑛 parallel lines in a plane. The number of
parallelograms formed, is
a) 𝑚−1 𝐶2 . 𝑛−1 𝐶2 b) 𝑚 𝐶2 . 𝑛 𝐶2 c) 𝑚−1 𝐶2 . 𝑛 𝐶2 d) 𝑚 𝐶2 . 𝑛−1 𝐶2

9. The value of 50
𝐶4 + ∑6𝑟=1 56−𝑟
𝐶3 is
a) 56 𝐶4 b) 56
𝐶3 c) 55
𝐶3 d) 55
𝐶4

10. The number of numbers of 4 digits which are not divisible by 5, are
a) 7200 b) 3600 c) 14400 d) 1800

11. 4 buses runs between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes
back to Gwalior by another bus, then the total possible ways are
a) 12 b) 16 c) 4 d) 8

12. The total number of different combinations of letters which can be made from the letters of the word
MISSISSIPPI is
a) 150 b) 148 c) 149 d) None of these

13. Six points in a plane be joined in all possible ways by indefinite straight lines and if no two of them be
coincident or parallel, and no three pass through the same point (with the exception of the original 6
points). The number of distinct points or intersection is equal to
a) 105 b) 45 c) 51 d) None of these

14. The total numbers of ways of dividing 15 things into groups of 8,4 and 3 respectively is
15 ! 15 ! 15 !
a) 8 !4 !(3 !)2 b) 8 !4 !3 ! c) 8 !4 ! d) None of these

15. In a circus there are ten cages for accommodating ten animals. Out of these four cages are so small that
five out of 10 animals cannot enter into them. In how many ways will it be possible to accommodate ten
animals in these ten cages?
a) 66400 b) 86400 c) 96400 d) None of these

16. Let 𝑇𝑛 denote the number of triangles which can be formed using the vertices of a regular polygon of 𝑛
sides. If 𝑇𝑛+1 − 𝑇𝑛 = 21, then 𝑛 equals
a) 5 b) 7 c) 6 d) 4

17. At an electron, a voter may vote for any number of candidates not greater than the number to be
elected. There are 10 candidates and 4 are to be elected. If a voter votes for at least one candidate, then
the number of ways in which he can vote, is
a) 6210 b) 385 c) 1110 d) 5040

18. All possible two factors products are formed from numbers 1, 2, 3, 4,…,200. The number of factors out
of the total obtained which are multiples of 5, is
a) 5040 b) 7180 c) 8150 d) None of these

19. If the total number of 𝑚 elements subsets of the set 𝐴 = {𝑎1 , 𝑎2 , 𝑎3 , … 𝑎𝑛 } is 𝜆 times the number of 3
elements subsets containing 𝑎4 ,then𝑛 is
a) (𝑚 − 1)𝜆 b) 𝑚𝜆 c) (𝑚 + 1)𝜆 d) 0

20. The number of natural numbers less than 1000, in which no two digits are replaced, is
a) 738 b) 792 c) 837 d) 720

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B D A A B A C B A A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A C C B B B B B B A

DPP No. 2
DPP Year : 2023
Topic : Permutations & Combinations

1. If 𝑛
𝐶𝑟 denotes the number of combinations of 𝑛 things takes 𝑟 at a time, then the expression
𝑛
𝐶𝑟+1 + 𝑛 𝐶𝑟−1 + 2 × 𝑛 𝐶𝑟 , equals
a) 𝑛+2 𝐶𝑟 b) 𝑛+2 𝐶𝑟+1 c) 𝑛+1 𝐶𝑟 d) 𝑛+1 𝐶𝑟+1

2 2 1 2𝑎
2. If 9! + 3!7! + 5!5! = 𝑏!
,where 𝑎, 𝑏, ∈ 𝑁, then the ordered pair (𝑎, 𝑏) is
a) (9, 10) b) (10, 9) c) (7, 10) d) (10, 7)

3. The number of diagonals that can be drawn by joining the vertices of an octagon is
a) 28 b) 48 c) 20 d) None of these

4. A father with 8 children takes 3 at a time to the zoological garden, as often as he can without taking the
same 3 children together more than once. The number of times he will go to the garden, is
a) 112 b) 56 c) 336 d) None of these

5. If 189 𝐶35 + 189

𝐶𝑥 = 190
𝐶𝑥 , then 𝑥 is equal to
a) 34 b) 35 c) 36 d) 37

6. The number of ways in which 𝑛 ties can be selected from a rack displaying 3 𝑛 different ties is
3𝑛 ! 3𝑛 !
a) 2𝑛 ! b) 3 × 𝑛 ! c) (3𝑛) ! d) 𝑛 !2𝑛 !

7. The number of permutations of 4 letters that can be made out of the letters of the word EXAMINATION
is
a) 2454 b) 2452 c) 2450 d) 1806

8. The number of ways in which 5 boys and 5 girls can be seated for a photograph so that no two girls sit
next to each other is
10! 10!
a) 6! .5! b) (5!)2 c) (5!) d) (5!)2

9. The number of diagonals of a polygon of 20 sides is

a) 210 b) 190 c) 180 d) 170

10. The value of 47

𝐶4 + ∑5𝑟=1 52−𝑟
𝐶3is equal to
a) 47 𝐶6 b) 52
𝐶5 c) 53
𝐶4 d) None of these

11. In how many ways can 21 English and 19 Hindi books be placed in a row so that no two Hindi books are
together?

a) 1540 b) 1450 c) 1504 d) 1405

12. In a group of boys, two boys are brothers and in this group, 6 more boys are there. In how many ways,
they can sit if the brothers are not to sit along with each other :
a) 4820 b) 1410 c) 2830 d) None of these

13. All possible four-digit numbers are formed using the digits 0,1,2,3 so that no number has repeated
digits. The number of even number among them is
a) 9 b) 18 c) 10 d) None of these

14. In how many ways can 4 prizes be distributed among 3 students, if each students can get all the 4
prizes?
a) 4! b) 34 c) 34 − 1 d) 33

15. In a chess tournament where the participants were to play one game with one another, two players fell
ill having played 6 games each, without playing among themselves. If the total number of games is 117,
then the number of participants at the beginning was
a) 15 b) 16 c) 17 d) 18

16. How many even numbers of 3 different digits can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9
(repetition of digits is not allowed)?
a) 224 b) 280 c) 324 d) None of these

17. If 𝑎 denotes the number of permutations of 𝑥 + 2 things taken all at a time, 𝑏 the number of
permutations of 𝑥 things taken 11 at a time and 𝑐 the number of permutations of 𝑥 − 11 things taken all
at a time such that 𝑎 = 182 𝑏𝑐, then the value of 𝑥 is
a) 15 b) 12 c) 10 d) 18

18. Eleven books consisting of 5 Mathematics, 4 physics and 2 Chemistry are places on a shelf. The number
of possible ways of arranging them on the assumption that the books of the same subject are all
together, is
a) 4! 2!b) 11!c) 5! 4! 3! 2!d) None of these

19. The number of mappings (functions) from the set 𝐴 = {1, 2, 3} into the set 𝐵 = {1, 2, 3, 4, 5, 6, 7} such
that 𝑓(𝑖) ≤ 𝑓(𝑗)whenever 𝑖 < 𝑗, is
a) 84 b) 90 c) 88 d) None of these

20. The number of ordered triplets of positives integers which are solutions of the equations of the
equation 𝑧 + 𝑦 + 𝓏 = 100, is
a) 6005 b) 4851 c) 5081 d) None of these

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B A C B C D A A D C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A D C B A A B C A B

DPP No. 4
DPP Year : 2023
Topic : Probability

1. The probability that the same number appear on throwing three dice simultaneously, is
a) 1/6 b) 1/36 c) 5/36 d) None of these

1
2. If 𝑃(𝐴) = 𝑃(𝐵) = 𝑥 and 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴′ ∩ 𝐵′ ) = 3, then𝑥 is equal to
1 1 1 1
a) 2 b) 4 c) 3 d) 6

3. One ticket is selected at random from 50 tickets numbered 00,01,02,….,49. Then, the
probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is
zero equals
1 1 5 1
a) 14 b) 7 c) 14 d) 50

4. If 𝑛 integers taken at random are multiplied together, then the probability that the last digit of the
product is 1,3,7 or 9, is
2𝑛 4𝑛 −2𝑛 4𝑛
a) b) c) d) None of these
5𝑛 5𝑛 5𝑛

5. Among the workers in a factory only 30% receive bonus and among those receiving bonus only 20% are
skilled. The probability that a randomly selected worker is skilled and is receiving bonus is
a) 0.03 b) 0.02 c) 0.06 d) 0.015

6. A box contains 10 good articles and 6 with defects, one article is chosen at random. What is the
probability that it is either good or has a defect?
24 40 49
a) 64 b) 64 c) 64 d) 1

7. A coin and six faced die. Both unbiased, are thrown simultaneously. The probability of getting a head on
the coin and an odd number on the die, is
1 3 1 2
a) 2 b) 4 c) 4 d) 3

8. An anti-aircraft gun can take a maximum of four shots at any plane moving away from it. The
probabilities of hitting the plane at the 1st, 2nd ,3rd and 4th shots are 0.4, 0.3, 0.2 and 0.1 respectively.
What is the probability that at least one shot hits the plane?
a) 0.6976 b) 0.3024 c) 0.72 d) 0.6431

9. A coin is tossed three times. The probability of getting head and tail alternatively, is
1 1 1
a) 8 b) 2 c) 4 d) None of these

10. A bag contains 4 tickets numbered 1,2,3,4 and another bag contains 6 tickets numbered 2,4,6,7,8,9. One
bag is chosen and a ticket is drawn. The probability that the ticket bears the number 4 is
a) 1/48 b) 1/8 c) 5/24 d) None of these

11. Six coins are tossed simultaneously. The probability of getting at least 4 heads is
a) 11/64 b) 11/32 c) 15/44 d) 21/32

12. Two cards are drawn successively with replacement from a well shuffled deck of 52 cards, then the
mean of the number of aces is
1 3 2
a) b) c) d) None of these
13 13 13

13. Given two mutually exclusive events 𝐴 and 𝐵 such that 𝑃(𝐴) = 0.45 and 𝑃(𝐵) = 0.35, 𝑃(𝐴 ∩ 𝐵) is equal
to
63 63
a) 400 b) 0.8 c) 200 d) 0

14. There is an objective type question with 4 answer choices exactly one of which is correct. A student has
not studied the topic on which the question has been set. The probability that the student guesses the
correct answer, is
1 1 1
a) 2 b) 4 c) 8 d) None of these

15. If 𝐸 and 𝐹 are two independent events such that 0 < 𝑃(𝐸) < 1 and 0 < 𝑃(𝐹) < 1, then
a) 𝐸and𝐹 𝑐 are independent b) 𝐸 𝑐 and𝐹 𝑐 are independent
𝐸 𝐸𝑐
c) 𝑃 ( ) + 𝑃 ( 𝑐 ) = 1 d) None of these
𝐹 𝐹

16. An integer is chosen at random from first two hundred numbers. Then, the probability that the integer
chosen is divisible by 6 or 8 is
1 2 3
a) 4 b) 4 c) 4 d) None of these

17. The mean and variance of a random variable 𝑋 having a binomial distribution are 4 and 2 respectively,
then 𝑃(𝑋 = 1) is
1 1 1 1
a) b) c) d)
32 16 8 4

18. One hundred identical coins, each with probability 𝑝 of showing heads are tossed once. If0 < 𝑝 < 1 and
the probability of head showing on 50 coins is equal to that of head showing on 51 coins, the value of 𝑝
is
1 51 49
a) 2 b) 101 c) 101 d) None of these

19. The probability of choosing a number divisible by 6 or 8 from among 1 to 90 is

1 1 1 23
a) 6 b) 90 c) 30 d) 90

20. An urn contains 6 white and 4 black balls. A fair die is rolled and that number of balls are chosen from
the urn. The probability that the balls selected are white is
a) 1/5 b) 1/6 c) 1/7 d) 1/8

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B A A A C D C A C C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B C D B C A A B D A

DPP No. 02
DPP Year : 2023
Topic : Minima & Maxima

1. The minimum value of 2𝑥 + 3𝑦, when 𝑥𝑦 = 6, is

a) 12 b) 9 c) 8 d) 6
2. If 𝑓 (𝑥) = (𝑥 − 𝑎) (𝑥 − 𝑏)
′ 2𝑛 2𝑚+1
where 𝑚, 𝑛 ∈ 𝑁, then
a) 𝑥 = 𝑏 is a point of minimum
b) 𝑥 = 𝑏 is a point of maximum
c) 𝑥 = 𝑏 is a point of inflexion
d) None of these
3. |𝑥|, 𝑓𝑜𝑟 0 < |𝑥| ≤ 2
Let 𝑓(𝑥) = { , then at 𝑥 = 0, 𝑓 has
1, 𝑓𝑜𝑟 𝑥 = 0
a) A local maximum b) A local minimum c) No local extremum d) No local maximum
4. Given 𝑃(𝑥) = 𝑥 + 𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥 + 𝑑 such that 𝑥 = 0 is the only real root of 𝑃 (𝑥) = 0. 𝐼𝑓 𝑃(−1) <
4 3 2 ′

𝑃(1),then in the interval[−1,1]

a) 𝑃(−1) is the minimum and 𝑃(1)is the maximum of 𝑃.
b) 𝑃(−1) is not minimum but 𝑃(1)is the maximum of 𝑃.
c) 𝑃(−1) is the minimum and 𝑃(1)is not the maximum of 𝑃.
d) Neither 𝑃(−1) is the minimum nor 𝑃(1)is not the maximum of 𝑃.
5. Let 𝑓(𝑥) = 1 + 2𝑥 2 + 22 𝑥 4 +. … … + 210 𝑥 20.Then, 𝑓(𝑥) has
a) More than one minimum b) Exactly one minimum
c) At least one maximum d) None of the above
6. The function 𝑓(𝑥) = 𝑥 4 − 62 𝑥 2 + 𝑎𝑥 + 9 attains its maximum value on the interval [0, 2] at 𝑥 = 1. Then,
the value of 𝑎 is
a) 120 b) −120 c) 52 d) 60
7. If for a function 𝑓(𝑥), 𝑓 ′ (𝑎)
= 0, 𝑓 ′′ (𝑎)
= 0, 𝑓 ′′′ (𝑎)
> 0, then at 𝑥 = 𝑎, 𝑓(𝑥) is
a) Minimum b) Maximum c) Not an extreme point d) Extreme point
8. The function 𝑓(𝑥) = 𝑥 + sin 𝑥 has
a) A minimum but no maximum b) A maximum but no minimum
c) Neither maximum nor minimum d) Both maximum and minimum
9. The point in the interval [0 ,2𝜋], where 𝑓(𝑥) = 𝑒 sin 𝑥 has maximum slope, is
𝑥

𝜋 𝜋 3𝜋
a) b) c) 𝜋 d)
4 2 2
10. The perimeter of a sector is𝑝. The area of the sector is maximum, when its radius is
1 𝑝 𝑝
a) √𝑝 b) c) d)
√𝑝 2 4
𝜋
11. The function 𝑓(𝑥) = 𝑎 cos 𝑥 + 𝑏 tan 𝑥 + 𝑥 has extreme values at 𝑥 = 0 and 𝑥 = , then
6
2 2 2 2
a) 𝑎 = − , 𝑏 = −1 b) 𝑎 = , 𝑏 = −1 c) 𝑎 = − , 𝑏 = 1 d) 𝑎 = , 𝑏 = 1
3 3 3 3
12. The function 𝑓(𝑥) = 2𝑥 3 − 15𝑥 2 + 36𝑥 + 4 is maximum at

a) 𝑥 = 2 b) 𝑥 = 4 c) 𝑥 = 0 d) 𝑥 = 3
13. The points of extremum of the function ϕ(𝑥) = ∫𝑥 𝑒 −𝑡 2 /2 (1 − 𝑡 2 )𝑑𝑡, are
1
a) 𝑥 = 0, 1 b) 𝑥 = 1, −1 c) 𝑥 = 1/2 d) 𝑥 = −1/2
14. A stone is thrown vertically upwards and the height 𝑥 ft reached by the stone in t seconds is given by 𝑥 =
80𝑡 − 16𝑡 2 . The stone reaches the maximum height in
a) 2s b) 2.5s c) 3s d) 1.5s
15. 2
If 𝑎𝑥 + 𝑏𝑥 + 4 attains its minimum value −1 at 𝑥 = 1, then the values of 𝑎and 𝑏 are respectively
a) 5, −10 b) 5, −5 c) 5, 5 d) 10, −5
16. 𝑥
Let 𝑓(𝑥) = 𝑒 sin 𝑥, slope of the curve 𝑦 = 𝑓(𝑥) is maximum at 𝑥 = 𝑎, if ‘a’ equals
a) 0 b) 𝜋/4 c) 𝜋/2 d) None of these
17. 4 4
If the function 𝑓(𝑥) = (2𝑎 − 3)(𝑥 + 2 sin 3) + (𝑎 − 1)(sin 𝑥 + cos 𝑥) + log 2 does not possess critical
points, then
a) 𝑎 ∈ (−∞, 4/3) ∪ (2, ∞)
b) 𝑎 ∈ ( 4/3,2)
c) 𝑎 ∈ (4/3, ∞)
d) 𝑎 ∈ (2, ∞)
18. The function 𝑓(𝑥) = 𝑥 −𝑥 , (𝑥 ∈ 𝑅) attains a maximum, value at 𝑥 which is
a) 2 b) 3 1 d) 1
c)
𝑒
19. The maximum slope of the curve 𝑦 = −𝑥 3 + 3𝑥 2 + 2𝑥 − 27 is
a) 5 b) −5 c) 1/5 d) None of these
20. The function 𝑓(𝑥) = 𝑥 + 2 has local minimum at
2 𝑥
a) 𝑥 = −2 b) 𝑥 = 0 c) 𝑥 = 1 d) 𝑥 = 2

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A A A B B A C B B D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A A B B A C A C A D

DPP No. 1
DPP Year : 2023
Topic : Probability

1. A random variable has the following probability distribution.

The value of 𝑝, is
x 0 1 2 3 4 5 6 7
p(x) 0 p 2p 2p 3p 𝑝2 2𝑝2 7𝑝2 + 𝑝
a) 1/10 b) −1 c) −1/10 d) None of these

2. Let 𝐸 and 𝐹 be two independent events. The probability that both 𝐸 and 𝐹 happen is 1/12 and the
probability that neither 𝐸 nor 𝐹 occurs is 1/2. Then,
1 1 1 1 1 1 1 2
a) 𝑃(𝐸) = , 𝑃(𝐹) = b) 𝑃(𝐸) = , 𝑃(𝐹) = c) 𝑃(𝐸) = , 𝑃(𝐹) = d) 𝑃(𝐸) = , 𝑃(𝐹) =
3 4 2 6 6 2 4 3

3. Six ordinary dice are rolled. The probability that at least half of them will show at least 3 is
24 24 24
a) 41 × 36 b) 36 c) 20 × 36 d) None of these

4. The probability that atleast one of 𝐴 and 𝐵 occurs is 0.6. If 𝐴 and 𝐵 occur simultaneously with
probability 0.3, then 𝑃(𝐴′ ) + 𝑃(𝐵′ ) is
a) 0.9 b) 0.15 c) 1.1 d) 1.2

5. A pair of dice is rolled together till a sum of either 5 or 7 is obtained. The probability that 5 comes
before 7 is
a) 2/5 b) 1/5 c) 3/5 d) None of these

1 1 1
6. If events are independent and𝑃(𝐴) = 3 , 𝑃(𝐵) = 3 , 𝑃(𝐶) = 4 , then P(𝐴′ ∩ 𝐵′ ∩ 𝐶 ′ )
is equal to
1 1 1 5
a) 4 b) 12 c) 3 d) 12

7. Three dice are thrown. The probability that the sum of the number appearing is 15, is
a) 1/216 b) 1/72 c) 5/108 d) 1/18

8. In a poisson distribution mean is 16, then standard deviation is

a) 16 b) 256 c) 128 d) 4

9. Six faces of an unbiased die are numbered with 2, 3, 5, 7, 11 and 13. If two such dice are thrown, then
the probability that the sum on the uppermost faces of the dice is an odd number, is
5 5 13 25
a) 18 b) 36 c) 18 d) 36

10. The mean and variance of a binomial distribution are 4 and 3 respectively, then the probability of
getting exactly six successes in this distribution is
1 10 3 6 1 6 3 10 1 10 3 6 1 6 3 6
a) 16
𝐶6 (4) (4) b) 16
𝐶6 (4) (4) c) 12
𝐶6 (4) (4) d) 12
𝐶6 (4) (4)

11. 𝐴and𝐵 are two independent events. The probability that both 𝐴 and 𝐵 occur is 1/6 and the probability
that neither of them occurs is 1/3. Then,
a) 𝑃(𝐴) = 1/2, 𝑃(𝐵) = 1/3
b) 𝑃(𝐴) = 1/2, 𝑃(𝐵) = 1/6
c) 𝑃(𝐴) = 1/3, 𝑃(𝐵) = 1/6
d) None of these

12. If 𝐴 and 𝐵 are independent events of a random experiments such that

1 1
𝑃(𝐴 ∩ 𝐵) = and 𝑃(𝐴̅ ∩ 𝐵̅) = , then 𝑃(𝐴) is equal to
6 3
1 1 5 2
a) b) c) d)
4 3 7 3

13. If the integers 𝑚 and 𝑛 are chosen at random between 1 and 100, then the probability that a number of
the form 7𝑚 + 7𝑛 is divisible by 5, equals
1 1 1 1
a) b) c) d)
4 7 8 49

14. Let ω be a complex cube root of unity withω ≠ 1. A fair die is thrown three times. If 𝑟1 , 𝑟2 and 𝑟3 are the
numbers obtained on the die, then the probability that ω𝑟1 + ω𝑟2 + ω𝑟3 = 0 is
1 1 2 1
a) 18 b) 9 c) 9 d) 36

15. A mapping is selected at random from the set of all the mappings of the set 𝐴 = {1,2, … , 𝑛} into itself.
The probability that the mapping selected is an injection is
1 1 (𝑛−1) ! 𝑛!
a) 𝑛𝑛 b) 𝑛 ! c) 𝑛𝑛−1
d) 𝑛𝑛−1

16. An urn contains five balls. Two balls are drawn and are found to be white. The probability that the balls
selected are white is
a) 3/4 b) 3/5 c) 3/10 d) 1/2

17. A single letter is selected at random from the word ‘PROBABILITY’. The probability that it is a vowel is
a) 3/11 b) 4/11 c) 2/11 d) None of these

18. A die is thrown. If it shows a six, we draw a ball from a bag consisting 2 black balls and 6 white balls. If it
does not show a six, then we toss a coin. Then, the sample space of this experiment consists of
a) 13 points b) 18 points c) 10 points d) None of these

19. For a binomial variate 𝑋with 𝑛 = 6, if 𝑃(𝑋 = 2) = 9𝑃(𝑋 = 4), then its variance is

8 1 9
a) 9 b) 4 c) 8 d) 4
20. Out of 13 applicants for a job, there are 5 women and 8 men. It is desired to select 2 persons for the job.
The probability that at least one of the selected persons will be a women is
a) 25/39 b) 14/39 c) 5/13 d) 10/13

