MATHS PRACTICE SET - 5

1. If 𝐴 and 𝐵 are finite sets and 𝐴 ⊂ 𝐵, then 12. If 𝑖 = √−1, then 1 + 𝑖 2 + 𝑖 3 − 𝑖 𝐵 + 𝑖 𝐵 is equal
a. 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐵) b. 𝑛(𝐴 ∩ 𝐵) = 𝑛(𝐵) to
c. 𝑛(𝐴 ∩ 𝐵) = 𝜙 d. 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) (a) 2 − i (b) 1
2. If X = {−2, −1,0,1,2,3,4,5,6,7,8} and A{x: |x − (c) 3 (d) -1
2| ≤ 3, x is an integer}, then X − A = 13. If 𝑎 and 𝛽 are the roots of the equation 4𝑥 2 +
a. {−2,6,7,8} b. {−2, −1,1,2,3,4,5,6} 1 1
3𝑥 + 𝑇 = 0, then 𝛼 + 𝛽 =
c. {−1,0,1,2,3,4,5,7,8} d. {−2, −1,2,3,6,7,8} 3 3
(a) − 7 (b) 7
3. The set (𝐴 ∣ 𝐵) ∪ (𝐵 ∣ 𝐴) is equal to 3 3
a. [𝐴 ∣ (𝐴 ∩ 𝐵)] ∩ [𝐵 ∣ (𝐴 ∩ 𝐵)] (c) − 5 (d) 5
b. (𝐴 ∪ 𝐵) ∣ (𝐴 ∩ 𝐵)
14. If the product of the roots of the equation
c. 𝐴 ∣ (𝐴 ∩ 𝐵)
{𝑎 + 1 ∣ 𝑥 2 + (2𝑥 + 3𝑥 + 𝑄𝑎 + 4) = 0 be 2 , then
d. (𝐴 ∩ 𝐵) ∣ (𝐴 ∪ 𝐵) the sum of roots is
4. Set A and B have 2 and 6 elements respectively. (a) 1 (b) -1
What can be the minimum number of elements in (c) 2 (d) -2
𝐴∪𝐵?
15. If 𝑧 + 𝛼√3 is a root of the equation 𝑥 2 + 𝛽𝑥 +
a. 18 b. 9
𝑞 = 0, where 𝑝 and 𝑞 are real, then ( 𝑝𝑞 =
c. 6 d. 3
(a) (−4,7) (b) (4, −7)
5. The symmetric difference of A and B is
(c) (4,7) (d) (−4, −7)
(a) (A − B) ∩ (B − A)
(b) (A − B) ∪ (B − A) 16. If the sum of the roots of the equation 𝜆𝑥 2 +
(c) (A ∪ B) − (A ∩ B) 2𝑥 + 3𝜆 = 0 be equal to their product, then 𝜆 =
(d) {(A ∪ B) − A} ∪ {(A ∪ B) − B} (a) 4 (b) -4
(c) 6 (d) None of these
6. If 𝐴 = {1,2,3,4,5}, then the number of proper
subsets of A is 17. If 𝛼 and 𝛽 are the roots of the equation 𝑥 2 +
(a) 120 (b) 30 6𝑥 + 𝜆 = 0 and 3𝛼 + 2𝛽 = −20, then 𝜆 =
(c) 31 (d) 32 (a) -8 (b) -16
(c) 16 (d) 8
7. If 𝐴 and 𝐵 are two given sets, then 𝐴 ∩ (𝐴 ∩ 𝐵)𝑐 18.In the expansion of (1 + 𝑥)5 the coefficient of
is equal to 𝑝ℎ and (𝑝 + 1) terms/are respectively 𝑝 and 𝑞.
(a) A (b) B Then 𝑝 + 𝑞
(c) Φ (d) 𝐴𝑐 ∩ 𝐵𝑐 (a) 𝑛 + 3 (b) 𝑛 + 1
1+𝑖 m
(c) n + 2 (d) 𝑛
8. If (1−𝑖) = 1, then the least integral value of 𝑚
19.If the middle term in the expansion of
is 1 4
(a) 2 (b) 4 (𝑥 2 + ) is 924𝑥 6 , then 𝑛 =
𝑥
(c) 8 (d) None of these (a) 10 (b) 12
𝑖 592 +𝑖 590 +𝑖 588 +𝑖 586 +𝑖 584 (c) 14 (d) None of these
9. The value of 𝑖582 +𝑖580 +𝑖578 +𝑖576 +𝑖574 − 1 =
20.The coefficient of middle term in the
(a) -1 (b) -2
expansion of (1 + 𝑥)13 is
(c) -3 (d) -4 10! 10!
(a) 516! (b) (5!)2
10. 1 + 𝑖 2 + 𝑖 4 + 𝑖 6 + ⋯ . . +𝑡 2𝑛 is 10!
(a) Positive (b) Negative (c) 5!7! (d) None of these
(c) Zero (d) Cannot be determined 3+5+7+⋯+𝑛
2 𝐴 𝐵
21.If 5+8+11+⋯+10 terms = 7, then the value of 𝑛 is
11. 𝑖 + 𝑖 + 𝑖 + ⋯ …. upto (2𝑛 + 1) terms =
(a) 1 (b) −𝑖 (a) 35 (b) 36
(c) 1 (d) -1 (c) 37 (d) 40
22.If sum of 𝑛 terms of an AP is 3𝑛2 + 5𝑛 and
𝑇𝑚 = 164, then 𝑚 is equal to