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A A A C A C C D A B

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A B A C C D B B C A

DPP No. 2
DPP Year : 2023
Topic : Probability

1. 𝐴and𝐵 are two independent events such that 𝑃(𝐴 ∪ 𝐵′ ) = 0.8 and 𝑃(𝐴) = 0.3. Then, 𝑃(𝐵) is
2 2 3 1
a) b) c) d)
7 3 8 8

2. Suppose that a die (with faces marked 1 to 6) is loaded in such a manner that for 𝐾 = 1,2,3, … .,6 the
probability of the face marked 𝐾 turning up when die is tossed is proportional to 𝐾. The probability of
the event that the outcome of a toss of the die will be an even number, is equal to
1 4 2 1
a) 2 b) 7 c) 5 d) 21

3. Three are six verities of a regular hexagon are chosen at random, then the possibility that the triangle
with three vertices is equilateral, is equal to
1 1 1 1
a) b) c) d)
2 3 10 20

4. If a committee of 3 is to be chosen from a group of 38 people of which you are a member. What is the
probability that you will be on the committee?
a) (38
3
) b) (37
2
) c) (37
2
) ∕ (38
3
) d) 666/8436

5. The probability that in a year of the 22nd century chosen at random there will be 53 Sundays, is
3 2 7 5
a) b) c) d)
28 28 28 28

6. Two cards are drawn without replacement from a well-shuffled pack. The probability that one of them
is an ace of heart, is
1 1 1
a) 25 b) 26 c) 52 d) None of these

7. A binary operation is chosen at random from the set of all binary operations on a set 𝐴 containing 𝑛
elements. The probability that the binary operation is commutative, is
𝑛𝑛 𝑛𝑛/2 𝑛𝑛/2
a) 2 b) 2 c) 2 d) None of these
𝑛𝑛 𝑛𝑛 𝑛𝑛 /2

8. A lot consists of 102 good pencils, 6 with minor defects and 2 with major defects. A pencil is chosen at
random. The probability that this pencil is not defective is
a) 3/5 b) 3/10 c) 4/5 d) 1/2

9. If 𝐴 and 𝐵 are events of the same experiments with 𝑃(𝐴) = 0.2, 𝑃(𝐵) = 0.5 ,then maximum value of
𝑃(𝐴′ ∩ 𝐵) is
a) 0.2 b) 0.5 c) 0.63 d) 0.25

10. Four tickets marked 00,01,10,11, respectively are placed in a bag. A ticket is drawn at random five
times, being replaced each time. The probability that the sum of the numbers on tickets thus drawn is
23, is
a) 25/256 b) 100/256 c) 231/256 d) None of these

11. Two dice are tossed 6 times. Then the probability that 7 will show an exactly four of the tosses is
225 116 125
a) 18442 b) 20003 c) 15552 d) None of these

12. Out of 3𝑛 consecutive natural numbers, 3 natural numbers are chosen at random without replacement.
The probability that the sum of the chosen numbers is divisible by 3, is
𝑛(3𝑛2 −3𝑛+2) (3𝑛2 −3𝑛+2) (3𝑛2 −3𝑛+2) 𝑛(3𝑛−1)(3𝑛−2)
a) 2
b) 2(3𝑛−1)(3𝑛−2) c) (3𝑛−1)(3𝑛−2) d) 3(𝑛−1)

13. 𝐴and𝐵 are two independent witnesses (𝑖𝑒, there is no collusion between them) in a case. The
probability that 𝐴 will speak the truth is 𝑥 and the probability that 𝐵 will speak the truth is 𝑦, 𝐴 and 𝐵
agree in a certain statement. The probability that the statement is true, is
𝑥−𝑦 𝑥𝑦 𝑥−𝑦 𝑥𝑦
a) 𝑥+𝑦 b) 1+𝑥+𝑦+𝑥𝑦 c) 1−𝑥−𝑦+2𝑥𝑦 d) 1−𝑥−𝑦+2𝑥𝑦

14. Five persons 𝐴, 𝐵, 𝐶, 𝐷 and 𝐸 are in queue of a shop. The probability that 𝐴 and 𝐸 always together, is
1 2 2 3
a) 4 b) 3 c) 5 d) 5

15. Three dice are thrown. The probability that the same number will appear on each of them, is
a) 1/6 b) 1/18 c) 1/36 d) None of these

16. A bag contains 8 red and 7 black balls. Two balls are drawn at random. The probability that both the
balls are of the same colour, is
14 11 7 4
a) 15 b) 15 c) 15 d) 15

17. A bag contains 10 white and 3 black balls. Balls are drawn one-by-one without replacement till all the
black balls are drawn. The probability that the procedure of drawing balls will come to an end at the
seventh draw is
105 15 181
a) 286 b) 286 c) 286 d) None of these

18. Two events 𝐴 and 𝐵 have probability 0.25 and 0.50 respectively. The probability that both 𝐴 and 𝐵
occur simultaneously is 0.14. Then, the probability that neither 𝐴 nor 𝐵 occur, is
a) 0.39 b) 0.25 c) 0.11 d) None of these

19. There are 9999 tickets bearing numbers 0001, 0002,….,9999. If one ticket is selected from these tickets
at random, the probability that the number on the ticket will consists of all different digits, is
5040 5000 5030
a) 9999 b) 9999 c) 9999 d) None of these

20. The probability of choosing randomly a number 𝑐 from the set {1, 2, 3,…..,9} such that the quadratic
equation 𝑥 2 + 4𝑥 + 𝑐 = 0 has real roots is
1 2 3 4
a) 9 b) 9 c) 9 d) 9

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A A C C D B C A B A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C C D C C C B A A D

DPP No. 2
DPP Year : 2023
Topic : Trigonometry

𝜋 3𝜋 5𝜋 7𝜋
1. (1 + cos 8 ) (1 + cos 8
) (1 + cos 8
) (1 + cos 8
) is equal to
1 𝜋 1 1+√2
a) 2 b) cos 8 c) 8 d) 2√2
𝐴 𝐴
2. If 2 sin 2 = √1 + sin 𝐴 + √1 − sin 𝐴, then 2 lies between
𝜋 3𝜋
a) 2𝑛 𝜋 + 4 and 2𝑛 𝜋 + 4
,𝑛 ∈ 𝑍
𝜋 𝜋
b) 2𝑛 𝜋 − 4
and 2𝑛 𝜋 + 4 , 𝑛 ∈ 𝑍
3𝜋 𝜋
c) 2𝑛 𝜋 − 4
and 2𝑛 𝜋 − 4 , 𝑛 ∈ 𝑍
d) −∞ and +∞
𝐶 𝐴 3𝑏
3. In a ∆𝐴𝐵𝐶, if 𝑎 cos 2 2 + 𝑐 cos 2 2 = 2
, then 𝑎, 𝑏, 𝑐 are in
a) A.P. b) G.P. c) H.P. d) None of these
4. The value of tan 5𝜃 is
5 tan 𝜃−10 tan3 𝜃+tan5 𝜃
a) 1−10 tan2 𝜃+5 tan4 𝜃
5 tan 𝜃+10 tan3 𝜃−tan5 𝜃
b) 1+10 tan2 𝜃−5 tan4 𝜃
5 tan5 𝜃−10 tan3 𝜃+tan 𝜃
c)
1−10 tan2 𝜃+5 tan4 𝜃
d) None of these
5. If the sides 𝑎, 𝑏 and 𝑐 of a ∆𝐴𝐵𝐶 are in A.P., then
𝐴 𝐶 𝐵
(tan 2 + tan 2 ) : cot 2 , is
a) 3 ∶ 2 b) 1 ∶ 2 c) 3 ∶ 4 d) None of these
6. If in a triangle 𝐴𝐵𝐶
cos 𝐴 cos 𝐵 cos 𝐶 𝑎 𝑏
2 + +2 = + ,
𝑎 𝑏 𝑐 𝑏𝑐 𝑐𝑎
then the value of the angle 𝐴 is
𝜋 𝜋 𝜋 𝜋
a) 3 b) 4 c) 2 d) 6
7. The value of tan 𝛼 + 2 tan(2𝛼) + 4 tan(4𝛼) +. . . +2𝑛−1 tan(2𝑛−1 𝛼) + 2𝑛 cot(2𝑛 𝛼) is
a) cot (2𝑛 𝛼) b) 2𝑛 tan(2𝑛 𝛼) c) 0 d) cot 𝛼
𝜋 𝜋
8. The maximum value ofcos 2 ( 3 − 𝑥) − cos2 ( 3 + 𝑥) is
√3 1 √3 3
a) − 2
b) 2 c) 2
d) 2
9. If 𝑎 = 2, 𝑏 = 3, 𝑐 = 5 in ∆𝐴𝐵𝐶, then 𝐶 =
𝜋 𝜋 𝜋
a) 6 b) 3 c) 2 d) None of these
𝑎 𝑏
10. If in a ∆𝐴𝐵𝐶, cos 𝐴 = cos 𝐵, then
a) 2 sin 𝐴 sin 𝐵 sin 𝐶 = 1
b) sin2 𝐴 + sin2 𝐵 = sin2 𝐶

c) 2 sin 𝐴 cos 𝐵 = sin 𝐶

d) None of these
𝜋
11. If 1 + sin 𝜃 + sin2 𝜃+. . . ∞ = 4 + 2√3, 0 < 𝜃 < 𝜋, 𝜃 ≠ 2 , then
𝜋 𝜋 𝜋 𝜋 𝜋 2𝜋
a) 𝜃 = 3
b) 𝜃 = 6
c) 𝜃 = 3 or 6 d) 𝜃 = 3 or 𝜃 = 3
12. In a ∆𝐴𝐵𝐶,
𝑎(𝑏 2 + 𝑐 2 ) cos 𝐴 + 𝑏(𝑐 2 + 𝑎2 ) cos 𝐵 + 𝑐(𝑎2 + 𝑏 2 ) cos 𝐶 is equal to
a) 𝑎𝑏𝑐 b) 2𝑎𝑏𝑐 c) 3𝑎𝑏𝑐 d) 4𝑎𝑏𝑐
π
13. If tan(π cos θ) = cot(π sin θ), then the value of cos (θ − ) is equal to
4
1 1 1 1
a) b) c) d)
2√2 √2 3√2 4√2
14. The number of points of intersection of the two curves 𝑦 = 2 sin 𝑥 and 𝑦 = 5𝑥 2 + 2𝑥 + 3, is
a) 0 b) 1 c) 2 d) ∞
15. If, in a ∆𝐴𝐵𝐶, + 𝑏 + 𝑐)(𝑏 + 𝑐 − 𝑎) = 𝜆 𝑏𝑐, then
(𝑎
a) 𝜆 < 0 b) 𝜆 > 4 c) 𝜆 > 0 d) 0 < 𝜆 < 4
16. The expression cosec 𝐴 cot 𝐴 − sec 𝐴 tan 𝐴 − (cot 𝐴 − tan 𝐴)(sec 𝐴 cosec 2 𝐴 − 1) is equal to
2 2 2 2 2 2 2

a) 1 b) −1 c) 0 d) 2
17. The sides of a triangle are in A.P. and its area is 3/5 times the area of an equilateral triangle of the same
perimeter. Then, the ratio of the sides is
a) 1 ∶ 2 ∶ 3 b) 3 ∶ 5 ∶ 7 c) 1 ∶ 3 ∶ 5 d) None of these
𝑏 𝜋 𝑎+𝑏 𝑎−𝑏
18. If tan 𝛼 = 𝑎 , 𝑎 > 𝑏 > 0 and if 0 < 𝛼 < 4 , then √𝑎−𝑏 − √𝑎+𝑏 is equal to
2 sin 𝛼 2 cos 𝛼 2 sin 𝛼 2 cos 𝛼
a) b) c) d)
√cos 2𝛼 √cos 2𝛼 √sin 2𝛼 √sin 2𝛼
1
19. If sin θ + cos θ = 𝑥, then sin θ + cos6 θ = 4 [4 − 3(𝑥 − 1)2 ] for
6 2

a) all real 𝑥 b) 𝑥 2 ≤ 2 c) 𝑥 2 > 2 d) None of these

sin 𝐴 sin(𝐴−𝐵)
20. If in a triangle 𝐴𝐵𝐶, sin 𝐶 = sin(𝐵−𝐶)
, then
a) 𝑎, 𝑏, 𝑐 are in A.P. b) 𝑎 , 𝑏 , 𝑐 are in A.P. c) 𝑎, 𝑏, 𝑐 are in H.P.
2 2 2
d) 𝑎2 , 𝑏 2 , 𝑐 2 are in H.P

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C A A A D C D C D C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. D C A A D C B A B B

DPP No. 02
DPP Year : 2023
Topic : Quadratic Equation

1. If 𝑏 > 𝑎, then the equation (𝑥 − 𝑎)(𝑥 − 𝑏) − 1 = 0 has

a) Both roots in [𝑎, 𝑏]
b) Both roots in (−∞, 𝑎)
c) Roots in (−∞, 𝑎) and other in (𝑏, ∞)
d) Both roots in (𝑏, ∞)
2. 𝑥 2 −3 𝑥 2 −3
The number of real solutions of the equation (5 + 2√6) + (5 − 2√6) = 10, is
a) 2 b) 4 c) 6 d) None of these
3. Number of roots of the equation 𝑥 −
2
=1−
2
is
𝑥−1 𝑥−1
a) One b) Two c) Infinite d) None of these
4. 2
If the sum of the roots of the equation 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 is equal to the sum of the squares of their
reciprocals of their reciprocals, then
a) 𝑐 2 𝑏, 𝑎2 𝑐, 𝑏 2 𝑎 are in A.P.
b) 𝑐 2 𝑏, 𝑎2 𝑐, 𝑏 2 𝑎 are in G.P.
𝑏 𝑎 𝑐
c) 𝑐 , 𝑏 , 𝑎 are in G.P.
𝑎 𝑏 𝑐
d) 𝑏 , 𝑐 , 𝑎 are in G.P.
𝛼
5. Let 𝛼, 𝛽 be the roots of the equation 𝑥 2 − 𝑝𝑥 + 𝑟 = 0 and 2 , 2𝛽 be the roots of the equation 𝑥 2 − 𝑞𝑥 + 𝑟 =
0. Then the value of 𝑟 is
2 2 2 2
a) (𝑝 − 𝑞)(2𝑞 − 𝑝) b) (𝑞 − 𝑝)(2𝑝 − 𝑞) c) (𝑞 − 2𝑝)(2𝑞 − 𝑝) d) (2𝑝 − 𝑞)(2𝑞 − 𝑝)
9 9 9 9
6. 2 7
If 𝜔 is an imaginary cube root of unity, then (1 + 𝜔 − 𝜔 ) equals
a) 128 𝜔 b) −128 𝜔 c) 128 𝜔 d) −128 𝜔2
7. If 𝑎 + 𝑏 + 𝑐 = 0, then the roots of the equation 4𝑎𝑥 2 + 3𝑏𝑥 + 2𝑐 = 0 are
a) Equal b) Imaginary c) Real d) None of these
8. 2 2
For how many values of 𝑘, 𝑥 + 𝑥 + 1 + 2𝑘 (𝑥 − 𝑥 − 1) = 0 is a perfect square?
a) 2 b) 0 c) 1 d) 3
9. The number of solutions of
log 5+log (𝑥 2 +1)
= 2 is
log(𝑥−2)
a) 2 b) 3 c) 1 d) None of these
10. The number of real roots of the equation |𝑥| − 3|𝑥| + 2 = 0 is
2

a) 4 b) 3 c) 2 d) 1
11. If the difference between the roots of 𝑥 2 + 𝑎𝑥 + 𝑏 = 0 and 𝑥 2 + 𝑏𝑥 + 𝑎 = 0 is same and 𝑎 ≠ 𝑏, then
a) 𝑎 + 𝑏 + 4 = 0 b) 𝑎 + 𝑏 − 4 = 0 c) 𝑎 − 𝑏 − 4 = 0 d) 𝑎 − 𝑏 + 4 = 0
12. If log √3 5 = 𝑎 and log √3 2 = 𝑏, then log √3 300 is equal to
a) 2(a + b) b) 2(𝑎 + 𝑏 + 1) c) 2(𝑎 + 𝑏 + 2) d) 𝑎 + 𝑏 + 4

13. If 𝑝, 𝑞, 𝑟, 𝑠, 𝑡 are numbers such that 𝑝 + 𝑞 < 𝑟 + 𝑠, 𝑞 + 𝑟 < 𝑠 + 𝑡, 𝑟 + 𝑠 < 𝑡 + 𝑝, 𝑠 + 𝑡 < 𝑝 + 𝑞, then the
largest and smallest numbers are
a) 𝑝 and 𝑞 respectively b) 𝑟 and 𝑡 respectively c) 𝑟 and 𝑞 respectively d) 𝑞 and 𝑝 respectively
14. The number of integral solutions of 𝑥+2 1
> is
𝑥 2 +1 2
a) 4 b) 5 c) 3 d) None of these
15. Let 𝛼, 𝛽 be the roots of the equation 𝑥 − 𝑥 + 𝑝 = 0 and 𝛾, 𝛿 be the roots of 𝑥 − 4𝑥 + 𝑞 = 0. If 𝛼, 𝛽, 𝛾, 𝛿 are
2 2

in GP, then integral values of 𝑝, 𝑞 are respectively

a) −2, −32 b) −2, 3 c) −6, 3 d) −6, −32
16. If 𝜔 is a complex cube root of unity, then
225 + (3𝜔 + 8𝜔2 )2 + (3𝜔2 + 8𝜔)2 is equal to
a) 72 b) 192 c) 200 d) 248
17. If the roots of 𝑥 − 12𝑥 + 39𝑥 − 28 = 0 are in A.P., then their common difference is
3 2

a) ± 1 b) ± 2 c) ± 3 d) ± 4
18. The solution set of the inequation 2
> 1, 𝑥 ≠ 4, is
|𝑥−4|
a) (2, 6) b) (2, 4) ∪ (4, 6) c) (−∞, 2) ∪ (6, ∞) d) None of these
19. If 𝑎, 𝑏, 𝑐 are all positive and in H.P., then the roots of 𝑎𝑥 2 + 2 𝑏𝑥 + 𝑐 = 0 are
a) Real b) Imaginary c) Rational d) Equal
20. If 𝛼 and 𝛽 be the roots of the equation 𝑥 2 + 𝑥√𝛼 + 𝛽 = 0, then
a) 𝛼 = 1 and 𝛽 = −1 b) 𝛼 = 1 and 𝛽 = −2
c) 𝛼 = 2 and 𝛽 = 1 d) 𝛼 = 2 and 𝛽 = −2

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C B D A D D C A D A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A B A C A D C B B D

DPP No. 02
DPP Year : 2023
Topic : Straight Line

1. If 𝑃𝑀 is the perpendicular from 𝑃(2, 3) onto the line 𝑥 + 𝑦 = 3, then the coordinates of 𝑀 are
a) (2,1) b) (−1, 4) c) (1,2) d) (4, −1)
2. A line through the point 𝐴(2, 0) which makes an angle of 30° with the positive direction of 𝑥-axis is
rotated about 𝐴 in clockwise direction through an angle of 15°. Then, the equation of the straight line in
the new position is
a) (2 − √3 )𝑥 + 𝑦 − 4 + 2√3 = 0 b) (2 − √3)𝑥 − 𝑦 − 4 + 2√3 = 0
c) (2 − √3)𝑥 − 𝑦 + 4 + 2√3 = 0 d) (2 − √3)𝑥 + 𝑦 + 4 + 2√3 = 0
3. The distance between the pair of parallel lines 𝑥 2 + 2𝑥𝑦 + 𝑦 2 − 8𝑎𝑥 − 8𝑎𝑦 − 9𝑎2 = 0 is
a) 2√5𝑎 b) √10𝑎 c) 10 𝑎 d) 5√2𝑎
4. One vertex of the equilateral triangle with centroid at the origin and one side as 𝑥 + 𝑦 − 2 = 0 is
a) (−1, −1) b) (2, 2) c) (−2, −2) d) None of these
5. The equation of straight line through the intersection of the lines 𝑥 − 2𝑦 = 1 and 𝑥 + 3𝑦 = 2 and
parallel to 3𝑥 + 4𝑦 = 0, is
a) 3𝑥 + 4𝑦 + 5 = 0 b) 3𝑥 + 4𝑦 − 10 = 0 c) 3𝑥 + 4𝑦 − 5 = 0 d) 3𝑥 + 4𝑦 + 6 = 0
6. The straight line 3𝑥 + 𝑦 = 9 divided the line segment joining the points (1, 3) and (2,7) in the ratio
a) 3:4 externally b) 3:4 internally c) 4:5 internally d) 5:6 externally
7. Orthocentre of the triangle whose sides are given by 4 𝑥 − 7 𝑦 + 10 = 0, 𝑥 + 𝑦 − 5 = 0 and
7 𝑥 + 4 𝑦 − 15 = 0 is
a) (−1, −2) b) (1, −2) c) (−1,2) d) (1,2)
8. The diagonals of the parallelogram whose sides are
𝑙𝑥 + 𝑚𝑦 + 𝑛 = 0, 𝑙𝑥 + 𝑚𝑦 + 𝑛′ = 0, 𝑚𝑥 + 𝑙𝑦 + 𝑛 = 0, 𝑚𝑥 + 𝑙𝑦 + 𝑛′ = 0 include an angle
𝑙 2 −𝑚2 2 𝑙𝑚
a) 𝜋/3 b) 𝜋/2 c) tan−1 (𝑙2 +𝑚2 ) d) tan−1 (𝑙2 +𝑚2 )
9. The centroid of an equilateral triangle is (0, 0). If two vertices of the triangle lie on 𝑥 + 𝑦 = 2√2, then
one of them will have its coordinates
a) (√2 + √6, √2 − √6) b) (√2 + √3, √2 − √3) c) (√2 + √5, √2 − √5) d) None of theses
10. If the lines 𝑎𝑥 + 2 𝑦 + 1 = 0, 𝑏𝑥 + 3 𝑦 + 1 = 0, 𝑐𝑥 + 4 𝑦 + 1 = 0 are concurrent, then a, b, c are in
a) AP b) GP c) HP d) None of these
11. Locus of the centroid of triangle whose vertices are (𝑎 cos 𝑡, 𝑎 sin 𝑡), (𝑏 sin 𝑡, −𝑏 cos 𝑡) and (1,0), where 𝑡
is a parameter, is
a) (3𝑥 − 1)2 + (3𝑦)2 = 𝑎2 − 𝑏 2
b) (3𝑥 − 1)2 + (3𝑦)2 = 𝑎2 + 𝑏 2
c) (3𝑥 + 1)2 + (3𝑦)2 = 𝑎2 + 𝑏 2
d) (3𝑥 + 1)2 + (3𝑦)2 = 𝑎2 − 𝑏 2

12. If 𝜃 is the acute angle between the lines given by 6𝑥 2 + 5𝑥𝑦 − 7𝑥 + 13𝑦 − 3 = 0, then the equation of
the line passing through the point of intersection of these lines and making angle 𝜃 with the positive 𝑥-
axis is
a) 2𝑥 + 11𝑦 + 13 = 0 b) 11𝑥 − 2𝑦 + 13 = 0 c) 2𝑥 − 11𝑦 + 2 = 0 d) 11𝑥 + 2𝑦 − 11 = 0
𝑥2 𝑦2 2𝑥𝑦
13. If + + = 0 represents a pair of straight lines such that slope of one line is twice the other,
𝑎 𝑏 ℎ
2
then 𝑎𝑏 ∶ ℎ is
a) 9 ∶ 8 b) 8 ∶ 9 c) 1 ∶ 2 d) 2 ∶ 1
14. The lines bisecting the angle between the bisectors of the angles between the lines
𝑎𝑥 2 + 2ℎ𝑥𝑦 + 𝑏𝑦 2 = 0 are given by
a) (𝑎 − 𝑏)(𝑥 2 − 𝑦 2 ) − 4 ℎ𝑥𝑦 = 0
b) (𝑎 − 𝑏)(𝑥 2 + 𝑦 2 ) + 4ℎ𝑥𝑦 = 0
c) (𝑎 − 𝑏)(𝑥 2 − 𝑦 2 ) + 4 ℎ𝑥𝑦 = 0
d) None of these
π 4
15. The line passing through (−1, 2 ) and perpendicular to √3 sin θ + 2 cos θ = 𝑟 is
a) 2 = √3𝑟 cos θ − 2𝑟 sin θ b) 5 = −2√3𝑟 sin θ + 4𝑟 cos θ
c) 2 = √3𝑟 cos θ + 2𝑟 sin θ d) 5 = 2√3𝑟 sin θ + 4𝑟 cos θ
16. Given a family of lines 𝑎(2𝑥 + 𝑦 + 4) + 𝑏(𝑥 − 2𝑦 − 3) = 0, the number of lines belonging to the family
at a distance √10 from 𝑃(2, −3) is
a) 0 b) 1 c) 2 d) 4
17. Let the perpendiculars from any point on the line 2𝑥 + 11𝑦 = 5 upon the lines 24𝑥 + 7𝑦 − 20 = 0 and
4𝑥 − 3𝑦 − 2 = 0 have the lengths 𝑝1 and 𝑝2 respectively. Then,
a) 2𝑝1 = 𝑝2 b) 𝑝1 = 𝑝2 c) 𝑝1 = 2𝑝2 d) None of these
18. The equation of bisectors of the angles between the lines |𝑥| = |𝑦| are
1 1
a) 𝑦 = ±𝑥 and 𝑥 = 0 b) 𝑥 = 2 and 𝑦 = 2 c) 𝑦 = 0 and 𝑥 = 0 d) None of these
19. The pairs of straight lines 𝑥 − 3𝑥𝑦 + 2𝑦 = 0 and 𝑥 − 3𝑥𝑦 + 2𝑦 + 𝑥 − 2 = 0 form a
2 2 2 2

a) Square but not rhombus b) Rhombus

c) Parallelogram d) Rectangle but not a square
20. The straight line whose sum of the intercepts on the axes is equal to half to the product of the
intercepts, passes through the point whose coordinates are
a) (1, 1) b) (2, 2) c) (3, 3) d) (4, 4)

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C B D C C B D B A A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B B A C A B B C C B

DPP No. 03
DPP Year : 2023
Topic : Straight Line

1. A straight line through 𝑃(1,2) is such that its intercept between the axes is bisected at 𝑃. Its equation is
a) 𝑥 + 2 𝑦 = 5 b) 𝑥 − 𝑦 + 1 = 0 c) 𝑥 + 𝑦 − 3 = 0 d) 2 𝑥 + 𝑦 − 4 = 0
2. The incentre of the triangle formed by the lines 𝑥 = 0, 𝑦 = 0 and 3𝑥 4𝑦 = 12 is at
a) (1/2,1/2) b) (1, 1) c) (1, 1/2) d) (1/2,1)
3. A pair of perpendicular straight lines passes through the origin and also through the point of
intersection of the curve 𝑥 2 + 𝑦 2 = 4 with 𝑥 + 𝑦 = 𝑎. The set containing the value of ‘𝑎’ is
a) {−2, 2} b) {−3, 3} c) {−4, 4} d) {−5, 5}
4. 2 2 2 2
If pairs straight lines 𝑥 − 2𝑝𝑥𝑦 − 𝑦 = 0 and 𝑥 − 2𝑞𝑥𝑦 − 𝑦 = 0 be such that each pair bisects the
angle between the other pair, then
a) 𝑝𝑞 = 1 b) 𝑝𝑞 = −1 c) 𝑝𝑞 = 2 d) 𝑝𝑞 = −2
5. In a rhombus 𝐴𝐵𝐶𝐷 the diagonals 𝐴𝐶 and 𝐵𝐷 intersect at the point (3,4). If the point 𝐴 is (1,2) the
diagonal 𝐵𝐷 has the equation
a) 𝑥 − 𝑦 − 1 = 0 b) 𝑥 + 𝑦 − 1 = 0 c) 𝑥 − 𝑦 + 1 = 0 d) 𝑥 + 𝑦 − 7 = 0
6. 2 2
The gradient of one of the lines of 𝑎𝑥 + 2ℎ𝑥𝑦 + 𝑏𝑦 = 0 is twice that of the other, then
a) ℎ2 = 𝑎𝑏 b) ℎ = 𝑎 + 𝑏 c) 8ℎ2 = 9𝑎𝑏 d) 9ℎ2 = 8𝑎𝑏
7. The family of lines making an angle 30° with the line √3𝑦 = 𝑥 + 1 is
a) 𝑥 = 𝜆(λ is parameter ) b) 𝑦 = −√3𝑥 + 𝜆(λ is parameter )
c) 𝑦 = √3𝑥 + 𝜆 d) None of the above
8. If the slope of one of the lines represented by 𝑎𝑥 2 + 2ℎ𝑥𝑦 + 𝑏𝑦 2 = 0 be the square of the other, then
𝑎+ℎ 8ℎ 2
+ is
ℎ 𝑎𝑏
a) 3 b) 4 c) 5 d) 6
9. The equation 𝑦 − 𝑥 + 2 𝑥 − 1 = 0, represents
2 2

a) A pair of st. lines b) A circle c) A parabola d) An ellipse

10. The vertices of a ∆ 𝑂𝐵𝐶 are (0, 0), 𝐵(−3, −1) and 𝐶(−1, −3). The equation of a line parallel to 𝐵𝐶 and
intersecting sides 𝑂𝐵 and 𝑂𝐶 whose distance from the origin is 1/2, is
1 1 1 1
a) 𝑥 + 𝑦 + = 0 b) 𝑥 + 𝑦 − = 0 c) 𝑥 + 𝑦 − =0 d) 𝑥 + 𝑦 + =0
2 2 √2 √2
11. The angle between the line joining the points (1, −2), (3, 2) and the line 𝑥 + 2𝑦 − 7 = 0 is
a) π b) π/2 c) π/3 d) π/6
12. The equation 𝑦 − 𝑥 + 2𝑥 − 1 = 0 represents
2 2

a) A hyperbola b) An ellipse
c) A pair of straight lines d) A rectangular hyperbola
13. The equation to the bisecting the join of (3, −4) and (5,2) and having its intercepts on the 𝑥-axis and the
𝑦-axis in the ratio 2 : 1 is
a) 𝑥 + 𝑦 − 3 = 0 b) 2𝑥 − 𝑦 = 9 c) 𝑥 + 2𝑦 = 2 d) 2𝑥 + 𝑦 = 7

14. 𝐴(−5,0) and 𝐵(3,0) are two of the vertices of a triangle 𝐴𝐵𝐶. Its area is 20 square cms. The vertex 𝐶 lies
on the line 𝑥 − 𝑦 = 2. The coordinates of 𝐶 are
a) (−7, −5) or (3,5) b) (−3, −5) or (−5,7) c) (7,5) or (3,5) d) (−3, −5) or (7,5)
15. The point of concurrence of the lines 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 and 𝑎, 𝑏, 𝑐 satisfy the relation 3𝑎 + 2𝑏 + 4𝑐 = 0 is
3 1 3 1 3 1 3 1
a) ( , ) b) ( , ) c) ( , ) d) ( , )
2 4 4 4 4 2 2 2
16. The angle between the straight line 𝑥 − 𝑦√3 = 5 and √3𝑥 + 𝑦 = 7 is
a) 90° b) 60° c) 75° d) 30°
17. The equation 𝑦 = ±√3𝑥, 𝑦 = 1 are the sides of
a) An equilateral triangle b) A right angled triangle
c) An isosceles triangle d) An obtuse triangle
18. A line passes through the point of intersection of the lines 3𝑥 + 𝑦 + 1 = 0 and 2𝑥 − 𝑦 + 3 = 0 and
makes equal intercepts with axes. Then, equation of the line is
a) 5𝑥 + 5𝑦 − 3 = 0 b) 𝑥 + 5𝑦 − 3 = 0 c) 5𝑥 − 𝑦 − 3 = 0 d) 5𝑥 + 5𝑦 + 3 = 0
19. The equation of the straight line which passes through the point (1, −2) and cuts off equal intercepts
from the axes will be
a) 𝑥 + 𝑦 = 1 b) 𝑥 − 𝑦 = 1 c) 𝑥 + 𝑦 + 1 = 0 d) 𝑥 − 𝑦 − 2 = 0
20. The orthocenter of a triangle formed by the lines 𝑥 + 𝑦 = 1, 2𝑥 + 3𝑦 = 6 and 4𝑥 − 𝑦 + 4 = 0 lies in the
a) Ist quadrant b) IInd quadrant c) IIIrd quadrant d) IVth quadrant

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. D B A B D C C D A D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B C C D C A A A C A

DPP No. 5
DPP Year : 2023
Topic : Trigonometry

1. If 𝜃 ∈ [0,5𝜋] and 𝑟 ∈ 𝑅 such that 2 sin 𝜃 = 𝑟 4 − 2𝑟 2 + 3, then the maximum number of values of the pair
(𝑟, 𝜃) is
a) 6 b) 8 c) 10 d) None of these
2. In a triangle 𝐴𝐵𝐶, 𝑟 =
𝐵 𝐵 𝐶 𝐶
a) (𝑠 − 𝑎) tan 2 b) (𝑠 − 𝑏) tan 2 c) (𝑠 − 𝑏) tan 2 d) (𝑠 − 𝑎) tan 2
3. If 𝑝1 , 𝑝2 , 𝑝3 are altitude of a triangle 𝐴𝐵𝐶 from the vertices 𝐴, 𝐵, 𝐶 and ∆, the area of the triangle, then
1 1 1
+ + 2=
𝑝12 𝑝22 𝑝3
cot 𝐴+cos 𝐵+cot 𝐶
a)
∆
∆
b) cot 𝐴+cot 𝐵+cot 𝐶
c) ∆(cot 𝐴 + cot 𝐵 + cot 𝐶)
d) None of these
4. Number of solutions of the equation sin 2𝜃 + 2 = 4 sin 𝜃 + cos 𝜃 lying in the interval [𝜋, 5 𝜋], is
a) 0 b) 2 c) 4 d) 5
5. If twice the square of the diameter of a circle is equal to half the sum of the squares of the sides of
inscribed triangle 𝐴𝐵𝐶, then sin2 𝐴 + sin2 𝐶 is equal to
a) 1 b) 2 c) 4 d) 8
6. tan 9° − tan 27° − tan 63° + tan 81° is equal to
a) 0 b) 1 c) −1 d) 4
𝜋
7. If sin 4𝐴 − cos 2𝐴 = cos 4𝐴 − sin 2𝐴, (0 < 𝐴 < 4 ), then the value of tan 4𝐴 is
1 √3−1
a) 1 b) c) √3 d)
√3 √3+1
8. In a ∆𝐴𝐵𝐶, sin 𝐴 and sin 𝐵 are the roots of the equation 𝑐 2 𝑥 2 − 𝑐(𝑎 + 𝑏)𝑥 + 𝑎𝑏 = 0, then sin 𝐶 =
a) 1/√2 b) 1/2 c) 1 d) 0
9. If sin(𝛼 + 𝛽) = 1, sin(𝛼 − 𝛽) = 1 /2; 𝛼, 𝛽 ∈ [0, 𝜋/2], then tan(𝛼 + 2𝛽) tan(2𝛼 + 𝛽) is equal to
a) 1 b) −1 c) 0 d) 1/2
√1−𝑎02
1
10. If 𝑎𝑛+1 = √2 (1 + 𝑎𝑛 ), then cos (𝑎 )=
1 𝑎2 𝑎3 …to ∞

a) 1 b) −1 c) 𝑎0 d) 1/𝑎0
11. If the angles of a triangle are in the ratio 1 ∶ 2 ∶ 7, then the ratio of the greatest side to the least side is
a) (√5 − 1) ∶ (√5 + 1) b) (√5 + 1) ∶ (√5 − 1) c) (√5 + 2) ∶ (√5 − 2) d) (√5 − 2) ∶ (√5 + 2)
2𝜋 9√3
12. In a ∆𝐴𝐵𝐶, 𝐴 = 3
,𝑏 − 𝑐 = 3√3 cm and ∆= 2
cm2 . Then, 𝑎 =
a) 6√3 cm b) 9 cm c) 18 cm d) 12 cm
13. If the radius of the incircle of a triangle with its sides 5𝑘, 6𝑘, and 5𝑘 is 6, then 𝑘 is equal to
a) 3 b) 4 c) 5 d) 6

14. The minimum value of 2sin 𝑥 + 2cos 𝑥 , is

1 1
− 1−
a) 1 b) 2 c) 2 √2 d) 2 √2
1
15. Minimum value of 3 sin 𝜃−4 cos 𝜃+7 is
1 5 7 1
a) 12 b) 12 c) 12 d) 6
𝑝+𝑞
16. If cosec 𝜃 = 𝑝−𝑞, then cot(𝜋/4 + 𝜃/2) =
𝑝 𝑞
a) √𝑞 b) √𝑝 c) √𝑝𝑞 d) 𝑝𝑞
1 𝑡
17. Suppose 0 < 𝑡 < 𝜋 andsin 𝑡 + cos 𝑡 = . Then, tan is equal to
5 2
1
a) 2 b) 3 c) 3 d) 5
18. For what and only what values of 𝛼 lying between 0 and 𝜋 is the inequality sin 𝛼 cos3 𝛼 > sin3 𝛼 cos 𝛼
valid?
a) 𝛼 ∈ (0, 𝜋/4) b) 𝛼 ∈ (0, 𝜋/2) c) 𝛼 ∈ ( 𝜋/4 , 𝜋/2) d) None of these
19. If 𝛼 + 𝛽 − 𝛾 = 𝜋, then sin2 𝛼 + sin2 𝛽 − sin2 𝛾 is equal to
a) 2 sin 𝛼 sin 𝛽 cos 𝛾 b) 2 cos 𝛼 cos 𝛽 cos 𝛾 c) 2 sin 𝛼 sin 𝛽 sin 𝛾 d) None of these
20. If sec 𝑥 cos 5𝑥 + 1 = 0, where 0 < 𝑥 < 2𝜋, then 𝑥 is equal to
𝜋 𝜋 𝜋 𝜋
a) , b) c) d) None of these
5 4 5 4

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A B A C C D C C A C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B B B D A B A A A C

DPP No. 4
DPP Year : 2023
Topic : Trigonometry

1. If the equation sin4 𝜃 + cos4 𝜃 = 𝑎 has a real solution then

1 1 1
a) 𝑎 ≤ 2 b) 𝑎 ≥ 2 c) 2 ≤ 𝑎 ≤ 1 d) 𝑎 ≥ 0
2. The general solution of the equation (√3 − 1) sin θ + ( √3 + 1) cos θ = 2 is
𝜋
𝜋
a) 2𝑛𝜋 ± +
4
12
𝜋 𝜋
b) 𝑛𝜋 + (−1)𝑛 +
4 12
𝜋 𝜋
c) 2𝑛𝜋 ± 4 − 12
𝜋 𝜋
d) 𝑛𝜋 + (−1)𝑛 4 − 12
1 1
3. If sin 𝐴 = 10 and sin 𝐵 = , where 𝐴 and 𝐵 are positive acute angles, then 𝐴 + 𝐵 is equal to
√ √5
𝜋 𝜋 𝜋
a) 𝜋 b) c) d)
2 3 4
4. 2
The general solution of sin θ sec θ + √3 tan θ = 0 is
𝜋
a) θ = 𝑛𝜋 + (−1)𝑛+1 3 , θ = 𝑛𝜋, 𝑛 ∈ 𝐼 b) θ = 𝑛𝜋, 𝑛 ∈ 𝐼
𝜋 𝑛𝜋
c) θ = 𝑛𝜋 + (−1)𝑛+1 , 𝑛 ∈ 𝐼 d) θ = ,𝑛 ∈𝐼
3 2
5. If 𝑦 + cos 𝜃 = sin 𝜃 has a real solution, then
a) −√2 ≤ 𝑦 ≤ √2 b) 𝑦 > √2 c) 𝑦 ≤ −√2 d) None of these
6. 2
If cos(θ − α) = 𝑎, sin(θ − 𝛽) = 𝑏, then cos (𝛼 − 𝛽) + 2𝑎𝑏 sin(𝛼 − 𝛽) is equal to
a) 4𝑎2 𝑏2 b) 𝑎2 − 𝑏 2 c) 𝑎2 + 𝑏 2 d) −𝑎2 𝑏 2
7. 2
The equation 8 sec 𝜃 − 6 sec 𝜃 + 1 = 0 has
a) Exactly two roots b) Exactly four roots c) Infinitely many roots d) No roots
8. If the sides a, b, c of a triangle ABC are the roots of the equation 𝑥 3 − 13𝑥 2 + 54𝑥 − 72 = 0, then the
cos 𝐴 cos 𝐵 cos 𝐶
value of 𝑎
+ 𝑏
+ 𝑐
is equal to
169 61 61 169
a) 144 b) 72 c) 144 d) 72
9. cos4 𝜃 − sin4 𝜃 is equal to
𝜃 𝜃
a) 1 + 2 sin2 (2 ) b) 2 cos2 𝜃 − 1 c) 1 − 2 sin2 (2 ) d) 1 + 2 cos 2 𝜃
1 1
10. The value of cos 15° cos 7 2 ° sin 7 2 ° is
1 1 1 1
a) 2 b) 8 c) 4 d) 16
1−sin θ 1+sin θ
11. If θ lies in the second quadrant, then the value of √1+sin θ + √ 1−sinθ is equal to
a) 2sec θ b) −2 sec θ c) 2 cosec θ d) None of these
12. The value of cos 2 𝐴 (3 − 4 cos2 𝐴)2 + sin2 𝐴 (3 − 4 sin2 𝐴)2 is equal to
a) cos 4𝐴 b) sin 4𝐴 c) 1 d) None of these
13. Let the angles 𝐴, 𝐵, 𝐶 of ∆𝐴𝐵𝐶 be in A.P. and let
a) 75° b) 45° c) 60° d) 15°

𝑏 𝑎+𝑏 𝑎−𝑏
14. If tan 𝑥 = 𝑎, then √𝑎−𝑏 + √𝑎+𝑏 =
2 sin 𝑥 2 cos 𝑥 2 cos 𝑥 2 sin 𝑥
a) b) c) d)
√sin 2𝑥 √cos 2𝑥 √sin 2𝑥 √cos 2𝑥
15. If sin 𝐴 + cos 𝐴 = 𝑚 and sin 𝐴 + cos 3 𝐴 = 𝑛, then
3

a) 𝑚3 − 3𝑚 + 𝑛 = 0 b) 𝑛3 − 3𝑛 + 2𝑚 = 0 c) 𝑚3 − 3𝑚 + 2𝑛 = 0 d) 𝑚3 + 3𝑚 + 2𝑛 = 0
16. The most general solutions of the equation sec 𝑥 − 1 = (√2 − 1) tan 𝑥 are given by
π π
a) 𝑛π + 8 b) 2𝑛π, 2𝑛π + 4 c) 2𝑛π d) None of these
17. If cos(𝜃 − 𝛼) = 𝑎 , cos(𝜃 − 𝛽) = 𝑏, then sin 2 (𝛼
− 𝛽) + 2𝑎𝑏 cos(𝛼 − 𝛽) is equal to
a) 𝑎 + 𝑏
2 2
b) 𝑎 − 𝑏
2 2
c) 𝑏 2 − 𝑎2 d) −𝑎2 − 𝑏 2
18. The sum S = sinθ + sin 2θ+. . . + sin 𝑛θ equals
1 𝑛θ θ 1 𝑛θ θ
a) sin 2 (𝑛 + 1)θ sin 2
/ sin 2 b) cos 2 (𝑛 + 1)θ sin 2
/ sin 2
1 𝑛θ θ 1 𝑛θ θ
c) sin 2 (𝑛 + 1)θ cos 2 / sin 2 d) cos 2 (𝑛 + 1)θ cos 2 / sin 2
19. The sides of an equilateral triangle, a square and a regular hexagon circumscribed in a circle are in
a) A.P. b) G.P. c) H.P. d) None of these
tan 3𝜃−1
20. If tan 3𝜃+1 = √3, then the general value of 𝜃 is
𝑛𝜋 𝜋 7𝜋 𝑛𝜋 7𝜋 𝜋
a) 3
− 12 b) 𝑛𝜋 + 12 c) 3
+ 36 d) 𝑛𝜋 + 12