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(a) 26 (b) 27 selection?

(c) 28 (d) None of these (A) 70 (B) 85
23.In a GP, if the (𝑚 + 𝑛) th term be 𝑝 and (𝑚 − (C) 80 (D) 18
𝑛) th term be 𝑞, then its 𝑚 th term is
33. In how many ways can a student choose
(a) √𝑝𝑞 (b) √𝑝/𝑞
(n - 2) courses out of n courses if 2 courses
(c) √𝑞/𝑝 (d) √𝑝 + 𝑞 are compulsory (n > 4)?
1
24.The sum of the first ' 𝑛 ' terms of the series 2 +
(A) (n - 3)(n - 4) (B) (n - 1)(n - 2)
3 7 15
4
+ 8 + 16 + ⋯ is (C) (n - 3)(n - 4)/2 (D) (n - 2)(n - 3)/2
(a) 2𝑛 − 𝑛 − 1 (b) 1 − 2−𝑛
(c) 𝑛 + 2−𝑛 − 1 (d) 2𝑛 − 1 34. Three perfect dice D1, D2 and D3 are
rolled. Let x, y and z represent the numbers
25.An AP consists of 𝑛 (odd terms) and its middle on D1, D2 and D3 respectively. What is the
term is 𝑚. Then, the sum of the AP is
1 number of possible outcomes such that x < y
(a) 2𝑚𝑛 (b) 2 𝑚𝑛 < z?
(c) 𝑚𝑛 (d) 𝑚𝑛2
(A) 20 (B) 18
26.Which term of the following sequence
1 1 1 1 (C) 14 (D) 10
, , … … is 19683 ?
3 9 27
(a) 3 (b) 9 35. The observations 4, 1, 4, 3, 6, 2, 1, 3, 4, 5,
(c) 6 (d) None of these 1, 6 are outputs of 12 dices thrown
27. The total number of terms in the expansion of simultaneously. If m and M are means of
(𝑥 + 𝑦)100 +(𝑥 − 𝑦)100 after simplification is lowest 8 observations and highest 4
a. 100 b. 50 observations respectively, then what is (2m +
c. 51 d. 202 M) equal to?
28. If one real root of the quadratic equation
(A) 10 (B) 12
81𝑥 2 + 𝑘𝑥 + 256 = 0 is cube of the other root,
then a value of 𝑘 is : (C) 17 (D) 21
(A) -81 (B) -300
(C) 100 (D) 144 36. A die is thrown 10 times and obtained the
following outputs: 1, 2, 1, 1, 2, 1, 4, 6, 5, 4.
29.If the complex numbers 𝑧1 , 𝑧2 , 𝑧3 are in A.P, What will be the mode of data so obtained?
then they lie on
(a) a circle (b) a parabola (A) 6 (B) 4
(c) a line (d) an ellipse (C) 2 (D) 1
30.The relation "congruence modulo 𝑚 " is
37.Consider the following frequency
(a) Reflexive only (b) Transitive only
(c) Symmetric only (d) An equivalence relation distribution

31. The value of 𝑥 if x 1 2 3 5

1 1 𝑥
+ 10! = 11! f 4 6 9 7
9!

(A) 11 (B) 121 (C) 140(D) 21 What is the value of median of the
distribution?
32. In a class there are 10 boys and 8 girls.
The teacher wants to select a boy and a girl to (A) 1 (B) 2
represent the class in a function. In how (C) 3 (D) 3.5
many ways can the teacher make this

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Consider the following for the next three (03)

items that follow: Frequency 17 𝑝+𝑞 32 𝑝 19
The algebraic sum of the deviations of a set of − 3𝑞
values 𝑥1 , 𝑥2 , 𝑥3 , … . . 𝑥𝑛 measured from 100 is -
20 and the algebraic sum of the deviation of
the same set of values measured from 92 is The total frequency is 120 . The mean is 50 .
140 .
44. What is the value of 𝑝 ?
38. What is the mean of the values (A) 25 (B) 26
(A) 91 (B) 96 (C) 27 (D) 28
(C) 98 (D) 99
45. What is the value of 𝑞 ?
39. What is the algebraic sum of the (A) 1 (B) 2
(C) 3 (D) 4
deviations of the same set of values measured
from 99? 46. If the frequency of each class is doubled, then
(A) 0 (B) 10 what would be the mean?
(C) 20 (D) 40 (A) 25 (B) 50
(C) 75 (D) 100
40. If the algebraic sum of the deviations of
the same set of values measured from 𝑦 is 47. The number of ways in which 5 boys and 3
180, then what is the value of 𝑦 ? girls can be seated in a row so that each girl is
(A) 80 (B) 85 between two boys:
(C) 90 (D) 95 (A) 2880 (B) 1880
Direction (Q. No. 41 to 43) (C) 3800 (D) 2800
Consider the following data for the next three (03)
items that follow: 48. Which of the following statement(s) is/are
correct?
The marks obtained by 51 students in a class are
in A.P. with its first term 4 and common (I)If nCr = 84, nCr-1 = 36 and nCr+1 = 126, then n = 7
difference 3 .
(II)If nC3 + nC4 > n+1C3, then n > 6
41. What is the mean of the marks?
(A) 67 (B) 71 (III) If 15C r+3 = 15C2 r-6, then the value of r is either
(C) 75 (D) 79 9 or 6

42. What is the median of the marks ? (A) (I) and (II) (B) (II) and (III)
(A) 79.5 (B) 79
(C) 78.5 (D) 77 (C) (I) and (III) (D) (I), (II), (III)

43. What is the sum of the deviations 49. A, B, C and D are mutually exclusive and
measured from the median ? exhaustive events. If 2P(A) = 3P(B) = 4P(C) =
(A) -1 (B) 0 5P(D), then what is 77P(A) equal to?
(C) 1 (D) 2 (A) 12 (B) 15
Direction (Q. No. 44 to 46) (C) 20 (D) 30
Consider the following frequency distribution for
50. Two distinct natural numbers from 1 to 9 are
the next three (03) items that follow:
picked at random. What is the probability that
Class their product has 1 in its unit place?
0 20 40 60 80
− 20 − 40 (A) 1/81 (B) 1/72
−60 −80 −100