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C A D B A C D C B B

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B C A B C B A A C C

DPP No. 3
DPP Year : 2023
Topic : Trigonometry

1. In a ∆𝐴𝐵𝐶, angles 𝐴, 𝐵, 𝐶 are in A.P., then

√3−4 sin 𝐴 sin 𝐶
lim |𝐴−𝐶|
is equal to
𝐴→𝐶
a) 1 b) 2 c) 3 d) 4
𝜋
2. For all values of 𝜃, the values of 3 − cos 𝜃 + cos (𝜃 + 3
) lie in the interval
a) [−2, 3] b) [−2, 1] c) [2, 4] d) [1, 5]
𝐴+𝐵 𝐵−𝐴
3. If cos 𝐴 = 𝑚 cos 𝐵 and cot 2 = 𝜆 tan 2 , then 𝜆 is
𝑚 𝑚+1 𝑚+1
a) 𝑚−1 b) 𝑚 c) 𝑚−1 d) None of these
𝜋 3𝜋 5𝜋 7𝜋
4. The value of cos 4 ( 8 ) + cos4 ( 8 ) + cos 4 ( 8 ) + cos 4 ( 8 ) is
1 3
a) 0 b) 2 c) 2 d) 1
12 𝜋 3 3𝜋
5. If sin 𝜃 = 13 , (0 < 𝜃 < 2 ) and cos ϕ = − 5 (𝜋 < ϕ < 2 ), then sin(𝜃 + ϕ) will be
a) −56/61b) −56/65c) 1/65d) −56
6. The quadratic equation whose roots are sec 𝜃 and cosec 2 𝜃 can be
2

a) 𝑥 2 − 2𝑥 + 2 = 0 b) 𝑥 2 + 5𝑥 + 5 = 0 c) 𝑥 2 − 4𝑥 + 4 = 0 d) None of these
1 1
7. If sec 𝜃 = 𝑚 and tan 𝜃 = 𝑛, then [(𝑚 + 𝑛) + ] is
𝑚 (𝑚+𝑛)
a) 2 b) 2𝑚 c) 2𝑛 d) 𝑚𝑛
8. If in a ∆ 𝐴𝐵𝐶, ∠𝐶 = 90°, then the maximum value of sin 𝐴 sin 𝐵 is
1
a) b) 1 c) 2 d) None of these
2
9. In a cyclic quadrilateral 𝐴𝐵𝐶𝐷, he value of cos 𝐴 + cos 𝐵 + cos 𝐶 + cos 𝐷, is
a) 1 b) 0 c) −1 d) None of these
10. If the angles of a triangle are in the ratio 1 ∶ 2 ∶ 3, the corresponding sides are in the ratio
a) 2 ∶ 3 ∶ 1 b) √3 ∶ 2 ∶ 1 c) 2 ∶ √3 ∶ 1 d) 1 ∶ √3 ∶ 2
𝜋
11. If sin(𝜋 cos 𝜃) = cos(𝜋 sin 𝜃), then the value of cos (𝜃 + 4 ) equals
1 1 1 1
a) b) 2 c) − 2 d) −
√2 √2 √2 √2
12. The most general solution of
2 3
21+|cos 𝑥|+cos 𝑥+|cos 𝑥|+⋯∞ = 4 is given by
𝜋
a) 𝑥 = 𝑛𝜋 ± , 𝑛 ∈ 𝑍
3
𝜋
b) 𝑥 = 2𝑛 𝜋 ± 3 , 𝑛 ∈ 𝑍
2𝜋
c) 𝑥 = 2𝑛 𝜋 ± 3
,𝑛 ∈𝑍
d) None of these
13. If cos 𝛼 + cos 𝛽 = 0 = sin 𝛼 + sin 𝛽, then cos 2 𝛼 + cos 2 𝛽 =
a) −2 sin(𝛼 + 𝛽) b) −2 cos(𝛼 + 𝛽) c) 2 sin(𝛼 + 𝛽) d) 2 cos(𝛼 + 𝛽)

sin2 𝑦 1+cos 𝑦 sin 𝑦

14. The value of the expression 1 − 1+cos 𝑦 + sin 𝑦
− 1−cos 𝑦 is equal to
a) 0 b) 1 c) sin 𝑦 d) cos 𝑦
15. In a Δ𝐴𝐵𝐶, 𝑎 = 2𝑏 and 𝐴 = 3𝐵, the 𝐴 =
a) 90° b) 60° c) 30° d) 45°
𝜋
16. If in a ∆𝐴𝐵𝐶, 𝐴 = and 𝐴𝐷 is the median, then
3
a) 2 𝐴𝐷 2 = 𝑏 2 + 𝑐 2 + 𝑏𝑐
b) 4 𝐴𝐷 2 = 𝑏 2 + 𝑐 2 + 𝑏𝑐
c) 6 𝐴𝐷 2 = 𝑏 2 + 𝑐 2 + 𝑏𝑐
d) None of these
17. If cos(θ − 𝛼) = 𝛼, cos(θ − 𝛽) = 𝑏, then sin2(𝛼 − 𝛽) + 2𝑎𝑏 cos(𝛼 − 𝛽) is equal to
a) 𝑎2 + 𝑏 2 b) 𝑎2 − 𝑏 2 c) 𝑏 2 − 𝑎2 d) −𝑎2 − 𝑏 2
𝑥 𝑥 𝑥 sin 𝑥
18. If cos . cos … . . cos 𝑛 = 𝑥 , then
2 22 2 2𝑛 sin 𝑛
2
1 𝑥 1 𝑥 1 𝑥
2
tan 2 + 22 tan 22 +. . . + 2𝑛 tan 2𝑛 is
𝑥 1 𝑥
a) cot 𝑥 − cot 2𝑛 b) 2𝑛 cot (2𝑛 ) − cot 𝑥
1 1 1 1 𝑥
c) 𝑛 tan ( 𝑛 ) − tan 𝑥 d) cot 𝑥 − cot ( 𝑛 )
2 2 2 2𝑛 2
19. In triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹, 𝐴𝐵 = 𝐷𝐸, 𝐴𝐶 = 𝐸𝐹 and ∠𝐴 = 2 ∠𝐸. Two triangles will have the same area if
angle 𝐴 is equal to
a) 𝜋/3 b) 𝜋/2 c) 2 𝜋/3 d) 5 𝜋/6
𝜋 5𝜋 7𝜋
20. The value of sin ( ) sin ( ) sin ( ), is
18 18 18
a) 1/2 b) 1/4 c) 1/8 d) 1/16

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A C C C B C A A B D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B A B D A B A B C C

DPP No. 1
DPP Year : 2023
Topic : Properties of Triangle

3
1. In a triangle ABC, 𝑎 = 5, 𝑏 = 7 and 𝑠𝑖𝑛 𝐴 = 4 how many such triangles are possible]
(a) 1 (b) 0
(c) 2 (d) Infinite
2. If in a triangle 𝐴𝐵𝐶, (𝑠 − 𝑎)(𝑠 − 𝑏) = 𝑠(𝑠 − 𝑐), then angle C is equal to
(a) 90𝑜 (b) 45𝑜
(c) 30𝑜 (d) 60𝑜
𝐴
3. In a Δ𝐴𝐵𝐶, if 2𝑠 = 𝑎 + 𝑏 + 𝑐and (𝑠 − 𝑏)(𝑠 − 𝑐) = 𝑥 𝑠𝑖𝑛2 2 , then x =
(a) bc (b) ca
(c) ab (d) abc
4. If the angles of a triangle 𝐴𝐵𝐶be in A.P., then
(a) 𝑐 2 = 𝑎2 + 𝑏 2 − 𝑎𝑏 (b)𝑏 2 = 𝑎2 + 𝑐 2 − 𝑎𝑐
(c) 𝑎2 = 𝑏 2 + 𝑐 2 − 𝑎𝑐 (d) 𝑏 2 = 𝑎2 + 𝑐 2
5. In triangle 𝐴𝐵𝐶, (𝑏 + 𝑐) 𝑐𝑜𝑠 𝐴 + (𝑐 + 𝑎) 𝑐𝑜𝑠 𝐵 +(𝑎 + 𝑏) 𝑐𝑜𝑠 𝐶 =
(a) 0 (b) 1
(c) 𝑎 + 𝑏 + 𝑐 (d) 2(𝑎 + 𝑏 + 𝑐)
𝑠𝑖𝑛 𝐵
6. In Δ𝐴𝐵𝐶, =
𝑠𝑖𝑛(𝐴+𝐵)
𝑏 𝑏
(a) 𝑎+𝑏 (b) 𝑐
𝑐
(c) 𝑏 (d) None of these
𝑠𝑖𝑛(𝐴−𝐵)
7. In Δ𝐴𝐵𝐶, 𝑠𝑖𝑛(𝐴+𝐵) =
𝑎2 −𝑏 2 𝑎2 +𝑏 2
(a) (b)
𝑐2 𝑐2
𝑐2 𝑐2
(c) 𝑎2 −𝑏2 (d) 𝑎2 +𝑏2
8. In a Δ𝐴𝐵𝐶, if 𝑏 + 𝑐 = 3𝑎2 , then 𝑐𝑜𝑡 𝐵 + 𝑐𝑜𝑡 𝐶 − 𝑐𝑜𝑡 𝐴 =
2 2

𝑎𝑏
(a) 1 (b)
4Δ
𝑎𝑐
(c) 0 (d) 4Δ
9. In a Δ𝐴𝐵𝐶, if 𝑐 + 𝑎 − 𝑏 2 = 𝑎𝑐, then ∠𝐵 =
2 2

𝜋 𝜋
(a) 6 (b) 4
𝜋
(c) 3 (d) None of these
𝐴 𝐵 𝐵 𝐴
10. In Δ𝐴𝐵𝐶, (𝑐𝑜𝑡 2 + 𝑐𝑜𝑡 2 ) (𝑎 𝑠𝑖𝑛2 2 + 𝑏 𝑠𝑖𝑛2 2 )=
(a) 𝑐𝑜𝑡 𝐶 (b) 𝑐 𝑐𝑜𝑡 𝐶

𝐶 𝐶
(c) 𝑐𝑜𝑡 2 (d) 𝑐 𝑐𝑜𝑡 2
𝐴 𝐵 𝐶
11. In Δ𝐴𝐵𝐶, if 𝑠𝑖𝑛2 2 , 𝑠𝑖𝑛2 2 , 𝑠𝑖𝑛2 2 be in H. P. then a, b, c will be in
(a) A. P. (b) G. P.
(c) H. P. (d) None of these
𝐶 𝐶
12. In Δ𝐴𝐵𝐶, (𝑎 − 𝑏)2 𝑐𝑜𝑠 2 2 + (𝑎 + 𝑏)2 𝑠𝑖𝑛2 2 =
(a) 𝑎2 (b) 𝑏 2
(c) 𝑐 2 (d) None of these
𝐵
13. In Δ𝐴𝐵𝐶, if 𝑎 = 16, 𝑏 = 24 and 𝑐 = 20, then 𝑐𝑜𝑠 2 =
(a) 3/4 (b) 1/4
(c) 1/2 (d) 1/3
1
14. InΔ𝐴𝐵𝐶, if 𝑐𝑜𝑠 𝐴 + 𝑐𝑜𝑠 𝐶 = 4 𝑠𝑖𝑛2 2 𝐵, then 𝑎, 𝑏, 𝑐 are in
(a) A. P. (b) G. P.
(c) H. P. (d) None of these
𝐴 𝐵
15. In Δ𝐴𝐵𝐶, 1 − 𝑡𝑎𝑛 2 𝑡𝑎𝑛 2 =
2𝑐 𝑎
(a) 𝑎+𝑏+𝑐 (b) 𝑎+𝑏+𝑐
2 4𝑎
(c) 𝑎+𝑏+𝑐 (d) 𝑎+𝑏+𝑐
16. In ABC , 𝑏 2 𝑐𝑜𝑠 2 𝐴 − 𝑎2 𝑐𝑜𝑠 2 𝐵 =
(a) 𝑏 2 − 𝑎2 (b) 𝑏 2 − 𝑐 2
(c) 𝑐 2 − 𝑎2 (d) 𝑎2 + 𝑏 2 + 𝑐 2
17. InΔ𝐴𝐵𝐶, a𝑠𝑖𝑛( 𝐵 − 𝐶) + 𝑏 𝑠𝑖𝑛( 𝐶 − 𝐴) + 𝑐 𝑠𝑖𝑛( 𝐴 − 𝐵) =
(a) 0 (b) 𝑎 + 𝑏 + 𝑐
(c) 𝑎 + 𝑏 + 𝑐
2 2 2
(d) 2(𝑎2 + 𝑏 2 + 𝑐 2 )
18. InΔ𝐴𝐵𝐶, if 𝑐𝑜𝑡 𝐴 , 𝑐𝑜𝑡 𝐵 , 𝑐𝑜𝑡 𝐶be in A. P., then 𝑎2 , 𝑏 2 , 𝑐 2 are in
(a) H. P. (b) G. P.
(c) A. P. (d) None of these
19. In Δ𝐴𝐵𝐶, if (𝑎 + 𝑏 + 𝑐)(𝑎 − 𝑏 + 𝑐) =3ac, then
(a) ∠𝐵 = 60𝑜 (b) ∠𝐵 = 30𝑜
(c) ∠𝐶 = 60 𝑜
(d) ∠𝐴 + ∠𝐶 = 90𝑜
20. InΔ𝐴𝐵𝐶, if 2(𝑏𝑐 𝑐𝑜𝑠 𝐴 + 𝑐𝑎 𝑐𝑜𝑠 𝐵 + 𝑎𝑏 𝑐𝑜𝑠 𝐶) =
(a) 0 (b) 𝑎 + 𝑏 + 𝑐
(c) 𝑎 + 𝑏 + 𝑐
2 2 2
(d) None of these

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A A A B C B A C C D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C C A A A A A C A C

DPP No. 1
DPP Year : 2023
Topic : Circle

1. A square is inscribed in the circle 𝑥 2 + 𝑦 2 − 2𝑥 + 4𝑦 + 3 = 0, whose sides are parallel to the coordinate
axes. One vertex of the square is
(a) (1 + √2, −2) (b) (1 − √2, −2)
(c) (1, −2 + √2) (d) None of these
2. If the line 𝑥 + 2𝑏𝑦 + 7 = 0 is a diameter of the circle 𝑥 2 + 𝑦 2 − 6𝑥 + 2𝑦 = 0, then 𝑏 =
(a) 3 (b) – 5
(c) –1 (d) 5
3. For all values of 𝜃, the locus of the point of intersection of the lines 𝑥 𝑐𝑜𝑠 𝜃 + 𝑦 𝑠𝑖𝑛 𝜃 = 𝑎 and 𝑥 𝑠𝑖𝑛 𝜃 −
𝑦 𝑐𝑜𝑠 𝜃 = 𝑏 is
(a) An ellipse (b) A circle
(c) A parabola (d) A hyperbola
4. If a circle whose centre is (1, –3) touches the line 3𝑥 − 4𝑦 − 5 = 0, then the radius of the circle is
(a) 2 (b) 4
5 7
(c) 2 (d) 2
5. If the circle 𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 touches x-axis, then
(a) 𝑔 = 𝑓 (b) 𝑔2 = 𝑐
2
(c) 𝑓 = 𝑐 (d) 𝑔2 + 𝑓 2 = 𝑐
6. The equation of the circle which touches both the axes and whose radius is a, is
(a) 𝑥 2 + 𝑦 2 − 2𝑎𝑥 − 2𝑎𝑦 + 𝑎2 = 0
(b) 𝑥 2 + 𝑦 2 + 𝑎𝑥 + 𝑎𝑦 − 𝑎2 = 0
(c) 𝑥 2 + 𝑦 2 + 2𝑎𝑥 + 2𝑎𝑦 − 𝑎2 = 0
(d) 𝑥 2 + 𝑦 2 − 𝑎𝑥 − 𝑎𝑦 + 𝑎2 = 0
7. The area of the circle whose centre is at (1, 2) and which passes through the point (4, 6) is
a) 5𝜋 (b) 10𝜋
(c) 25𝜋 (d) None of these
8. The centres of the circles 𝑥 2 + 𝑦 2 = 1, 𝑥 2 + 𝑦 2 + 6𝑥 − 2𝑦 = 1 and 𝑥 2 + 𝑦 2 − 12𝑥 + 4𝑦 = 1 are
(a) Same (b) Collinear
(c) Non-collinear (d) None of these
9. If a circle passes through the point (0, 0), (a, 0), (0, b), then its centre is
(a) (𝑎, 𝑏) (b) (𝑏, 𝑎)
𝑎 𝑏 𝑏 𝑎
(c) (2 , 2) (d) (2 , − 2 )
10. The equation of the circle whose centre is (1, –3) and which touches the line 2𝑥 − 𝑦 − 4 = 0 is
(a) 5𝑥 2 + 5𝑦 2 − 10𝑥 + 30𝑦 + 49 = 0
(b) 5𝑥 2 + 5𝑦 2 + 10𝑥 − 30𝑦 + 49 = 0
(c) 5𝑥 2 + 5𝑦 2 − 10𝑥 + 30𝑦 − 49 = 0
(d) None of these
11. The circle 𝑥 2 + 𝑦 2 + 4𝑥 − 4𝑦 + 4 = 0 touches
(a) x-axis (b) y-axis
(c) x-axis and y-axis (d) None of these

12. The equation of the circle which touches both axes and whose centre is (𝑥1 , 𝑦1 ) is
(a) 𝑥 2 + 𝑦 2 + 2𝑥1 (𝑥 + 𝑦) + 𝑥12 = 0
(b) 𝑥 2 + 𝑦 2 − 2𝑥1 (𝑥 + 𝑦) + 𝑥12 = 0
(c) 𝑥 2 + 𝑦 2 = 𝑥12 + 𝑦12
(d) 𝑥 2 + 𝑦 2 + 2𝑥𝑥1 + 2𝑦𝑦1 = 0
13. The equation of the circle whose radius is 5 and which touches the circle 𝑥 2 + 𝑦 2 − 2𝑥 − 4𝑦 − 20 = 0
externally at the point (5, 5), is
(a) 𝑥 2 + 𝑦 2 − 18𝑥 − 16𝑦 − 120 = 0
(b) 𝑥 2 + 𝑦 2 − 18𝑥 − 16𝑦 + 120 = 0
(c) 𝑥 2 + 𝑦 2 + 18𝑥 + 16𝑦 − 120 = 0
(d) 𝑥 2 + 𝑦 2 + 18𝑥 − 16𝑦 + 120 = 0
14. The lines 2𝑥 − 3𝑦 = 5 and 3𝑥 − 4𝑦 = 7 are the diameters of a circle of area 154 square units. The
equation of the circle is
(a) 𝑥 2 + 𝑦 2 + 2𝑥 − 2𝑦 = 62 (b) 𝑥 2 + 𝑦 2 − 2𝑥 + 2𝑦 = 47
(c) 𝑥 2 + 𝑦 2 + 2𝑥 − 2𝑦 = 47 (d) 𝑥 2 + 𝑦 2 − 2𝑥 + 2𝑦 = 62
15. A circle touches the y-axis at the point (0, 4) and cuts the x-axis in a chord of length 6 units. The radius
of the circle is
(a) 3 (b) 4
(c) 5 (d) 6
16. The number of circle having radius 5 and passing through the points (– 2, 0) and (4, 0) is
(a) One (b) Two
(c) Four (d) Infinite
17. The equation of the circle which touches x-axis and whose centre is (1, 2), is
(a) 𝑥 2 + 𝑦 2 − 2𝑥 + 4𝑦 + 1 = 0
(b) 𝑥 2 + 𝑦 2 − 2𝑥 − 4𝑦 + 1 = 0
(c) 𝑥 2 + 𝑦 2 + 2𝑥 + 4𝑦 + 1 = 0
(d) 𝑥 2 + 𝑦 2 + 4𝑥 + 2𝑦 + 4 = 0
18. The locus of the centre of the circle which cuts off intercepts of length 2𝑎 and 2𝑏 from x-axis and y-axis
respectively, is
(a) 𝑥 + 𝑦 = 𝑎 + 𝑏 (b) 𝑥 2 + 𝑦 2 = 𝑎2 + 𝑏 2
2 2 2 2
(c) 𝑥 − 𝑦 = 𝑎 − 𝑏 (d) 𝑥 2 + 𝑦 2 = 𝑎2 − 𝑏 2
19. If the lines 3𝑥 − 4𝑦 + 4 = 0 and 6𝑥 − 8𝑦 − 7 = 0 are tangents to a circle, then the radius of the circle is
(a) 3/2 (b) 3/4
(c) 1/10 (d) 1/20
20. If the radius of the circle 𝑥 2 + 𝑦 2 −18𝑥 + 12𝑦 + 𝑘 = 0 be 11, then 𝑘 =
(a) 347 (b) 4
(c) −4 (d) 49

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. D D B A B A C B C A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C B B B C B B C B C

DPP No. 2
DPP Year : 2023
Topic : Circle

1. Centre of circle (𝑥 − 𝑥1 )(𝑥 − 𝑥2 ) +(𝑦 − 𝑦1 )(𝑦 − 𝑦2 ) = 0 is

𝑥1 +𝑦1 𝑥2 +𝑦2 𝑥1 −𝑦1 𝑥2 −𝑦2
(a) ( , ) (b) ( , )
2 2 2 2
𝑥1 +𝑥2 𝑦1 +𝑦2 𝑥1 −𝑥2 𝑦1 −𝑦2
(c) ( , ) (d) ( , )
2 2 2 2
2. ABC is a triangle in which angle C is a right angle. If the coordinates of A and B be (–3, 4) and (3, –4) respectively, then the
equation of the circumcircle of triangle ABC is
(a) 𝑥 2 + 𝑦 2 − 6𝑥 + 8𝑦 = 0
(b) 𝑥 2 + 𝑦 2 = 25
(c) 𝑥 2 + 𝑦 2 − 3𝑥 + 4𝑦 + 5 = 0
(d) None of these
3. The equation of the circle in the first quadrant touching each coordinate axis at a distance of one unit from the origin is
(a) 𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 + 1 = 0
(b) 𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 − 1 = 0
(c) 𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 = 0
(d) None of these
4. The number of circles touching the line 𝑦 − 𝑥 = 0 and the y-axis is
(a) Zero (b) One
(c) Two (d) Infinite
5. The equation of the circle passing through the point (−1, −3) and touching the line 4𝑥 + 3𝑦 − 12 = 0 at the point (3, 0), is
(a) 𝑥 2 + 𝑦 2 − 2𝑥 + 3𝑦 − 3 = 0
(b) 𝑥 2 + 𝑦 2 + 2𝑥 − 3𝑦 − 5 = 0
(c) 2𝑥 2 + 2𝑦 2 − 2𝑥 + 5𝑦 − 8 = 0
(d) None of these
6. If the vertices of a triangle be (2, −2), (−1, −1) and (5, 2), then the equation of its circumcircle is
(a) 𝑥 2 + 𝑦 2 + 3𝑥 + 3𝑦 + 8 = 0
(b) 𝑥 2 + 𝑦 2 − 3𝑥 − 3𝑦 − 8 = 0
(c) 𝑥 2 + 𝑦 2 − 3𝑥 + 3𝑦 + 8 = 0
(d) None of these
7. The equation of a circle which touches both axes and the line 3𝑥 − 4𝑦 + 8 = 0 and whose centre lies in the third quadrant is
(a) 𝑥 2 + 𝑦 2 − 4𝑥 + 4𝑦 − 4 = 0
(b) 𝑥 2 + 𝑦 2 − 4𝑥 + 4𝑦 + 4 = 0
(c) 𝑥 2 + 𝑦 2 + 4𝑥 + 4𝑦 + 4 = 0
(d) 𝑥 2 + 𝑦 2 − 4𝑥 − 4𝑦 − 4 = 0
8. If one end of a diameter of the circle 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 + 11 = 0be (3, 4), then the other end is
(a) (0, 0) (b) (1, 1)
(c) (1, 2) (d) (2, 1)
9. If the equation 𝑝𝑥 + (2 − 𝑞)𝑥𝑦 + 3𝑦 2 −6𝑞𝑥 + 30𝑦 + 6𝑞 = 0 represents a circle, then the values of p and q are
2

(a) 3, 1 (b) 2, 2
(c) 3, 2 (d) 3, 4
10. The equation of the circle passing through the origin and cutting intercepts of length 3 and 4 units from the positive axes, is
(a) 𝑥 2 + 𝑦 2 + 6𝑥 + 8𝑦 + 1 = 0
(b) 𝑥 2 + 𝑦 2 − 6𝑥 − 8𝑦 = 0
(c) 𝑥 2 + 𝑦 2 + 3𝑥 + 4𝑦 = 0
(d) 𝑥 2 + 𝑦 2 − 3𝑥 − 4𝑦 = 0
11. Circle 𝑥 2 + 𝑦 2 + 6𝑦 = 0 touches

(a) y-axis at the origin (b) x-axis at the origin

(c) x-axis at the point (3, 0) (d) The line 𝑦 + 3 = 0
12. The circle represented by the equation 𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 will be a point circle, if
(a) 𝑔2 + 𝑓 2 = 𝑐 (b) 𝑔2 + 𝑓 2 > 𝑐
(c) 𝑔 + 𝑓 + 𝑐 = 0
2 2 (d) None of these
13. The equation of the circle having centre (1, −2) and passing through the point of intersection of lines 3𝑥 + 𝑦 = 14, 2𝑥 + 5𝑦 =
18 is
(a) 𝑥 2 + 𝑦 2 − 2𝑥 + 4𝑦 − 20 = 0
(b) 𝑥 2 + 𝑦 2 − 2𝑥 − 4𝑦 − 20 = 0