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𝜋
(C) 1/18 (D) 1/36 𝑥 2 cos⁡ 𝑥
57. The value of ∫ 2
𝜋 𝑑𝑥 is equal to
− 1+𝑒 𝑥
2
51. Two dice are thrown. What is the probability 𝜋2 𝜋2
that the difference of numbers on them is 2 or 3? (a) −2 (b) +2
4 4
2 −𝜋/2
(c) 𝜋 − 𝑒 (d) 𝜋 + 𝑒 𝜋/2
2
(A) 7/36 (B) 7/18
58. The value of ∫ (𝑒 log⁡ 𝑥 + sin⁡ 𝑥)cos⁡ 𝑥𝑑𝑥
(C) 5/18 (D) 11/36 cos⁡ 2𝑥
(a) 𝑥sin⁡ 𝑥 + cos⁡ 𝑥 − +𝐶
4
52. Suppose that there is a chance for a newly (b) 𝑥cos⁡ 𝑥 + sin⁡ 𝑥 −
sin⁡ 2𝑥
+𝐶
constructed building to collapse, whether the 4
sin⁡ 2𝑥
design is faulty or not. The chance that the design (c) sin⁡ 𝑥 + 𝑥cos⁡ 𝑥 − 2 + 𝐶
is faulty is 10%. The chance that the building sin⁡ 2𝑥
(d) cos⁡ 𝑥 + 𝑥sin⁡ 𝑥 − 2 + 𝐶
collapses is 95% if the design is faulty, otherwise it
is 45%. If it is seen that the building has collapsed, 𝑥
then what is the probability that it is due to faulty 59. The function 𝑓(𝑥) = cos⁡ + {𝑥}, where {𝑥} =
2
design? the fractional part of 𝑥 is a
(a) periodic function with period 4𝜋
(A) 0.10 (B) 0.19 (b) periodic function with period 1
(c) periodic function with interminate period
(C) 0.45 (D) 0.95 (d) None of the above
53. A fair coin is tossed 6 times. What is the 60. The value
probability of getting a result in the 6th toss
which is different from those obtained in the first 2
1+𝑥 1 − 𝑥 −2
five tosses? ∫ {𝑝ln⁡ ( ) + 𝑞ln⁡ ( ) + 𝑟} 𝑑𝑥 depends
−2 1−𝑥 1+𝑥
(A) 7/16 (B) 1/16
on the value of
(C) 1/32 (D) 1/64 (a) 𝑝 (b) 𝑞
(c) 𝑟 (d) 𝑝 and 𝑞
54. In a class, there are n students including the 𝑥 1−𝑥
student P and Q. What is the probability that P 61. The function 𝑓(𝑥) = ∫0 log⁡ (1+𝑥) 𝑑𝑥 is
and Q sit together if seats are assigned randomly? (a) an even function (b) an odd function
(c) a periodic function (d) None of these
(A) 1/n (B) 2/n
62. The domain of function 𝑓(𝑥) =
(C) 4/n (D) 1/2n
𝑥−1 1
√log 0.4 ⁡ (
𝑥+5
) + 𝑥 2 −36 is
55. One bag contains 3 white and 2 black balls,
another bag contains 2 white and 3 black balls. (a) (−∞, 0) − {−6} (b) (0, ∞) − {1,6}
Two balls are drawn from the first bag and put it (c) (1, ∞) − {6, −6} (d) [1, ∞) − {6}
into the second bag and then a ball is drawn from
the second bag. What is the probability that it is 63. If the function 𝑓: [1, ∞) → [1, ∞) is defined by
white? 𝑓(𝑥) = 2𝑥(𝑥−1) , then 𝑓 −1 (𝑥) is
1
(a) (2) (1 − √1 + 4log 2 ⁡ 𝑥)
(A) 6/7 (B) 32/70 1
(b) 2
(C) 3/10 (D) 1/70 1
(c) 2 (1 + √1 + 4log 2 ⁡ 𝑥)
2 1 𝑥(𝑥+1)
56. ∫0 |𝑥 2 + 2𝑥 − 3|𝑑𝑥 is equal to (d) (2)
(a) 4 (b) 6
𝑑𝑦
(c) 3 (d) 2 64. The solution of the differential equation 𝑑𝑥 +
1 = 𝑒 𝑥+𝑦 is

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(a) 𝑥𝑒 𝑥+𝑦 + 𝑦 = 𝑐 (b) 𝑥 + 𝑒 𝑥+𝑦 = 𝑐 1+𝑥 2 +𝑥 4 𝑑𝑦

72. If 𝑦 = 1+𝑥+𝑥 2 and 𝑑𝑥 = 𝑎𝑥 + 𝑏, then
(c) −𝑥𝑒 𝑥+𝑦 + 𝑦 = 𝑐 (d) 𝑥 + 𝑒 −(𝑥+𝑦) = 𝑐
(a) 𝑎 = 2, 𝑏 = 1 (b) 𝑎 = −2, 𝑏 = 1
65. The function (c) 𝑎 = 2, 𝑏 = −1 (d) 𝑎 = −2, 𝑏 = −1