(d) 𝑥 2 + 𝑦 2 + 2𝑥 + 4𝑦 − 20 = 0

14. For the circle 𝑥 2 + 𝑦 2 + 6𝑥 − 8𝑦 + 9 = 0, which of the following statements is true

(a) Circle passes through the point (−3, 4)
(b) Circle touches x-axis
(c) Circle touches y-axis
(d) None of these
15. Equation of the circle which touches the lines 𝑥 = 0, 𝑦 = 0 and 3𝑥 + 4𝑦 = 4 is
(a) 𝑥 2 − 4𝑥 + 𝑦 2 + 4𝑦 + 4 = 0
(b) 𝑥 2 − 4𝑥 + 𝑦 2 − 4𝑦 + 4 = 0
(c) 𝑥 2 + 4𝑥 + 𝑦 2 + 4𝑦 + 4 = 0
(d) 𝑥 2 + 4𝑥 + 𝑦 2 − 4𝑦 + 4 = 0
16. For the line 3𝑥 + 2𝑦 = 12 and the circle 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 + 3 = 0, which of the following statements is true
(a) Line is a tangent to the circle
(b) Line is a chord of the circle
(c) Line is a diameter of the circle
(d) None of these
17. The locus of the centre of the circle which cuts a chord of length 2a from the positive x-axis and passes through a point on
positive y-axis distant b from the origin is
(a) 𝑥 2 + 2𝑏𝑦 = 𝑏 2 + 𝑎2 (b) 𝑥 2 − 2𝑏𝑦 = 𝑏 2 + 𝑎2
(c) 𝑥 2 + 2𝑏𝑦 = 𝑎2 − 𝑏 2 (d) 𝑥 2 − 2𝑏𝑦 = 𝑏 2 − 𝑎2
18. The equation of circle passing through (4, 5) and having the centre at (2, 2), is
(a) 𝑥 2 + 𝑦 2 + 4𝑥 + 4𝑦 − 5 = 0
(b) 𝑥 2 + 𝑦 2 − 4𝑥 − 4𝑦 − 5 = 0
(c) 𝑥 2 + 𝑦 2 − 4𝑥 = 13
(d) 𝑥 2 + 𝑦 2 − 4𝑥 − 4𝑦 + 5 = 0
19. A circle touches x-axis and cuts off a chord of length 2l from y-axis. The locus of the centre of the circle is
(a) A straight line (b) A circle
(c) An ellipse (d) A hyperbola
20. Radius of circle (𝑥 − 5)(𝑥 − 1) + (𝑦 − 7)(𝑦 − 4) = 0 is
(a) 3 (b) 4
(c) 5/2 (d) 7/2

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C B A D A B C C C D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B A A B B C C B D C

DPP No. 1
DPP Year : 2023
Topic : Definite Integration

1
1. ∫0 𝑒 2In𝑥 𝑑𝑥 =
1
(a) 0 (b) 2
1 1
(c) 3 (d) 4

𝜋/4
2. ∫0 𝑡𝑎𝑛2 𝑥 𝑑𝑥 =
𝜋 𝜋
(a) 1 − (b) 1 +
4 4
𝜋 𝜋
(c) 4
−1 (d) 4

𝜋/2 𝑥+𝑠𝑖𝑛 𝑥
3. ∫0 1+𝑐𝑜𝑠 𝑥 𝑑𝑥 =
(a) − 𝑙𝑜𝑔 2 (b) 𝑙𝑜𝑔 2
𝜋
(c) (d) 0
2

𝜋/2
4. ∫0 𝑒 𝑥 𝑠𝑖𝑛 𝑥 𝑑𝑥 =
1 1
(a) (𝑒 𝜋/2 − 1) (b) (𝑒 𝜋/2 + 1)
2 2
1
(c) (1 − 𝑒 𝜋/2 ) (d) 2(𝑒 𝜋/2 + 1)
2

2 1 1
5. ∫1 𝑒 𝑥 (𝑥 − 𝑥 2 ) 𝑑𝑥 =
𝑒2 𝑒2
(a) +𝑒 (b) 𝑒 −
2 2
𝑒2
(c) 2
−𝑒 (d) None of these

𝜋/2 𝑐𝑜𝑠 𝑥
6. ∫0 (1+𝑠𝑖𝑛 𝑥)(2+𝑠𝑖𝑛 𝑥)
𝑑𝑥 =
4 1
(a) 𝑙𝑜𝑔 3 (b) 𝑙𝑜𝑔 3
3
(c) 𝑙𝑜𝑔 (d) None of these
4

𝜋/2 √1+𝑐𝑜𝑠 𝑥
7. ∫𝜋/3 5 𝑑𝑥 =
(1−𝑐𝑜𝑠 𝑥)2
5 3
(a) 2 (b) 2
1 2
(c) 2 (d) 5

2 1 −1
8. ∫1𝑥2
𝑒 𝑥 𝑑𝑥 =
(a) √𝑒 + 1 (b) √𝑒 − 1

√𝑒+1 √𝑒−1
(c) 𝑒
(d) 𝑒

1 2𝑥
9. ∫0 𝑠𝑖𝑛−1 (1+𝑥 2 ) 𝑑𝑥 =
𝜋 𝜋
(a) 2 − 2 𝑙𝑜𝑔 √2 (b) 2 + 2 𝑙𝑜𝑔 √2
𝜋 𝜋
(c) − 𝑙𝑜𝑔 √2 (d) + 𝑙𝑜𝑔 √2
4 4

𝑒 𝑒𝑥
10. ∫1 𝑥
(1 + 𝑥 𝑙𝑜𝑔 𝑥)𝑑𝑥 =
𝑒
(a) 𝑒 (b) 𝑒 𝑒 − 𝑒
(c) 𝑒 𝑒 + 𝑒 (d) None of these

2
11. The value of ∫−2(𝑎𝑥 3 + 𝑏𝑥 + 𝑐) depends on
(a) The value of 𝑎 (b) The value of 𝑏
(c) The value of 𝑐 (d) The values of 𝑎 and 𝑏

𝜋/4
12. ∫𝜋/6 cosec2𝑥𝑑𝑥 =
(a) 𝑙𝑜𝑔 3 (b) 𝑙𝑜𝑔 √3
(c) 𝑙𝑜𝑔 9 (d) None of these

𝜋/2
13. ∫0 √𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛3 𝜃 𝑑𝜃 =
20 8
(a) (b)
21 21
−20 −8
(c) 21
(d) 21

𝜋/4
14. ∫0 𝑠𝑒𝑐 7 𝜃 𝑠𝑖𝑛3 𝜃 𝑑𝜃 =
1 3
(a) 12 (b) 12
5
(c) 12 (d) None of these

𝑏 𝑙𝑜𝑔 𝑥
15. ∫𝑎 𝑑𝑥 =
𝑥
𝑙𝑜𝑔 𝑏 𝑏
(a) 𝑙𝑜𝑔 ( ) (b) 𝑙𝑜𝑔( 𝑎𝑏) 𝑙𝑜𝑔 ( )
𝑙𝑜𝑔 𝑎 𝑎
1 𝑏 1 𝑎
(c) 2 𝑙𝑜𝑔( 𝑎𝑏) 𝑙𝑜𝑔 (𝑎) (d) 2 𝑙𝑜𝑔( 𝑎𝑏) 𝑙𝑜𝑔 (𝑏 )

1
16. ∫0 𝑡𝑎𝑛−1 𝑥 𝑑𝑥 =
𝜋 1 1
(a) 4 − 2 𝑙𝑜𝑔 2 (b) 𝜋 − 2 𝑙𝑜𝑔 2
𝜋
(c) 4
− 𝑙𝑜𝑔 2 (d) 𝜋 − 𝑙𝑜𝑔 2

1 𝑑𝑥
17. ∫0 [𝑎𝑥+𝑏(1−𝑥)]2
=

𝑎 𝑏
(a) 𝑏 (b) 𝑎
1
(c) 𝑎𝑏 (d) 𝑎𝑏

𝑘 𝑑𝑥 𝜋
18. If ∫0 2+8𝑥 2
= 16, then 𝑘 =
1
(a) 1 (b) 2
1
(c) 4 (d) None of these

𝜋/2
19. ∫𝜋/4 𝑐𝑜𝑠 𝜃 cosec 2 𝜃𝑑𝜃 =
(a) √2 − 1 (b) 1 − √2
(c) √2 + 1 (d) None of these

1/√2 𝑠𝑖𝑛−1 𝑥
20. ∫0 (1−𝑥 2 )3/2
𝑑𝑥 =
𝜋 1 𝜋 1
(a) + 𝑙𝑜𝑔 2 (b) − 𝑙𝑜𝑔 2
4 2 4 2
𝜋 𝜋
(c) 2 + 𝑙𝑜𝑔 2 (d) 2 − 𝑙𝑜𝑔 2

𝜋/2
21. The correct evaluation of ∫0 𝑠𝑖𝑛 𝑥 𝑠𝑖𝑛 2 𝑥 is
4 1
(a) 3
(b) 3
3 2
(c) 4
(d) 3

𝜋/2 𝑑𝑥
22. ∫0 2+𝑐𝑜𝑠 𝑥
=
1 −1 1
(a) 𝑡𝑎𝑛 ( 3) (b) √3 𝑡𝑎𝑛−1 (√3)
√3 √
2 −1 1
(c) 𝑡𝑎𝑛 ( 3) (d) 2√3 𝑡𝑎𝑛−1(√3)
√3 √

1 𝑡𝑎𝑛−1 𝑥
23. ∫0 1+𝑥 2
𝑑𝑥 =
𝜋2 𝜋2
(a) 8
(b) 16
𝜋2 𝜋2
(c) 4
(d) 32

2 𝑐𝑜𝑠( 𝑙𝑜𝑔 𝑥)
24. ∫1 𝑥
𝑑𝑥 =
(a) 𝑠𝑖𝑛( 𝑙𝑜𝑔 3) (b) 𝑠𝑖𝑛( 𝑙𝑜𝑔 2)
(c) 𝑐𝑜𝑠( 𝑙𝑜𝑔 3) (d) None of these

𝑎 𝑥𝑑𝑥
25. ∫0 =
√𝑎 2 +𝑥 2
(a) 𝑎(√2 − 1) (b) 𝑎(1 − √2)
(c) 𝑎(1 + √2) (d) 2𝑎√3

2/𝜋 𝑠𝑖𝑛(1/𝑥)
26. The value of integral ∫1/𝜋 𝑑𝑥 =
𝑥2
(a) 2 (b) −1
(c) 0 (d) 1

𝑎𝑥 4 𝑑𝑥
27. ∫0 (𝑎 2 +𝑥 2 )4
=
1 𝜋 1 1 𝜋 1
(a) 16𝑎3 ( 4 − 3) (b) 16𝑎3 ( 4 + 3)
1 𝜋 1 1 𝜋 1
(c) 16 𝑎3 ( 4 − 3) (d) 16 𝑎3 ( 4 + 3)

2𝜋 𝑥 𝜋
28. ∫0 𝑒 𝑥/2 . 𝑠𝑖𝑛 (2 + 4 ) 𝑑𝑥 =
(a) 1 (b) 2√2
(c) 0 (d) None of these

1 𝑒 −𝑥
29. ∫0 𝑑𝑥 =
1+𝑒 −𝑥
1+𝑒 1 1+𝑒 1
(a) 𝑙𝑜𝑔 ( 𝑒 ) − 𝑒
+1 (b) 𝑙𝑜𝑔 ( 2𝑒 ) − 𝑒 + 1
1+𝑒 1
(c) 𝑙𝑜𝑔 ( 2𝑒 ) + 𝑒
−1 (d) None of these

𝜋/4 𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 𝑥

30. ∫0 𝑑𝑥 =
9+16 𝑠𝑖𝑛 2𝑥
1
(a) 𝑙𝑜𝑔 3 (b) 𝑙𝑜𝑔 3
20
1
(c) 20
𝑙𝑜𝑔 5 (d) None of these

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C A C B C A B D A A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C D B C C A D B A B

Ques. 21 22 23 24 25 26 27 28 29 30

Ans. D C D B A D A C B A

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DPP No. 3
DPP Year : 2023
Topic : Circle

1. The equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the
straight line 𝑦 − 4𝑥 + 3 = 0, is
(a) 𝑥 2 + 𝑦 2 + 4𝑥 − 10𝑦 + 25 = 0
(b) 𝑥 2 + 𝑦 2 − 4𝑥 − 10𝑦 + 25 = 0
(c) 𝑥 2 + 𝑦 2 − 4𝑥 − 10𝑦 + 16 = 0
(d) 𝑥 2 + 𝑦 2 − 14𝑦 + 8 = 0
2. The equation of the circle with centre at (1, –2) and passing through the centre of the given circle 𝑥 2 +
𝑦 2 + 2𝑦 − 3 = 0, is
(a) 𝑥 2 + 𝑦 2 − 2𝑥 + 4𝑦 + 3 = 0
(b) 𝑥 2 + 𝑦 2 − 2𝑥 + 4𝑦 − 3 = 0
(c) 𝑥 2 + 𝑦 2 + 2𝑥 − 4𝑦 − 3 = 0
(d) 𝑥 2 + 𝑦 2 + 2𝑥 − 4𝑦 + 3 = 0
3. The equation of the circle concentric with the circle 𝑥 2 + 𝑦 2 + 8𝑥 + 10𝑦 − 7 = 0 and passing through
the centre of the circle 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 = 0 is
(a) 𝑥 2 + 𝑦 2 + 8𝑥 + 10𝑦 + 59 = 0
(b) 𝑥 2 + 𝑦 2 + 8𝑥 + 10𝑦 − 59 = 0
(c) 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 + 87 = 0
(d) 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 − 87 = 0
4. The equation of the circle passing through the points (0, 0), (0, b) and (a, b) is
(a) 𝑥 2 + 𝑦 2 + 𝑎𝑥 + 𝑏𝑦 = 0 (b) 𝑥 2 + 𝑦 2 − 𝑎𝑥 + 𝑏𝑦 = 0
(c) 𝑥 2 + 𝑦 2 − 𝑎𝑥 − 𝑏𝑦 = 0 (d) 𝑥 2 + 𝑦 2 + 𝑎𝑥 − 𝑏𝑦 = 0
5. The equation 𝑎𝑥 2 + 𝑏𝑦 2 + 2ℎ𝑥𝑦 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 will represent a circle, if
(a) 𝑎 = 𝑏 = 0 and 𝑐 = 0 (b) 𝑓 = 𝑔 and ℎ = 0
(c) 𝑎 = 𝑏 ≠ 0 and ℎ = 0 (d) 𝑓 = 𝑔 and 𝑐 = 0
6. The equations of the circles touching both the axes and passing through the point (1, 2) are
(a) 𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 + 1 = 0, 𝑥 2 + 𝑦 2 − 10𝑥 − 10𝑦 + 25 = 0
(b) 𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 − 1 = 0, 𝑥 2 + 𝑦 2 − 10𝑥 − 10𝑦 − 25 = 0
(c) 𝑥 2 + 𝑦 2 + 2𝑥 + 2𝑦 + 1 = 0, 𝑥 2 + 𝑦 2 + 10𝑥 + 10𝑦 + 25 = 0
(d) None of these
7. Which of the following line is a diameter of the circle 𝑥 2 + 𝑦 2 − 6𝑥 − 8𝑦 − 9 = 0
(a) 3𝑥 − 4𝑦 = 0 (b) 4𝑥 − 3𝑦 = 9
(c) 𝑥 + 𝑦 = 7 (d) 𝑥 − 𝑦 = 1
8. A circle is concentric with the circle 𝑥 2 + 𝑦 2 − 6𝑥 + 12𝑦 + 15 = 0 and has area double of its area. The
equation of the circle is
(a) 𝑥 2 + 𝑦 2 − 6𝑥 + 12𝑦 − 15 = 0
(b) 𝑥 2 + 𝑦 2 − 6𝑥 + 12𝑦 + 15 = 0
(c) 𝑥 2 + 𝑦 2 − 6𝑥 + 12𝑦 + 45 = 0
(d) None of these
9. If the radius of the circle 𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 be r, then it will touch both the axes, if
(a) 𝑔 = 𝑓 = 𝑟 (b) 𝑔 = 𝑓 = 𝑐 = 𝑟

(c) 𝑔 = 𝑓 = √𝑐 = 𝑟 (d) 𝑔 = 𝑓 and 𝑐 2 = 𝑟

10. The equation of the circle with centre on the x-axis, radius 4 and passing through the origin, is
(a) 𝑥 2 + 𝑦 2 + 4𝑥 = 0 (b) 𝑥 2 + 𝑦 2 − 8𝑦 = 0
(c) 𝑥 2 + 𝑦 2 ± 8𝑥 = 0 (d) 𝑥 2 + 𝑦 2 + 8𝑦 = 0
11. The equation of the circle passing through the point (2, 1) and touching y-axis at the origin is
(a) 𝑥 2 + 𝑦 2 − 5𝑥 = 0 (b) 2𝑥 2 + 2𝑦 2 − 5𝑥 = 0
(c) 𝑥 2 + 𝑦 2 + 5𝑥 = 0 (d) None of these
12. The equation of the circle which passes through the origin and cuts off intercepts of 2 units length from
negative coordinate axes, is
(a) 𝑥 2 + 𝑦 2 − 2𝑥 + 2𝑦 = 0
(b) 𝑥 2 + 𝑦 2 + 2𝑥 − 2𝑦 = 0
(c) 𝑥 2 + 𝑦 2 + 2𝑥 + 2𝑦 = 0
(d) 𝑥 2 + 𝑦 2 − 2𝑥 − 2𝑦 = 0
13. For the circle 𝑥 2 + 𝑦 2 + 3𝑥 + 3𝑦 = 0, which of the following relations is true
(a) Centre lies on x-axis
(b) Centre lies on y-axis
(c) Centre is at origin
(d) Circle passes through origin
14. The equation of the circle with centre on x-axis, radius 5 and passing through the point (2, 3), is
(a) 𝑥 2 + 𝑦 2 + 4𝑥 − 21 = 0 (b)𝑥 2 + 𝑦 2 + 4𝑥 + 21 = 0
(c) 𝑥 2 + 𝑦 2 − 4𝑥 − 21 = 0 (d)𝑥 2 + 𝑦 2 + 5𝑥 − 21 = 0
15. The equation of the circle which touches x-axis at (3, 0) and passes through (1, 4) is given by
(a) 𝑥 2 + 𝑦 2 − 6𝑥 − 5𝑦 + 9 = 0
(b)𝑥 2 + 𝑦 2 + 6𝑥 + 5𝑦 − 9 = 0
(c) 𝑥 2 + 𝑦 2 − 6𝑥 + 5𝑦 − 9 = 0
(d) 𝑥 2 + 𝑦 2 + 6𝑥 − 5𝑦 + 9 = 0
16. If the lines 𝑥 + 𝑦 = 6 and 𝑥 + 2𝑦 = 4 be diameters of the circle whose diameter is 20, then the
equation of the circle is
(a) 𝑥 2 + 𝑦 2 − 16𝑥 + 4𝑦 − 32 = 0 (b) 𝑥 2 + 𝑦 2 + 16𝑥 + 4𝑦 − 32 = 0
(c) 𝑥 2 + 𝑦 2 + 16𝑥 + 4𝑦 + 32 = 0 (d)𝑥 2 + 𝑦 2 + 16𝑥 − 4𝑦 + 32 = 0
17. The number of circles touching the lines 𝑥 = 0, 𝑦 = 𝑎 and 𝑦 = 𝑏 is
(a) One (b) Two
(c) Four (d) Infinite
18. The equation of the circle whose diameters have the end points (a, 0) (0, b) is given by
(a) 𝑥 2 + 𝑦 2 − 𝑎𝑥 − 𝑏𝑦 = 0 (b)𝑥 2 + 𝑦 2 + 𝑎𝑥 − 𝑏𝑦 = 0
(c) 𝑥 2 + 𝑦 2 − 𝑎𝑥 + 𝑏𝑦 = 0 (d)𝑥 2 + 𝑦 2 + 𝑎𝑥 + 𝑏𝑦 = 0
19. The centre and radius of the circle 2𝑥 2 + 2𝑦 2 − 𝑥 = 0 are
1 1 1 1
(a) (4 , 0) and 4 (b) (− 2 , 0) and 2
1 1 1 1
(c) (2 , 0) and 2 (d) (0, − 4) and 4
20. Centre of the circle (𝑥 − 3)2 + (𝑦 − 4)2 = 5 is
(a) (3, 4) (b) (−3, −4)
(c) (4, 3) (d) (−4, −3)

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B A B C C A C A C C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B C D A A A B A A A

DPP No. 2
DPP Year : 2023
Topic : Integral
𝜋
1. The value of the integral ∫0 log(1 + cos 𝑥) 𝑑𝑥 is
𝜋
a) 2 log 2 b) −𝜋 log 2 c) 𝜋 log 2 d) None of these

−1 𝑥 1+𝑥+𝑥 2
2. ∫ 𝑒 tan (
1+𝑥 2
) 𝑑𝑥 is equal to
−1 −1 𝑥 1 −1 𝑥
a) 𝑥 𝑒 tan 𝑥
+𝐶 b) 𝑥 2 𝑒 tan +𝐶 c) 𝑒 tan +𝐶 d) None of these
𝑥

𝜋−𝑎 𝜋−𝑎
3. Let 𝐼1 = ∫𝑎 𝑥𝑓(sin 𝑥) 𝑑𝑥, 𝐼2 = ∫𝑎 𝑓 (sin 𝑥) 𝑑𝑥, then 𝐼2 is equal to
π 2
a) 𝐼1 b) π𝐼1 c) 𝐼1 d) 2𝐼1
2 π

1
4. ∫ cos−1 (𝑥) 𝑑𝑥equals
a) 𝑥 sec −1 𝑥 + cosh−1 𝑥 + 𝑐 b) 𝑥 sec −1 𝑥 − cosh−1 𝑥 + 𝑐
c) 𝑥 sec −1 𝑥 − sin−1 𝑥 + 𝑐 d) None of these

1 𝑑𝑥
5. The value of ∫0 is
𝑥+√1−𝑥 2
π π 1 π
a) b) c) d)
3 2 2 4

√ 𝑛
6. For any natural number 𝑛, the value of the integral ∫0 [𝑥 2 ]𝑑𝑥, is
a) 𝑛√𝑛 + ∑𝑛𝑟=1 √𝑟 b) 𝑛√𝑛 − ∑𝑛𝑟=1 √𝑟 c) ∑𝑛𝑟=1 √𝑟 − 𝑛√𝑛 d) None of these

1
7. ∫ 𝑥 (log 𝑒𝑥 𝑒)𝑑𝑥 is equal to
a) log 𝑒 (1 − log 𝑒 𝑥) + 𝑐 b) log 𝑒 (log 𝑒 𝑒𝑥 − 1) + 𝑐
c) log 𝑒 (log 𝑒 𝑥 − 1) + 𝑐 d) log 𝑒 (1 + log 𝑒 𝑥) + 𝑐

8. Let 𝑓 be integrable over [0, 𝑎] for any real 𝑎.If we define

𝜋/2
𝐼1 = ∫ cos θ𝑓(sin θ + cos 2 θ) 𝑑θ
0
𝜋/2
And 𝐼2 = ∫0 sin 2θ𝑓(sin θ + cos2 θ) 𝑑θ, then
a) 𝐼1 = 𝐼2 b) 𝐼1 = −𝐼2 c) 𝐼1 = 2𝐼2 d) 𝐼1 = −2𝐼2

9. Consider the following statements:

𝜋/2 3
1. ∫−𝜋/2 √cos 𝑥 − cos 3 𝑥 𝑑𝑥 = 4
4
2. ∫0 (|𝑥 − 1| + |𝑥 − 3|)𝑑𝑥 = 10

Which of these is/are correct?

a) Only (1) b) Only (2) c) Both of these d) None of these

𝜋⁄4
10. The value of the integral ∫−𝜋⁄4 sin−4 𝑥 𝑑𝑥 is
8 3 8
a) − b) c) d) None of these
3 2 3

𝑥 2
(∫0 𝑒 𝑥 𝑑𝑥 )
11. The value of lim 𝑥 3 , is
𝑥→∞ ∫0 𝑒 2𝑥 𝑑𝑥
a) 1 b) 2 c) 3 d) 0

1 1
12. If ∫sin 𝑥 𝑡 2 𝑓(𝑡)𝑑𝑡 = 1 − sin 𝑥 , ∀ 𝑥 ∈ [0, 𝜋⁄2], then 𝑓 ( ) is
√3
1
a) 3 b) √3 c) 3 d) None of these

2𝑎
13. If 𝑎 is fixed real number such that 𝑓(𝑎 − 𝑥) + 𝑓(𝑎 + 𝑥) = 0, then ∫0 𝑓(𝑥) 𝑑𝑥 =
𝑎 𝑎
a) b) 0 c) − d) 2𝑎
2 2

5−𝑥
15. Let ∫ √2+𝑥 𝑑𝑥 equal

𝑥+2
a) √𝑥 + 2√5 − 𝑥 + 3 sin−1 √ 3
+ 𝐶

𝑥+2
b) √𝑥 + 2√5 − 𝑥 + 7 sin−1 √ + 𝐶
7

𝑥+2
c) √𝑥 + 2√5 − 𝑥 + 5 sin−1 √ + 𝐶
5
d) None of these

2𝑥 2 +3 𝑥+1 𝑥
16. ∫ (𝑥 2 −1)(𝑥2 +4) 𝑑𝑥 = 𝑎 log (𝑥−1) + 𝑏 tan−1 2, then (𝑎, 𝑏) is
a) (−1/2, 1/2) b) (1/2, 1/2) c) (−1, 1) d) (1, −1)

(𝑥−1)
17. ∫ 𝑒 𝑥 𝑑𝑥 is equal to
𝑥2
𝑒𝑥 −𝑒 𝑥 𝑒𝑥 −𝑒 𝑥
a) +𝑐 b) +𝑐 c) +𝑐 d) +𝑐
𝑥2 𝑥2 𝑥 𝑥

3𝛼
18. The value of the integral ∫0 cosec(𝑥 − 𝛼)cosec(𝑥 − 2𝛼)𝑑𝑥, is
1 1
a) 2 sec 𝛼 log (2 cosec 𝛼) b) 2 sec 𝛼 log (2 sec 𝛼)
1
c) 2 cosec 𝛼 log(sec 𝛼) d) 2 cosec 𝛼 log ( sec 𝛼)
2

3 3𝑥+1
19. ∫0 𝑑𝑥 is equal to
𝑥 2 +9
𝜋 𝜋 𝜋 𝜋
a) log(2√2) + b) log(2√2) + c) log(2√2) + d) log(2√2) +
12 2 6 3

𝑑𝑥
20. ∫ is equal to
√(1−𝑥)(𝑥−2)
−1 (2𝑥
a) sin − 3) + 𝑐 b) sin−1 (2𝑥 + 5) + 𝑐
−1 (3
c) sin − 2𝑥) + 𝑐 d) sin−1 (5 − 2𝑥) + 𝑐

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B A C B D B D A B A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. D A B A B A C D A A

DPP No. 3
DPP Year : 2023
Topic : Integral

1. If 𝑓(𝑥) = lim [2𝑥 + 4𝑥 3 +. . . +2𝑛𝑥 2𝑛−1 ](0 < 𝑥 < 1), then ∫ 𝑓(𝑥)𝑑𝑥 is equal to
𝑛→∞
1 1 1
a) −√1 − 𝑥 2 b) c) 𝑥 2 −1 d) 1−𝑥 2
√1−𝑥 2

𝑑𝑥
2. ∫ sin 𝑥−cos 𝑥+√2equals
1 𝑥 𝜋 1 𝑥 𝜋 1 𝑥 𝜋 1 𝑥 𝜋
a) − tan (2 + 8 ) + 𝑐 b) tan (2 + 8) + 𝑐 c) cot (2 + 8) + 𝑐 d) − cot (2 + 8 ) + 𝑐
√2 √2 √2 √2

𝜋⁄2 cos 𝑥
3. ∫0 𝑑𝑥 is equal to
1+sin 𝑥
1
a) log 2 b) 2 log 2 c) (log 2)2 d) log 2
2

−1 𝑥
1 2 sin
4. The integral ∫0 2
𝑑𝑥 equals
𝑥
𝜋/6 𝑥 𝜋/6 2𝑥 𝜋/2 2𝑥 𝜋/6 𝑥
a) ∫0 𝑑𝑥 b) ∫0 𝑑𝑥 c) ∫0 𝑑𝑥 d) ∫0 𝑑𝑥
tan 𝑥 tan 𝑥 tan 𝑥 sin 𝑥

𝑒 1 1 𝑏
5. If ∫2 ( − (log ) 𝑑𝑥 = 𝑎 + , then
log 𝑥 𝑥)2 log 2
a) 𝑎 = 𝑒, 𝑏 = −2 b) 𝑎 = 𝑒, 𝑏 = 2 c) 𝑎 = −𝑒, 𝑏 = 2 d) None of these

8
6. The value of ∫0 |𝑥 − 5| 𝑑𝑥 is
a) 17 b) 12 c) 9 d) 18

1 𝑥 𝑑𝑥
7. ∫0 is equal to
[𝑥+√1−𝑥 2 ]√1−𝑥2
𝜋 𝜋2
a) 0 b) 1 c) d)
4 2

1
8. ∫0 cot −1 (1 − 𝑥 + 𝑥 2 ) 𝑑𝑥 is equal to
𝜋 𝜋
a) 𝜋 − log 2 b) 𝜋 + log 2 c) + log 2 d) − log 2
2 2

15 𝑑𝑥
9. ∫8 (𝑥−3)√𝑥+1
is equal to
1 5 1 5 1 3 1 3
a) 2 log 3 b) 3 log 3 c) 2 log 5 d) 5 log 5

10. ∫{1 + 2 tan 𝑥 (tan 𝑥 + sec 𝑥)}1/2 𝑑𝑥 is equal to

a) log sec 𝑥 (sec 𝑥 − tan 𝑥) + 𝐶

b) log cosec(sec 𝑥 + tan 𝑥) + 𝐶
c) log sec 𝑥 (sec 𝑥 + tan 𝑥 + 𝐶)
d) log(sec 𝑥 + tan 𝑥) + 𝐶

∞ 1 ∞ 𝑥2 𝐼
11. If 𝐼1 = ∫0 1+𝑥 4
𝑑𝑥and 𝐼2 = ∫0 1+𝑥 4
𝑑𝑥. Then 𝐼1 =
2
a) 1 b) 2 c) 1/2 d) None of these

sin 𝑥 𝑑𝑥
12. ∫ 3+4 cos2 𝑥 is equal to
1 cos 𝑥
a) log(3 + 4 cos2 𝑥) + 𝑐 b) 2 tan−1 ( 3 ) + 𝑐
√3 √
1 2cos 𝑥 1 −1 2cos 𝑥
c) − 2 tan−1 ( 3 ) +𝑐 d) 2 3 tan ( 3 ) + 𝑐
√3 √ √ √

13. For any integer 𝑛, the integral

𝜋 2𝑥
∫0 𝑒 cos cos3 (2𝑛 + 1)𝑥 𝑑𝑥 has the value
a) 𝜋 b) 1 c) 0 d) None of these

𝑑 1 𝑑
14. If {𝑓(𝑥) = 1+𝑥 2, then 𝑑𝑥 {𝑓(𝑥 3 )} is
𝑑𝑥
3𝑥 3𝑥 2 −6𝑥 5 −6𝑥 5
a) b) c) d)
1+𝑥 3 1+𝑥 6 (1+𝑥 6 )2 1+𝑥 6

𝜋
15. ∫0 [cot 𝑥] 𝑑𝑥, [. ]denotes the greatest integer function, is equal to
𝜋 𝜋
a) b) 1 c) −1 d) −
2 2

2
16. ∫−3{|𝑥 + 1| + |𝑥 + 2| + |𝑥 − 1|} 𝑑𝑥 is equal to
31 35 47 39
a) b) c) d)
2 2 2 2

3
17. ∫0 |𝑥 3 + 𝑥 2 + 3𝑥|𝑑𝑥 is equal to
171 171 170 170
a) b) c) d)
2 4 4 3

2𝑛𝜋 1
19. ∫0 {| sin 𝑥| − | sin 𝑥|} 𝑑𝑥equals
2
a) 𝑛b) 2𝑛c) −2𝑛d) None of these

𝑏 𝑏 2
20. If ∫𝑎 𝑥 3 𝑑𝑥 = 0 and if ∫𝑎 𝑥 2 𝑑𝑥 = , then the values of 𝑎 and 𝑏 are respectively
3
a) 1,1 b) −1, −1 c) 1, −1d) −1,1

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. D C A B A A C D A C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A C C B D C B B B D

DPP No. 1 [Integral]

DPP Year : 2023
Topic : Continuity & Differentiability
𝑥
1. Let 𝑓(𝑥) = ∫0 |𝑥 − 2|𝑑𝑥 , 𝑥 ≥ 0. Then, 𝑓′(𝑥) is
a) Continuous and non differentiable at 𝑥 = 2
b) Discontinuous at 𝑥 = 4
c) Neither continuous nor differentiable at 𝑥 = 2
d) Non-differentiable at 𝑥 = 4

𝑥
2. If 𝑓(𝑡) is an odd function, then ∫0 𝑓(𝑡)𝑑𝑡 is
a) An odd function b) An even function
c) Neither even nor odd d) 0

π θ sin θ
4. ∫0 dθis equal to
1+cos2 θ
𝜋2 𝜋3 𝜋2
a) b) c) 𝜋 2 d)
2 3 4

sin2 𝜃 cos2 𝜃
5. The value of [∫0 sin−1 √ϕ 𝑑ϕ + ∫0 cos −1 √ϕ 𝑑ϕ] is equal to
a) 𝜋 b) 𝜋⁄2 c) 𝜋⁄3 d) 𝜋⁄4

𝜋/2 𝑑𝑥
6. ∫0 is equal to
1+tan3 𝑥
𝜋 𝜋 3𝜋
a) 𝜋 b) c) d)
2 4 2

1 sin 𝑥 1 cos 𝑥
7. Let 𝐼 = ∫0 𝑑𝑥 and 𝐽 = ∫0 𝑑𝑥 . Then, which one of the following is true?
√𝑥 √𝑥
2 2 2 2
a) 𝐼 > and 𝐽 < 2 b) 𝐼 > and 𝐽 > 2 c) 𝐼 < and 𝐽 < 2 d) 𝐼 < and 𝐽 > 2
3 3 3 3

|𝑥|, −1 ≤ 𝑥 ≤ 1 3
8. If 𝑓(𝑥) = { then ∫−1 𝑓(𝑥)𝑑𝑥 is equal to
|𝑥 − 2|, 1 < 𝑥 ≤ 3′
a) 0 b) 1 c) 2 d) 4

1
9. ∫ 𝑥 (𝑥𝑛 +1) 𝑑𝑥 is equal to

1 𝑥𝑛 1 𝑥 𝑛 +1 𝑥𝑛
a) log ( )+𝐶 b) log ( ) c) log ( )+ 𝐶 d) None of these
𝑛 𝑥 𝑛 +1 𝑛 𝑥𝑛 𝑥 𝑛 +1

1/2
10. ∫0 |sin 𝜋 𝑥| 𝑑𝑥 is equal to
a) 0 b) 𝜋 c) −𝜋 d) 1/𝜋

𝜋 1
11. The value of the integral ∫0 𝑎 2 −2𝑎 cos 𝑥+1
𝑑𝑥(𝑎 > 1), is
𝜋 𝜋 2𝜋 2𝜋
a) 1−𝑎2 b) 𝑎2 −1 c) 𝑎2 −1 d) 1−𝑎2

1
12. If 𝐼 = ∫0 √1 + 𝑥 3 𝑑𝑥 then
√5 √7
a) 𝐼 > 2 b) 𝐼 ≠ 2
c) 𝐼 > 2
d) None of these

1 𝑏𝑐 𝑥
13. Assuming that 𝑓 is everywhere continuous, 𝑐 ∫𝑎𝑐 𝑓 (𝑐 ) 𝑑𝑥 is equal to
1 𝑏 𝑏 𝑏 𝑏𝑐 2
a) ∫𝑎 𝑓(𝑥) 𝑑𝑥 b) ∫𝑎 𝑓(𝑥) 𝑑𝑥 c) 𝑐 ∫𝑎 𝑓(𝑥) 𝑑𝑥 d) ∫𝑎𝑐 2 𝑓(𝑥) 𝑑𝑥
𝑐

1−𝑥 2
14. The value of the integral ∫ 𝑒 𝑥 (1+𝑥) 𝑑𝑥 is
1−𝑥 1+𝑥 𝑒𝑥
a) 𝑒 𝑥 (1+𝑥 2 ) + 𝑐 b) 𝑒 𝑥 (1+𝑥 2 ) + 𝑐 c) 1+𝑥 2 + 𝑐 d) 𝑒 𝑥 (1 − 𝑥) + 𝑐

𝑑 𝑒 sin 𝑥 43 3
15. Let 𝑑𝑥 (𝐹(𝑥)) = 𝑥
,𝑥 > 0. If ∫1 𝑥 𝑒 sin 𝑥 𝑑𝑥 = 𝐹(𝑘) − 𝑓(1), then one of the possible values of 𝑘, is
a) 64 b) 15 c) 16 d) 63

𝑎
16. The values of ′𝑎′ for which ∫0 (3𝑥 2 + 4𝑥 − 5) 𝑑𝑥 < 𝑎3 − 2 are
1 1 1
a) 2 < 𝑎 < 2 b) 2 ≤ 𝑎 ≤ 2 c) 𝑎 ≤ 2 d) 𝑎 ≥ 2

log(𝑥+1)−log 𝑥
17. The value of the integral ∫ 𝑥(𝑥+1)
𝑑𝑥 is
1 1
a) 2 [log(𝑥 + 1)]2 + 2 (log 𝑥)2 + log(𝑥 + 1) log 𝑥 + 𝐶
b) −[{log(𝑥 + 1)}2 − (log 𝑥)2 ] + log(𝑥 + 1) ∙ log 𝑥 + 𝐶
1
c) 2 [log(1 + 1/𝑥)]2 + 𝐶
d) None of these

2π
18. The value of ∫0 [2 sin 𝑥] 𝑑𝑥, where [∙] represents the greatest integral functions, is
5π 5π
a) − 3
b) −π c) 3
d) −2π

1 𝑑 2𝑥
19. ∫0 [sin−1 (1+𝑥2 )] 𝑑𝑥 is equal to
𝑑𝑥

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π π
a) 0 b) π c) d)
2 4
20. Let 𝑓(𝑥) be a function satisfying𝑓 ′ (𝑥) = 𝑓(𝑥) with 𝑓(0) = 1 and g(𝑥) be a function that
1
satisfies𝑓(𝑥) + g(𝑥) = 𝑥 2 . Then, the value of the integral∫0 𝑓(𝑥) g(𝑥)𝑑𝑥, is
𝑒2 5 𝑒2 3 𝑒2 3 𝑒2 5
a) 𝑒 − −2 b) 𝑒 + −2 c) 𝑒 − −2 d) 𝑒 + +2
2 2 2 2

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. A B C D D C C C A D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. B C B C A A A A C C

DPP No. 4
DPP Year : 2023
Topic : Integral

𝜋/2
1. The value of ∫0 cosec (𝑥 − 𝜋/3) cosec(𝑥 − 𝜋/6)𝑑𝑥, is
a) 2 log 3 b) −2 log 3 c) log 3 d) None of these

√(𝑎 2 −𝑥 2 )
2. The primitive function of the function 𝑓(𝑥) = 𝑥4
is
3/2 3/2
√𝑎 2 −𝑥 2 (𝑎 2 −𝑥 2 ) (𝑎 2 −𝑥 2 )
a) 𝑐 + 3𝑎 2 𝑥 3
b) 𝑐 − 2𝑎2 𝑥 2
c) 𝑐 − 3𝑎2 𝑥 3
d) None of these

𝑥, for 𝑥 < 1 2
3. If 𝑓(𝑥) = { , then ∫0 𝑥 2 𝑓(𝑥)𝑑𝑥 is equal to
𝑥 − 1, for 𝑥 ≥ 1
4 5 5
a) 1 b) c) d)
3 3 2
𝑥 2 +𝑥−6
4. ∫ (𝑥−2)(𝑥−1) 𝑑𝑥
a) 𝑥 + 2 log(𝑥 − 1) + 𝑐 b) 2𝑥 + 2 log(𝑥 − 1) + 𝑐
c) 𝑥 + 4 log(1 − 𝑥) + 𝑐 d) 𝑥 + 4 log(𝑥 − 1) + 𝑐

𝑥3
5. ∫ (1+𝑥 2 )1/3 𝑑𝑥 is equal to
20
a) (1 + 𝑥 2 )2/3 (2𝑥 2 − 3) + 𝐶
3
3
b) 20 (1 + 𝑥 2 )2/3 (2𝑥 2 − 3) + 𝐶
3
c) 20 (1 + 𝑥 2 )2/3 (2𝑥 2 + 3) + 𝐶
d) None of these

𝜋/4 𝑏 sin 𝑥
6. The equation ∫−𝜋/4 {𝑎|sin 𝑥| + 1+cos 𝑥 + 𝑐} 𝑑𝑥 = 0, where 𝑎, 𝑏, 𝑐 are constants, gives a relation between
a) 𝑎, 𝑏and c b) 𝑎 and 𝑐 c) 𝑎 and 𝑏 d) 𝑏 and 𝑐

4
7. The value of ∫2 {|𝑥 − 2| + |𝑥 − 3|}𝑑𝑥 is
a) 1 b) 2 c) 3 d) 5

𝑥
8. If g(𝑥) = ∫0 cos4 𝑡 𝑑𝑡, then g(𝑥 + 𝜋) equals
g(𝑥)
a) g(𝑥) + g(𝜋) b) g(𝑥) − g(𝜋) c) g(𝑥)g(𝜋) d) g(𝜋)

9. ∫ cos3 𝑥 𝑒 log(sin 𝑥) 𝑑𝑥 is equal to

sin4 𝑥 cos4 𝑥 𝑒 sin 𝑥
a) − 4
+𝐶 b) − 4
+𝐶 c) 4
+𝐶 d) None of these

15 𝑑𝑥
10. ∫8 is equal to
(𝑥−3)√𝑥+1
1 5 1 5 1 3 1 3
a) log b) log c) log d) log
2 3 3 3 5 5 2 5

𝑎𝑥 2 −𝑏
11. The value of ∫ 𝑑𝑥, is
𝑥 √𝑐 2 𝑥 2 −(𝑎𝑥 2 +𝑏)2
𝑏 𝑏 𝑏
𝑎𝑥+ 𝑎𝑥 2 + 2 𝑎𝑥+𝑏/𝑥 𝑎𝑥 2 + 2
a) sin−1 ( 𝑐
𝑥
)+𝑘 b) sin−1 ( 𝑐
𝑥
)+ 𝑘 c) cos −1 ( 𝑐
)+ 𝑘 d) cos −1 ( 𝑐
𝑥
)+𝑘

𝑥
12. If 𝑓(𝑥) = ∫−1|𝑡|𝑑𝑡, then for any 𝑥 ≥ 0, 𝑓(𝑥) equals
1 1 1
a) (1 − 𝑥 2 ) b) 𝑥 2 c) (1 + 𝑥 2 ) d) None of these
2 2 2

2 1 21
13. Let 𝐼1 = ∫1 𝑑𝑥and 𝐼2 = ∫1 𝑑𝑥. Then
√1+𝑥 2 𝑥
a) 𝐼1 > 𝐼2 b) 𝐼2 > 𝐼1 c) 𝐼1 = 𝐼2 d) 𝐼1 > 2𝐼2

𝜋
14. The value of ∫−𝜋(1 − 𝑥 2 ) sin 𝑥 cos 2 𝑥 𝑑𝑥 is
𝜋3 7
a) 0 b) 𝜋 − c) 2𝜋 − 𝜋 3 d) − 2𝜋 3
3 2

𝜋/2
15. If 𝐼𝑛 = ∫0 𝑥 𝑛 sin 𝑥 𝑑𝑥, then 𝐼4 + 12𝐼2 is equal to
𝜋 3 𝜋 2 𝜋 3
a) 4𝜋 b) 3 ( 2 ) c) ( 2 ) d) 4 ( 2 )

2
16. The value of the integral ∫0 𝑥[𝑥] 𝑑𝑥, is
7 3 5
a) 2 b) 2 c) 2 d) None of these

0 𝑑𝑥
17. ∫−1 𝑥 2 +2𝑥+2 is equal to
a) 0 b) π/4 c) π/2 d) −π/4

π/4
18. The value of ∫–π/4 𝑥 3 sin4 𝑥 𝑑𝑥 is
π π π
a) 4 b) 2 c) 8 d) 0
𝑘 𝑘
19. Let 𝑓 be apositive function. Let 𝐼1 = ∫1−𝑘 𝑥𝑓{𝑥(1 − 𝑥)}, 𝐼2 = ∫1−𝑘 𝑓{𝑥(1 − 𝑥)}
𝐼
𝑑𝑥 where2𝑘 − 1 > 0.Then, 𝐼1 is
2
1
a) 2 b) 𝑘 c) d) 1
2

𝜋/4
20. If 𝐼𝑛 = ∫0 tan𝑛 𝑥 𝑑𝑥, then lim 𝑛 (𝐼𝑛+1 + 𝐼𝑛−1 ) equals
𝑛→∞

a) 1 b) 2 c) 𝜋/4 d) 𝜋

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B C C D B B C A B A

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A C B A C B B D C A

DPP No. 5
DPP Year : 2023
Topic : Integral

𝑒1
1. ∫1 𝑥 𝑑𝑥 is equal to
a) ∞ b) 0 c) 1 d) log(1 + 𝑒)

199 +299 +⋯+𝑛99

2. lim 𝑛100
=
𝑛→∞
99 1 1 1
a) 100 b) 100 c) 99 d) 101

cos 2𝑥
3. ∫ cos 𝑥
𝑑𝑥 is equal to
a) 2 sin 𝑥 + log(sec 𝑥 − tan 𝑥) + 𝐶
b) 2 sin 𝑥 − log(sec 𝑥 − tan 𝑥) + 𝐶
c) 2 sin 𝑥 + log(sec 𝑥 + tan 𝑥) + 𝐶
d) None of these

4. ∫ 𝑓 ′ (𝑎𝑥 + 𝑏){𝑓(𝑎𝑥 + 𝑏)}𝑛 𝑑𝑥 is equal to

1
a) 𝑛+1 {𝑓(𝑎𝑥 + 𝑏)}𝑛+1 + 𝐶, for all 𝑛 except 𝑛 = −1
1
b) {𝑓(𝑎𝑥 + 𝑏)}𝑛+1 + 𝐶, for all 𝑛
𝑛+1
1
c) {𝑓(𝑎𝑥 + 𝑏)}𝑛+1 + 𝐶, for all 𝑛 except 𝑛 = −1
𝑎(𝑛+1)
1
d) 𝑎(𝑛+1) {𝑓(𝑎𝑥 + 𝑏)}𝑛+1 + 𝐶, for all 𝑛

1 1 2 2
5. If 𝐼1 = ∫0 2𝑥 2 𝑑𝑥, 𝐼2 = ∫0 2𝑥 3 𝑑𝑥, 𝐼3 = ∫1 2𝑥 2 𝑑𝑥 and𝐼4 = ∫1 2𝑥 3 𝑑𝑥, then
a) 𝐼3 > 𝐼4 b) 𝐼3 = 𝐼4 c) 𝐼1 > 𝐼2 d) 𝐼2 > 𝐼1

√tan 𝑥 𝑘𝜋
6. ∫ sin 𝑥 cos 𝑥 𝑑𝑥 =. . . +𝑐; 𝑥 ≠ and tan 𝑥 > 0
2
1
a) b) √2 tan 𝑥 c) 2√tan 𝑥 d) √tan 𝑥
2√tan 𝑥

1 𝑟
7. lim 𝑛 ∑2𝑛
𝑟=1 √𝑛2 equals
𝑛→∞ +𝑟 2
a) 1 + √5 b) −1 + √5 c) −1 + √2 d) 1 + √2

∞ 𝑥 log 𝑥 𝑑𝑥
8. ∫0 (1+𝑥 2 )2
is equal to
a) 0 b) 1 c) ∞ d) None of these

1
9. The value of ∫−1[𝑥[1 + sin π𝑥] 𝑑𝑥 is ([∙] denotes the greatest integer)
a) 2 b) 0 c) 1 d) None of these

∞ 2 𝜋 ∞ 2
10. If ∫0 𝑒 −𝑥 𝑑𝑥 = √ then ∫0 𝑒 −𝑎𝑥 𝑑𝑥, 𝑎 > 0 is
2
√𝜋 √𝜋 √𝜋 1 √𝜋
a) b) c) 2 d)
2 2𝑎 𝑎 2 𝑎

2𝑥
11. If ∫ 𝑑𝑥 = 𝐾 sin−1(2𝑥 ) + 𝐶, then 𝐾 is equal to
√1−4 𝑥
1 1 1
a) log 2 b) log 2 c) d)
2 2 log 2

𝜋
12. The value of the integral ∫−𝜋(cos 𝑎𝑥 − sin 𝑏𝑥)2 𝑑𝑥, where (𝑎 and 𝑏 integers), is
a) −𝜋 b) 0 c) 𝜋 d) 2𝜋

𝑚𝑥 𝑚+2𝑛−1 −𝑛𝑥 𝑛−1

13. ∫ 𝑥 2𝑚+2𝑛 +2𝑥𝑚+𝑛 +1 𝑑𝑥 is equal to
𝑥𝑚 𝑥𝑛 𝑥 𝑚+𝑛 −1 𝑥𝑛
a) 𝑥 𝑚+𝑛 +1 + 𝑐 b) 𝑥 𝑚+𝑛 +1 + 𝑐 c) 𝑥 𝑚+𝑛 +1 + 𝑐 d) − 𝑥 𝑚+𝑛 +1 + 𝑐

16 𝜋/3
14. The value of ∫0 |sin 𝑥|𝑑𝑥, is
a) 21 b) 21/2 c) 10 d) 11

15. If 𝑓(𝑥) and 𝑔(𝑥), 𝑥𝜖𝑅 are continuous functions, then value of integral
𝜋⁄2
∫−𝜋⁄2[{𝑓(𝑥) + 𝑓(−𝑥)}{𝑔(𝑥) − 𝑔(−𝑥)}] 𝑑𝑥is
𝜋
a) 𝜋 b) c) 1 d) 0
2

√tan 𝑥
16. ∫ sin 𝑥 cos 𝑥 𝑑𝑥 is equal to
√tan 𝑥
a) 2√tan 𝑥 + 𝐶 b) 2√cot 𝑥 + 𝐶 c) 2
+𝐶 d) None of these

1 1+𝑥
17. ∫0 sin {2 tan−1 √ } 𝑑𝑥 =
1−𝑥

a) 𝜋/6 b) 𝜋/4 c) 𝜋/2 d) 𝜋

𝜋/3 𝑎 𝑏 tan 𝑥
18. If ∫−𝜋/3 (3 |tan 𝑥| + 1+sec 𝑥) 𝑑𝑥 = 0 where 𝑎, 𝑏, 𝑐 are constants, then 𝑐 =
𝑎 𝑎 2𝑎
a) 𝑎 ln 2 b) 𝜋 ln 2 c) − 𝜋 ln 2 d) 𝜋 ln 2

𝜋 𝜋
19. If the tangent to the graph function 𝑦 = 𝑓(𝑥) makes angles and with the 𝑥-axis is at the point
4 3
4
𝑥 = 2 and 𝑥 = 4 respectively, the value of ∫2 𝑓 ′ (𝑥)𝑓 ′′ (𝑥) 𝑑𝑥
a) 𝑓(4)𝑓(2) b) 𝑓(4) c) 𝑓(2) d) 1

𝑑𝑥
20. ∫ 3 is equal to
cos √2 sin 2𝑥
tan5/2 𝑥 2
a) √tan 𝑥 + 5
+𝑐 b) √tan 𝑥 + 5 tan5/2 𝑥 + 𝑐
2
c) 2√tan 𝑥 + 5 tan5/2 𝑥 +𝑐 d) None of these

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C B A C C C B A A D

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. D D D B D A B C D A

DPP No. 1
DPP Year : 2023
Topic : Parabola

1. If a double ordinate of the parabola 𝑦 2 = 4𝑎𝑥 be of length 8𝑎, then the angle between the lines
joining the vertex of the parabola to the ends of this double ordinate is
(a) 30o (b) 60o (c) 90o (d) 120o
2. PQ is a double ordinate of the parabola 𝑦 2 = 4𝑎𝑥. The locus of the points of trisection of PQ is
(a) 9𝑦 2 = 4𝑎𝑥 (b) 9𝑥 2 = 4𝑎𝑦
(c) 9𝑦 2 + 4𝑎𝑥 = 0 (d) 9𝑥 2 + 4𝑎𝑦 = 0
3. If the vertex of a parabola be at origin and directrix be 𝑥 + 5 = 0, then its latus rectum is
(a) 5 (b) 10
(c) 20 (d) 40
4. The latus rectum of a parabola whose directrix is 𝑥 + 𝑦 − 2 = 0 and focus is (3, – 4), is
(a) −3√2 (b) 3√2
(c) −3/√2 (d) 3/√2
5. The equation of the lines joining the vertex of the parabola 𝑦 2 = 6𝑥 to the points on it whose
abscissa is 24, is
(a) 𝑦 ± 2𝑥 = 0 (b) 2𝑦 ± 𝑥 = 0
(c) 𝑥 ± 2𝑦 = 0 (d) 2𝑥 ± 𝑦 = 0
2
6. The points on the parabola 𝑦 = 36𝑥 whose ordinate is three times the abscissa are
(a) (0, 0), (4, 12) (b) (1, 3), (4, 12)
(c) (4, 12) (d) None of these
2
7. The points on the parabola 𝑦 = 12𝑥 whose focal distance is 4, are
(a) (2, √3), (2, −√3) (b) (1, 2√3), (1, −2√3)
(c) (1, 2) (d) None of these
2
8. The focal distance of a point on the parabola 𝑦 = 16𝑥 whose ordinate is twice the abscissa, is
(a) 6 (b) 8
(c) 10 (d) 12
9. The co-ordinates of the extremities of the latus rectum of the parabola 5𝑦 2 = 4𝑥 are
(a) (1/5, 2/5), (−1/5, 2/5) (b) (1/5, 2/5), (1/5, −2/5)
(c) (1/5, 4/5), (1/5, −4/5) (d) None of these
10. A parabola passing through the point (−4, −2) has its vertex at the origin and y-axis as its axis.
The latus rectum of the parabola is
(a) 6 (b) 8
(c) 10 (d) 12
11. The focus of the parabola 𝑥 2 = −16𝑦 is
(a) (4, 0) (b) (0, 4)

(c) (–4, 0) (d) (0, –4)

12. If (2, 0) is the vertex and y-axis the directrix of a parabola, then its focus is
(a) (2, 0) (b) (–2, 0)
(c) (4, 0) (d) (–4, 0)
13. If the parabola 𝑦 2 = 4𝑎𝑥 passes through (–3, 2), then length of its latus rectum is
(a) 2/3 (b) 1/3
(c) 4/3 (d) 4
2
14. The ends of latus rectum of parabola 𝑥 + 8𝑦 = 0 are
(a) (–4, –2) and (4, 2) (b) (4, –2) and (–4, 2)
(c) (–4, –2) and (4, –2) (d) (4, 2) and (–4, 2)
2
15. The end points of latus rectum of the parabola 𝑥 = 4𝑎𝑦 are
(a) (𝑎, 2𝑎), (2𝑎, −𝑎) (b) (−𝑎, 2𝑎), (2𝑎, 𝑎)
(c) (𝑎, −2𝑎), (2𝑎, 𝑎) (d) (−2𝑎, 𝑎), (2𝑎, 𝑎)
16. The equation of the parabola with its vertex at the origin, axis on the y-axis and passing through
the point (6, –3) is
(a) 𝑦 2 = 12𝑥 + 6 (b) 𝑥 2 = 12𝑦
(c) 𝑥 2 = −12𝑦 (d) 𝑦 2 = −12𝑥 + 6
17. Focus and directrix of the parabola 𝑥 2 = −8𝑎𝑦 are
(a) (0, −2𝑎) and 𝑦 = 2𝑎 (b) (0, 2𝑎) and 𝑦 = −2𝑎
(c) (2𝑎, 0) and 𝑥 = −2𝑎 (d) (−2𝑎, 0) and 𝑥 = 2𝑎
18. The equation of the parabola with focus (3, 0) and the directirx 𝑥 + 3 = 0 is
(a) 𝑦 2 = 3𝑥 (b) 𝑦 2 = 2𝑥
(c) 𝑦 2 = 12𝑥 (d) 𝑦 2 = 6𝑥
19. Locus of the poles of focal chords of a parabola is of parabola
(a) The tangent at the vertex (b) The axis
(c) A focal chord (d) The directrix
2
20. The parabola 𝑦 = 𝑥 is symmetric about
(a) x-axis (b) y-axis
(c) Both x-axis and y-axis (d) The line 𝑦 = 𝑥
2
21. The point on the parabola 𝑦 = 18𝑥, for which the ordinate is three times the abscissa, is
(a) (6, 2) (b) (–2, –6)
(c) (3, 18) (d) (2, 6)
22. The equation of latus rectum of a parabola is 𝑥 + 𝑦 = 8 and the equation of the tangent at the
vertex is 𝑥 + 𝑦 = 12, then length of the latus rectum is
(a) 4√2 (b) 2√2
(c) 8 (d) 8√2
23. Vertex of the parabola 𝑦 2 + 2𝑦 + 𝑥 = 0 lies in the quadrant
(a) First (b) Second

(c) Third (d) Fourth

2 2
24. The equation 𝑥 − 2𝑥𝑦 + 𝑦 + 3𝑥 + 2 = 0 represents
(a) A parabola (b) An ellipse
(c) A hyperbola (d) A circle
2
25. 𝑥 − 2 = 𝑡 , 𝑦 = 2𝑡 are the parametric equations of the parabola
(a) 𝑦 2 = 4𝑥 (b) 𝑦 2 = −4𝑥
(c) 𝑥 2 = −4𝑦 (d) 𝑦 2 = 4(𝑥 − 2)
26. The equation of the latus rectum of the parabola 𝑥 2 + 4𝑥 + 2𝑦 = 0 is
(a) 2𝑦 + 3 = 0 (b) 3𝑦 = 2
(c) 2𝑦 = 3 (d) 3𝑦 + 2 = 0
2
27. Vertex of the parabola 9𝑥 − 6𝑥 + 36𝑦 + 9 = 0 is
(a) (1/3, −2/9) (b) (−1/3, −1/2)
(c) (−1/3, 1/2) (d) (1/3, 1/2)
28. The equation of the parabola whose axis is vertical and passes through the points (0, 0), (3, 0)
and (–1, 4) is
2
(a) 𝑥 − 3𝑥 − 𝑦 = 0 (b) 𝑥 2 + 3𝑥 + 𝑦 = 0
(c) 𝑥 2 − 4𝑥 + 2𝑦 = 0 (d) 𝑥 2 − 4𝑥 − 2𝑦 = 0
29. The equation of the parabola whose vertex is (–1, –2), axis is vertical and which passes through
the point (3, 6), is
(a) 𝑥 2 + 2𝑥 − 2𝑦 − 3 = 0 (b) 2𝑥 2 = 3𝑦
(c) 𝑥 2 − 2𝑥 − 𝑦 + 3 = 0 (d) None of these
30. Axis of the parabola 𝑥 2 − 4𝑥 − 3𝑦 + 10 = 0 is
(a) 𝑦 + 2 = 0 (b) 𝑥+2=0
(c) 𝑦 − 2 = 0 (d) 𝑥−2=0

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. C A C B B,C A D B B B

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. D C C C D C A C D A

Ques. 21 22 23 24 25 26 27 28 29 30

Ans. D D D A D C A A A D

DPP No. 2
DPP Year : 2023
Topic : Parabola

1. Equation of the parabola whose directrix is 𝑦 = 2𝑥 − 9 and focus (–8, –2) is

(a) 𝑥 2 + 4𝑦 2 + 4𝑥𝑦 + 16𝑥 + 2𝑦 + 259 = 0
(b) 𝑥 2 + 4𝑦 2 + 4𝑥𝑦 + 116𝑥 + 2𝑦 + 259 = 0
(c) 𝑥 2 + 𝑦 2 + 4𝑥𝑦 + 116𝑥 + 2𝑦 + 259 = 0
(d) None of these
2. The equation of the parabola with (–3, 0) as focus and 𝑥 + 5 = 0 as directirx, is
(a) 𝑥 2 = 4(𝑦 + 4) (b) 𝑥 2 = 4(𝑦 − 4)
(c) 𝑦 2 = 4(𝑥 + 4) (d) 𝑦 2 = 4(𝑥 − 4)
3. The equation of the parabola whose vertex and focus lies on the x-axis at distance a and a’ from
the origin, is
(a) 𝑦 2 = 4(𝑎' − 𝑎)(𝑥 − 𝑎) (b) 𝑦 2 = 4(𝑎' − 𝑎)(𝑥 + 𝑎)
(c) 𝑦 2 = 4(𝑎' + 𝑎)(𝑥 − 𝑎) (d) 𝑦 2 = 4(𝑎' + 𝑎)(𝑥 + 𝑎)
4. The focus of the parabola 𝑦 2 = 4𝑦 − 4𝑥 is
(a) (0, 2) (b) (1, 2)
(c) (2, 0) (d) (2, 1)
5. Vertex of the parabola 𝑥 2 + 4𝑥 + 2𝑦 − 7 = 0 is
(a) (–2, 11/2) (b) (–2, 2)
(c) (–2, 11) (d) (2, 11)
6. If the axis of a parabola is horizontal and it passes through the points (0, 0), (0, –1) and (6, 1),
then its equation is
(a) 𝑦 2 + 3𝑦 − 𝑥 − 4 = 0 (b) 𝑦 2 − 3𝑦 + 𝑥 − 4 = 0
(c) 𝑦 2 − 3𝑦 − 𝑥 − 4 = 0 (d) None of these
7. The equation of the latus rectum of the parabola represented by equation
𝑦 2 + 2𝐴𝑥 + 2𝐵𝑦 + 𝐶 = 0 is
𝐵2 +𝐴2 −𝐶 𝐵2 −𝐴2 +𝐶
(a) 𝑥 = (b) 𝑥 =
2𝐴 2𝐴
𝐵2 −𝐴2 −𝐶 𝐴2 −𝐵2 −𝐶
(c) 𝑥 = (d) 𝑥 =
2𝐴 2𝐴
8. The parametric equation of the curve 𝑦 2 = 8𝑥are
(a) 𝑥 = 𝑡 2 , 𝑦 = 2𝑡 (b) 𝑥 = 2𝑡 2 , 𝑦 = 4𝑡
(c) 𝑥 = 2𝑡, 𝑦 = 4𝑡 2 (d) None of these
𝑡 𝑡2
9. The equations 𝑥 = 4 , 𝑦 = represents
4
(a) A circle (b) A parabola
(c) An ellipse (d) A hyperbola

10. The equation of parabola whose vertex and focus are (0, 4) and (0, 2) respectively, is
(a) 𝑦 2 − 8𝑥 = 32 (b) 𝑦 2 + 8𝑥 = 32
(c) 𝑥 2 + 8𝑦 = 32 (d) 𝑥 2 − 8𝑦 = 32
11. Curve 16𝑥 2 + 8𝑥𝑦 + 𝑦 2 − 74𝑥 − 78𝑦 + 212 = 0 represents
(a) Parabola (b) Hyperbola
(c) Ellipse (d) None of these
12. The length of the latus rectum of the parabola 9𝑥 2 − 6𝑥 + 36𝑦 + 19 = 0
(a) 36 (b) 9
(c) 6 (d) 4
13. The axis of the parabola 9𝑦 2 − 16𝑥 − 12𝑦 − 57 = 0 is
(a) 3𝑦 = 2 (b) 𝑥 + 3𝑦 = 3
(c) 2𝑥 = 3 (d) 𝑦 = 3
14. The vertex of a parabola is the point (a, b) and latus rectum is of length l. If the axis of the
parabola is along the positive direction of y-axis, then its equation is
𝑙 𝑙
(a) (𝑥 + 𝑎)2 = 2 (2𝑦 − 2𝑏) (b) (𝑥 − 𝑎)2 = 2 (2𝑦 − 2𝑏)
𝑙 𝑙
(c) (𝑥 + 𝑎)2 = 4 (2𝑦 − 2𝑏) (d) (𝑥 − 𝑎)2 = 8 (2𝑦 − 2𝑏)
15. If the vertex of the parabola 𝑦 = 𝑥 2 − 8𝑥 + 𝑐 lies on x-axis, then the value of c is
(a) –16 (b) –4
(c) 4 (d) 16
16. The points of intersection of the curves whose parametric equations are 𝑥 = 𝑡 2 + 1, 𝑦 = 2𝑡 and
2
𝑥 = 2𝑠, 𝑦 = 𝑠 is given by
(a) (1, −3) (b) (2, 2)
(c) (–2, 4) (d) (1, 2)
17. The latus rectum of the parabola 𝑦 2 = 5𝑥 + 4𝑦 + 1 is
5
(a) 4 (b) 10
5
(c) 5 (d) 2
18. The equation of the locus of a point which moves so as to be at equal distances from the point
(a, 0) and the y-axis is
(a) 𝑦 2 − 2𝑎𝑥 + 𝑎2 = 0 (b)𝑦 2 + 2𝑎𝑥 + 𝑎2 = 0
(c) 𝑥 2 − 2𝑎𝑦 + 𝑎2 = 0 (d)𝑥 2 + 2𝑎𝑦 + 𝑎2 = 0
19. The focus of the parabola 𝑥 2 = 2𝑥 + 2𝑦 is
3 −1 −1
(a) (2 , ) (b) (1, )
2 2
(c) (1, 0) (d) (0, 1)
20. Latus rectum of the parabola 𝑦 2 − 4𝑦 − 2𝑥 − 8 = 0 is
(a) 2 (b) 4
(c) 8 (d) 1

𝑥 𝑦
21. The equation of the parabola with focus (a, b) and directrix 𝑎 + 𝑏 = 1 is given by
(a) (𝑎𝑥 − 𝑏𝑦)2 − 2𝑎3 𝑥 − 2𝑏 3 𝑦 + 𝑎4 + 𝑎2 𝑏 2 + 𝑏 4 = 0
(b) (𝑎𝑥 + 𝑏𝑦)2 − 2𝑎3 𝑥 − 2𝑏 3 𝑦 − 𝑎4 + 𝑎2 𝑏 2 − 𝑏 4 = 0
(c) (𝑎𝑥 − 𝑏𝑦)2 + 𝑎4 + 𝑏 4 − 2𝑎3 𝑥 = 0
(d) (𝑎𝑥 − 𝑏𝑦)2 − 2𝑎3 𝑥 = 0
22. The length of latus rectum of the parabola 4𝑦 2 + 2𝑥 − 20𝑦 + 17 = 0 is
(a) 3 (b) 6
1
(c) 2 (d) 9
23. Eccentricity of the parabola 𝑥 2 − 4𝑥 − 4𝑦 + 4 = 0 is
(a) 𝑒 = 0 (b) 𝑒 = 1
(c) 𝑒 > 4 (d) 𝑒 = 4
24. The vertex of the parabola 3𝑥 − 2𝑦 2 − 4𝑦 + 7 = 0 is
(a) (3, 1) (b) (–3, –1)
(c) (–3, 1) (d) None of these
25. The focus of the parabola 4𝑦 2 − 6𝑥 − 4𝑦 = 5 is
(a) (–8/5, 2) (b) (–5/8, 1/2)
(c) (1/2, 5/8) (d) (5/8, –1/2)
26. The vertex of the parabola 𝑥 2 + 8𝑥 + 12𝑦 + 4 = 0 is
(a) (–4, 1) (b) (4, –1)
(c) (–4, –1) (d) (4, 1)
27. Focus of the parabola (𝑦 − 2)2 = 20(𝑥 + 3) is
(a) (3, –2) (b) (2, –3)
(c) (2, 2) (d) (3, 3)
28. The length of the latus rectum of the parabola 𝑥 2 − 4𝑥 − 8𝑦 + 12 = 0 is
(a) 4 (b) 6
(c) 8 (d) 10
29. The focus of the parabola 𝑦 = 2𝑥 2 + 𝑥 is
1 1
(a) (0, 0) (b) ( , )
2 4
1 1 1
(c) (− 4 , 0) (d) (− 4 , 8)
30. The focus of the parabola 𝑦 2 − 𝑥 − 2𝑦 + 2 = 0 is
(a) (1/4, 0) (b) (1, 2)
(c) (3/4, 1) (d) (5/4, 1)

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. B C A A A D C B B C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. A D A B D B C A C A

Ques. 21 22 23 24 25 26 27 28 29 30

Ans. A C B B B A C C C D

DPP No. 01
DPP Year : 2023
Topic : Vector Algebra

1. A unit vector in 𝑥𝑦-plane that makes an angle 45° with the vector (î + ĵ) and an angle of 60° with the
vector (3𝐢̂ − 4𝐣̂), is
1 1 d) None of these
a) 𝐢̂ b) (𝐢̂ − 𝐣̂) c) (𝐢̂ + 𝐣̂)
√2 √2
2. Let 𝐚⃗ and 𝐛 be unit vectors inclined at an angle 2𝛼(0 ≤ 𝛼 ≤ 𝜋) each other, then | 𝐚⃗ + 𝐛| < 1, if
𝜋 𝜋 2𝜋 𝜋 2𝜋
a) 𝛼 = b) 𝛼 < c) 𝛼 > d) < 𝛼 <
2 3 3 3 3
3. The cartesian from of the plane 𝐫 = (𝑠 − 2𝑡)𝐢̇̂ + (3 − 𝑡)𝐣̇̂ + (2𝑠 + 𝑡)𝐤 ̂ is
a) 2𝑥 − 5𝑦 − 𝑧 − 15 = 0 b) 2𝑥 − 5𝑦 + 𝑧 − 15 = 0
c) 2𝑥 − 5𝑦 − 𝑧 + 15 = 0 d) 2𝑥 + 5𝑦 − 𝑧 + 15 = 0
4. If 𝐚⃗ = 4𝐢̂ + 6𝐣̂ and 𝐛 = 3𝐣̂ + 4𝐤̂ , the vector form of the component of 𝐚⃗ along 𝐛 is
18 18 36 19
a) (3𝐢̂ + 4𝐤̂ ) b) (3𝐣̂ + 4𝐤 ̂ ) c) (3𝐣̂ + 4𝐤 ̂ ) d) (2𝐢̂ + 3𝐣̂ )
5 25 25 18
5. A force 𝐅 = 2𝐢̂ + 𝐣̂ − 𝐤̂ acts at a point 𝐴, whose position vectors is 2𝐢̂ − 𝐣̂. The moment of 𝐅 about the
origin is
a) 𝐢̂ + 2𝐣̂ − 4𝐤̂ b) 𝐢̂ − 2𝐣̂ − 4𝐤 ̂ c) 𝐢̂ + 2𝐣̂ + 4𝐤 ̂ d) 𝐢̂ − 2𝐣̂ + 4𝐤̂
6. If 𝑎, 𝑏⃗, 𝑐 are linearly independent vectors, then
(𝑎 + 2𝑏⃗) × (2𝑏⃗ + 𝑐) ∙ (5𝑐 + 𝑎)
is equal to
𝑎 ∙ (𝑏⃗ × 𝑐)
a) 10 b) 14 c) 18 d) 12
7. If 𝐚⃗, 𝐛 and 𝐜 are perpendicular to 𝐛 + 𝐜, 𝐜 + 𝐚⃗ and 𝐚⃗ + 𝐛 respectively and if |𝐚⃗ + 𝐛| = 6, |𝐛 + 𝐜| = 8 and
|𝐜 + 𝐚⃗| = 10, then |𝐚⃗ + 𝐛 + 𝐜| is equal to
a) 5√5 b) 50 c) 10√2 d) 10
8. If 𝑎, 𝑏⃗, 𝑐 are three mutually perpendicular vectors of equal magnitude, then the angle 𝜃 which 𝑎 + 𝑏⃗ + 𝑐
makes with any one of three given vectors is given by
1 1 2 d) None of these
a) cos−1 b) cos−1 c) cos−1
√3 3 √3
9. Forces 3 𝑂𝐴, 5 𝑂𝐵 ⃗ act along 𝑂𝐴 and 𝑂𝐵. If their resultant passes through 𝐶 on 𝐴𝐵, then
a) 𝐶 is a mid-point of 𝐴𝐵
b) 𝐶 divides 𝐴𝐵 in the ratio 2 : 1
c) 3 𝐴𝐶 = 5 𝐶𝐵
d) 2 𝐴𝐶 = 3 𝐶𝐵
10. The centre of the circle given by 𝐫 ∙ (𝐢̇̂ + 2𝐣̇̂ + 2𝐤
̂ ) = 15 and 𝐫 − (𝐣̇̂ + 2𝐤
̂ ) = 4 is
a) (1,2,4) b) (3,1,4) c) (1,3,4) d) None of these

11. Consider a tetrahedron with faces 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 . Let 𝑣1 , 𝑣2 , 𝑣3 , 𝑣4 be the vectors whose magnitudes are
respectively equal to areas of 𝐹1 , 𝐹2 , 𝐹3 , 𝐹4 and whose directions are perpendicular to these faces in
outward direction. Then, |𝑣1 + 𝑣2 + 𝑣3 + 𝑣4 | equals
a) 1 b) 4 c) 0 d) None of these
12. The volume of the tetrahedron having the edges 𝐢̇̂ + 2𝐣̇̂ − 𝐤 ̂ , 𝐢̇̂ + 𝐣̇̂ + 𝐤 ̂ as coterminous is 2 cu
̂ , 𝐢̇̂ − 𝐣̇̂ + 𝜆𝐤
3
unit. Then, λ equals
a) 1 b) 2 c) 3 d) 4
13. If 𝐚⃗, 𝐛, 𝐜 are three non-coplanar vectors then the vector equation
𝐫 = (1 − 𝑝 − 𝑞)𝐚 ⃗ + 𝑝𝐛 + 𝑞𝐜 represent a
a) Straight line b) Plane
c) Plane passing through the origin d) Sphere
14. A force of magnitude 5 units acting along the vector 2𝑖̂ − 2𝑗̂ + 𝑘̂ displaces the point of application from
the point (1, 2, 3) to the point (5, 3, 7), then the work done by the force is
50 50 25 25
a) 7
units b) 3
units c) 3
units d) 4
units
15. If α
⃗ = 2𝐢̇̂ + 3𝐣̇̂ − 𝐤 ⃗ = −𝐢̇̂ + 2𝐣̇̂ − 4𝐤
̂, β ̂, γ ̂ , then what is the value of (𝐚⃗ × 𝐛) ∙ (α
⃗ = 𝐢̇̂ + 𝐣̇̂ + 𝐤 ⃗ ×γ ⃗ )?
a) 47 b) 74 c) −74 d) None of these
16. The line of intersection of the planes 𝐫 ∙ (𝐢̇̂ − 3𝐣̇̂ + 𝐤 ̂ ) = 1 and 𝐫 ∙ (2𝐢̇̂ + 5𝐣̇̂ − 3𝐤
̂ ) = 2 is parallel to the
vector
a) −4𝐢̇̂ + 5𝐣̇̂ + 11𝐤̂ ̂
b) 4𝐢̇̂ + 5𝐣̇̂ + 11𝐤 ̂
c) −4𝐢̇̂ − 5𝐣̇̂ + 11𝐤 d) −4𝐢̇̂ + 5𝐣̇̂ − 11𝐤 ̂
17. 𝐴𝐵𝐶𝐷𝐸𝐹 is a regular hexagon with centre at the origin such that
⃗⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
𝐀𝐃 ⃗⃗⃗⃗⃗ = 𝜆 𝐄𝐃
𝐄𝐁 + 𝐅𝐂 ⃗⃗⃗⃗⃗ . Then, 𝜆 is equal to
a) 2 b) 4 c) 6 d) 3
18. If 𝐚⃗ and 𝐛 are two non-zero, non-collinear vectors, then
̂ ]𝐤
2[𝐚⃗ 𝐛 𝐢̇̂]𝐢̇̂ + 2[𝐚⃗ 𝐛 ̂𝐣̇]𝐣̇̂ + 2[𝐚⃗ 𝐛 𝐤 ̂ + [𝐚⃗ 𝐛 𝐚⃗ ] is equal to
a) 2(𝐚⃗ × 𝐛) b) 𝐚⃗ × 𝐛 c) 𝐚⃗ + 𝐛 d) None of these
19. If 𝐚⃗ = (𝐢̂ + 𝐣̂ + 𝐤
̂ ), 𝐚⃗ ∙ 𝐛 = 1 and 𝐚⃗ × 𝐛 = 𝐣̂ − 𝐤
̂ , then 𝐛 is
̂
a) 𝐢̂ − 𝐣̂ + 𝐤 ̂
b) 2𝐣̂ − 𝐤 c) 𝐢̂ d) 2𝐢̂
20. Let 𝐚⃗, 𝐛 and 𝐜 be three non-coplanar vectors and let 𝐩 ⃗ ,𝐪
⃗ and 𝐫 be vector defined by the relations.
𝐛×𝐜 ⃗
𝐜×𝐚 ⃗ ×𝐛
𝐚
⃗ =
𝐩 ⃗ =
,𝐪 and 𝐫 = . Then, the value of the expression
⃗ 𝐛𝐜]
[𝐚 ⃗ 𝐛𝐜]
[𝐚 ⃗ 𝐛𝐜]
[𝐚

(𝐚⃗ + 𝐛) ∙ 𝐩 ⃗ + (𝐛 + 𝐜) ∙ 𝐪 ⃗ + (𝐜 + 𝐚⃗) ∙ 𝐫 is equal to

a) 0 b) 1 c) 2 d) 3
21. If 𝑚1 , 𝑚2 , 𝑚3 and 𝑚4 are respectively the magnitudes of the vectors
𝐚 ̂ , ⃗⃗⃗⃗
⃗⃗⃗⃗1 = 2𝐢̇̂ − 𝐣̇̂ + 𝐤 ̂ ,𝐚
𝐚2 = 3𝐢̇̂ − 4𝐣̇̂ + 4𝐤 ̂ and 𝐚
⃗⃗⃗⃗3 = 𝐢̇̂ + 𝐣̇̂ − 𝐤 ̂ , then the correct order of
̂ + 3𝐣̇̂ + 𝐤
⃗⃗⃗⃗4 = −𝐢̇
𝑚1 , 𝑚2 , 𝑚3 and 𝑚4 is
a) 𝑚3 < 𝑚1 < 𝑚4 < 𝑚2 b) 𝑚3 < 𝑚1 < 𝑚2 < 𝑚4 c) 𝑚3 < 𝑚4 < 𝑚1 < 𝑚2 d) 𝑚3 < 𝑚4 < 𝑚2 < 𝑚1
22. If 𝐚⃗, 𝐛, 𝐜 are non-coplanar vectors and λ is a real number, then [λ(𝐚⃗ + 𝐛)λ2 𝐛 𝜆 𝐜] = [𝐚⃗ 𝐛 + 𝐜 𝐛] for
a) exactly two values of λ b) exactly three values of λ
c) no real values of λ d) exactly one values of λ

23. Let 𝑎, 𝑏 and 𝑐 be distinct non-negative numbers. If the vectors 𝑎𝐢̇̂ + 𝑎𝐣̇̂ + 𝑐𝐤
̂ , 𝐢̇̂ + 𝐤
̂ and 𝑐𝐢̇̂ + 𝑐𝐣̇̂ + 𝑏𝐤
̂ lie in
a plane, then 𝑐 is
a) The harmonic mean of 𝑎 and 𝑏 b) Equal to zero
c) The arithmetic mean of 𝑎 and 𝑏 d) The geometric mean of 𝑎 and 𝑏
24. In a trapezium 𝐴𝐵𝐶𝐷 the vector 𝐵𝐶 = 𝜆 𝐴𝐷. If 𝑃⃗ = 𝐴𝐶 + 𝐵 ⃗ 𝐷 is collinear with 𝐴𝐷 such that 𝑝 = 𝜇 𝐴𝐷,
then
a) 𝜇 = 𝜆 + 1 b) 𝜆 = 𝜇 + 1 c) 𝜆 + 𝜇 = 1 d) 𝜇 = 2 + 𝜆
25. If 𝑎, ⃗⃗𝑏, 𝑐 are three vectors such that 𝑎 + 𝑏⃗ + 𝑐 = 0 and |𝑎| = 2, |𝑏⃗| = 3, |𝑐| = 4, then the value of 𝑎 ∙
𝑏⃗ + 𝑏⃗ ∙ 𝑐 + 𝑐 ∙ 𝑎 is equal to
a) 29 b) −29 c) 29/2 d) −29/2
26. If |𝑎| = 3, |𝐛| = 4, then a value of λ for which 𝐚⃗ + 𝜆𝐛 is perpendicular to 𝐚⃗ − 𝜆𝐛 is
9 3 3 4
a) b) c) d)
16 4 2 3
27. 𝐮 ̂ × (𝐚⃗ × 𝐤
⃗ = 𝐢̇̂ × (𝐚⃗ × 𝐢̇̂) + 𝐣̇̂ × (𝐚⃗ × 𝐣̇̂) + 𝐤 ̂ ) is equal
a) 𝐚⃗ b) 2𝐚⃗ c) 3𝐚⃗ d) None of these
28. The locus of a point equidistant from two points whose position vectors are 𝐚⃗ and 𝐛, is
1
a) {𝐫 − ( 𝐚⃗ + 𝐛)} ( 𝐚⃗ − 𝐛) = 0 b) {𝐫 − ( 𝐚⃗ + 𝐛)} ∙ 𝐛 = 0
2
1 1
c) {𝐫 − ( 𝐚⃗ + 𝐛)} ∙ 𝐚⃗ = 0 d) {𝐫 − ( 𝐚⃗ − 𝐛)} ∙ ( 𝐚⃗ + 𝐛) = 0
2 2
29. If 𝐚⃗ and 𝐛 are two vectors such that |𝐚⃗| + 3√3, 𝐛 = 4 and |𝐚⃗ + 𝐛| = √7, then the angle between 𝐚⃗ and
𝐛 is
a) 120° b) 60° c) 30° d) 150°
30. If 𝐚⃗ = 2𝐢̇̂ + 3𝐣̇̂ − 𝐤
̂ , 𝐛 = 𝐢̇̂ + 2𝐣̇̂ − 5𝐤
̂, ̂ ̂ ̂
𝐜 = 3𝐢̇ + 5𝐣̇ − 𝐤, then a vector perpendicular to 𝐚⃗ and in the
plane containing 𝐛 and 𝐜 is
̂
a) −17𝐢̇̂ + 21𝐣̇̂ − 97𝐤 ̂
b) 17𝐢̇̂ + 21𝐣̇̂ − 123𝐤 ̂
c) −17𝐢̇̂ − 21𝐣̇̂ + 97𝐤 ̂
d) −17𝐢̇̂ − 21𝐣̇̂ − 97𝐤
31. If 𝑎, 𝑏⃗, 𝑐 are three mutually perpendicular vectors each of magnitude unity, then |𝑎 + 𝑏⃗ + 𝑐| is equal to
a) 3 b) 1 c) √3 d) None of these
32. (𝐚⃗ ∙ 𝐢̇̂)𝐢̇̂ + (𝐚⃗ ∙ ̂𝐣̇)𝐣̇̂ + (𝐚⃗ ∙ 𝐤
̂ )𝐤
̂ is equal to
a) 𝐚⃗ b) 2 𝐚⃗ c) 3 𝐚⃗ d) 𝟎
⃗
33. If 𝐚⃗ ∙ 𝐢̂ = 𝐚⃗ ∙ (𝐢̂ + 𝐣̂) = 𝐚⃗ ∙ (𝐢̂ + 𝐣̂ + 𝐤
̂ ), then 𝐚⃗ is equal to
a) 𝐢̂ ̂
b) 𝐤 c) 𝐣̂ ̂
d) 𝐢̂ + 𝐣̂ + 𝐤
34. Let 𝐚⃗ = 𝐢̇̂ + ̂𝐣̇ − 𝐤 ̂ , 𝐛 = 𝐢̇̂ − ̂𝐣̇ + 3𝐤
̂ and 𝐜 be a unit victor perpendicular to 𝐚⃗ and coplanar with 𝐚⃗ and
𝐛, then 𝐜 is
1 1 1 1
a) ̂)
(𝐣̇̂ + 𝐤 b) ̂)
(𝐣̇̂ − 𝐤 c) ̂)
(𝐢̇̂ − 2𝐣̇̂ + 𝐤 d) ̂)
(2𝐢̇̂ − ̂𝐣̇ + 𝐤
√2 √2 √6 √6
35. The plane through the point (−1, −1, −1) and containing the line of intersection of the planes 𝐫 ∙
̂ ) = 0 and 𝐫 ∙ (𝐣̇̂ + 2𝐤
(𝐢̇̂ + 3𝐣̇̂ − 𝐤 ̂ ) = 0 is
̂ ) = 0 b) 𝐫 ∙ (𝐢̇̂ + 4𝐣̇̂ + 𝐤
a) 𝐫 ∙ (𝐢̇̂ + 2𝐣̇̂ − 3𝐤 ̂)=0 ̂ ) = 0 d) 𝐫 ∙ (𝐢̇̂ + 𝐣̇̂ + 3𝐤
c) 𝐫 ∙ (𝐢̇̂ + 5𝐣̇̂ − 5𝐤 ̂)=0

36. If a parallelogram is constructed on the vectors 𝑎 = 3𝜇 − 𝑣 , 𝑏⃗ = 𝑢 ⃗ + 3𝑣 and |𝑢

⃗ | = |𝑣 | = 2 and the
angle between 𝑢 ⃗ is 𝜋/3, then the ratio of the lengths of the sides is
a) √7: √13 b) √6: √2 c) √3: √5 d) None of these
37. Let 𝑎, 𝑏⃗, 𝑐 be the position vectors of the vertices 𝐴, 𝐵, 𝐶 respectively of ∆𝐴𝐵𝐶. The vector area of ∆𝐴𝐵𝐶
is
1
a) {𝑎 × (𝑏⃗ × 𝑐) + 𝑏⃗ × (𝑐 × 𝑎) + 𝑐 × (𝑎 × 𝑏⃗)}
2
1
b) (𝑎 × 𝑏⃗ + 𝑏⃗ × 𝑐 + 𝑐 × 𝑎)
2
1
c) (𝑎 + 𝑏⃗ + 𝑐)
2
1
d) {(𝑏⃗. 𝑐)𝑎 + (𝑐. 𝑎)𝑏⃗ + (𝑎. 𝑏⃗)𝑐}
2
38. The work done in moving an object along a vector 𝑑 = 3𝑖̂ + 2𝑗̂ − 5𝑘̂ if the applied force is 𝐹 = 2𝑖̂ − 𝑗̂ −
𝑘̂ is
a) 12 units b) 11 units c) 10 units d) 9 units
39. 𝐚⃗ × [𝐚⃗ × (𝐚⃗ × 𝐛)] is equal to
a) (𝐚⃗ × 𝐚⃗) ∙ (𝐛 × 𝐚⃗) b) 𝐚⃗ ∙ (𝐛 × 𝐚⃗) − 𝐛(𝐚⃗ × 𝐛)
c) [𝐚⃗ ∙ (𝐚⃗ × 𝐛)]𝐚⃗ d) (𝐚⃗ ∙ 𝐚⃗)(𝐛 × 𝐚⃗)
40. 𝐚⃗ ∙ 𝐚⃗ 𝐚⃗ ∙ 𝐛 𝐚⃗ ∙ 𝐜
If 𝐚⃗, 𝐛, 𝐜 are non-coplanar vectors, then |𝐛 ∙ 𝐚⃗ 𝐛 ∙ 𝐛 𝐛 ∙ 𝐜| is equal to
𝐜 ∙ 𝐚⃗ 𝐜 ∙ 𝐛 𝐜 ∙ 𝐜
a) [ 𝐚⃗ 𝐛 𝐜 ]
2
b) [ 𝐚⃗ 𝐛 𝐜 ] c) [ 𝐚⃗ 𝐛 𝐜 ]
1/3 d) None of these
41. If |𝐚⃗| = 10, |𝐛| = 2 and 𝐚⃗ ∙ 𝐛 = 12 then |𝐚⃗ × 𝐛| is equal to
a) 12 b) 14 c) 16 d) 18
42. If |𝑎| = 4, |𝑏⃗| = 4 and |𝑐| = 5 such that 𝑎 ⊥ (𝑏⃗ + 𝑐 ), 𝑏⃗ ⊥ (𝑐 + 𝑎) and 𝑐 ⊥ (𝑎 + 𝑏⃗), then |𝑎 + 𝑏⃗ + 𝑐| is
a) 7 b) 5 c) 13 d) √57
43. The summation of two unit vectors is a third unit vector, then the modulus of the difference of the unit
vectors is
a) √3 b) 1 − √3 c) 1 + √3 d) −√3
44. If 𝜃 is the angle between vectors 𝑎 and 𝑏⃗ such that 𝑎. 𝑏⃗ ≥ 0, then
𝜋 𝜋 𝜋
a) 0 ≤ 𝜃 ≤ 𝜋 b) ≤ 𝜃 ≤ 𝜋 c) 0 ≤ 𝜃 ≤ d) 0 < 𝜃 <
2 2 2
45. The vectors 2𝑖̂ + 3𝑗̂, 5𝑖̂ + 6𝑗̂ and 8𝑖̂ + 𝜆𝑗̂ have their initial points at (1, 1). The value of 𝜆 so that the
vectors terminate on one straight line, is
a) 0 b) 3 c) 6 d) 9
46. If 𝑝th, 𝑞th, 𝑟th term of a GP are the positive numbers 𝑎, 𝑏, 𝑐 then angle between the vectors
̂ and (𝑞 − 𝑟)𝐢̂ + (𝑟 − 𝑝)𝐣̂ + (𝑝 − 𝑞)𝐤
log 𝑎3 𝐢̂ + log 𝑏 3 𝐣̂ + log 𝑐 3 𝐤 ̂ is
𝜋 𝜋
a) b)
6 2

𝜋 1
c) d) sin−1 ( )
3 √𝑎2 + 𝑏 2 + 𝑐 2
47. The value of 𝜆, for which the four points 2𝐢̂ + 3𝐣̂ − 𝐤 ̂ , 𝐢̂ + 2𝐣̂ + 3𝐤
̂ , 3𝐢̂ + 4𝐣̂ − 2𝐤
̂ , 𝐢̂ − λ𝐣̂ + 6𝐤
̂ are coplanar,
is
a) −2 b) 8 c) 6 d) 0
48. Given that |𝐚⃗| = 3, |𝐛| = 4, |𝐚⃗ × 𝐛| = 10, then |𝐚⃗ ∙ 𝐛|2 equals
a) 88 b) 44 c) 22 d) None of these
49. If the diagonals of a parallelogram are 3𝑖̂ + 𝑗̂ − 2𝑘̂ and 𝑖̂ − 3𝑗̂ + 4𝑘̂, then the lengths of its sides are
a) √8, √10 b) √6, √14 c) √5, √12 d) None of these
50. If 𝐚⃗ × 𝐛 = 𝐜 and 𝐛 × 𝐜 = 𝐚⃗, then
a) |𝐚⃗|= 1, |𝐛|= |𝐜| b) |𝐜|= 1, |𝐚⃗| = 1 c) |𝐛|= 2, |𝐛| = 2|𝐚⃗| d) |𝐛|= 1, |𝐜|= |𝐚⃗|

Answer Key
Ques. 1 2 3 4 5 6 7 8 9 10

Ans. D D C B C D D A C C

Ques. 11 12 13 14 15 16 17 18 19 20

Ans. C A B B C B B A C D

Ques. 21 22 23 24 25 26 27 28 29 30

Ans. A C D A D B B A D D

Ques. 31 32 33 34 35 36 37 38 39 40

Ans. C A A D A A B D D A

Ques. 41 42 43 44 45 46 47 48 49 50

Ans. C D A C D B A B B D

21, Lakhanpur , Kanpur, Uttar Pradesh 208024 Page No. 6 8400072444, 7007286637
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