𝜋𝑥 𝜋𝑥 73. lim−
√𝑥−𝑏−√𝑎−𝑏
, (𝑎 > 𝑏) is
𝑓(𝑥) = sin⁡ ( ) − cos⁡ ( ) is 𝑥→𝑎 (𝑥 2 −𝑎2 )
𝑛! (𝑛 + 1)! 1 1
(a) 4𝑎 (b) 𝑎 𝑎−𝑏
√
(a) 2(𝑛 + 1) ! 1
(c) 2𝑎 𝑎−𝑏
1
(d) 4𝑎 𝑎−𝑏
(b) periodic, with period 2(𝑛!) √ √
(c) non-periodic log⁡(1+𝑎𝑥)−log⁡(1−𝑏𝑥)
(d) periodic, with (𝑛 + 1) 74. The function 𝑓(𝑥) = is not
𝑥
defined at 𝑥 = 0. The value which should be
66. If Δ(𝑥) = assigned to 𝑓 at 𝑥 = 0, so that it is continuous at
1 cos⁡ 𝑥 1 − cos⁡ 𝑥 𝑥 = 0, is
|1 + sin⁡ 𝑥 cos⁡ 𝑥 1 + sin⁡ 𝑥 − cos⁡ 𝑥 | then (a) 𝑎 − 𝑏 (b) 𝑎 + 𝑏
𝜋
sin⁡ 𝑥 sin⁡ 𝑥 1 (c) log⁡ 𝑎 + log⁡ 𝑏 (d) None of the above
∫0 Δ(𝑥)𝑑𝑥 equals
2

1 −1 75. The values of 𝑎 and 𝑏 so that the function

(a) − (b)
4 2 𝑥 + 𝑎√2sin⁡ 𝑥, 0 ≤ 𝑥 < 𝜋/4
(c) 0 (d) None of these 𝑓(𝑥) = {2𝑥cot⁡ 𝑥 + 𝑏, 𝜋/4 ≤ 𝑥 ≤ 𝜋/2 is
1 𝑎cos⁡ 2𝑥 − 𝑏sin⁡ 𝑥, 𝜋/2 < 𝑥 ≤ 𝜋
67. If 𝑓(𝑥) = 64𝑥 3 + 𝑥 3 and 𝑎, 𝑏 are the roots of
continuous for 𝑥 ∈ [0, 𝜋], are
1 𝜋 𝜋 𝜋 𝜋
4𝑥 + = 3, then (a) 𝑎 = 6 , 𝑏 = − 6 (b) 𝑎 = − 6 , 𝑏 = 12
𝑥
(a) 𝑓(𝑎) = 𝑓(𝑏) (b) 𝑓(𝑎) = 11 𝜋
(c) 𝑎 = 6 , 𝑏 = − 12
𝜋
(d) None of these
(c) 𝑓(𝑏) = 8 (d) None of these
𝑥 4 −1 𝑥 3 −𝑘 3 76. If the function f be given by 𝑓(𝑥) = 𝑥 3 − 3𝑥 +
68. If lim = lim 2 2, then 𝑘 is 3, then
𝑥→1 𝑥−1 𝑥→𝑘 𝑥 −𝑘
4
(a) 3 (b) 8
3 I. 𝑥 = ±2 are the only critical points for local
3 8
maxima or local minima.
(c) (d) II. x = 1 is a point of local minima.
2 3
III. local minimum value is 2 .
𝑥 2 +𝑥+1
69. If lim𝑥→∞ ( 𝑥+1 − 𝑎𝑥 − 𝑏) = 4, then IV. local maximum value is 5 .
(a) Only I and II are true
(a) 𝑎 = 1, 𝑏 = 4 (b) 𝑎 = 1, 𝑏 = −4 (b) Only II and III are true
(c) 𝑎 = 2, 𝑏 = −3 (d) 𝑎 = 2, 𝑏 = 3 (c) Only I, II and III are true
(d) Only II and IV are true
70. Let 𝑓 be a twice differentiable function such
that 𝑓 ′′ (𝑥) = −𝑓(𝑥) and 𝑓 ′ (𝑥) = 𝑔(𝑥). If ℎ(𝑥) = 77. If 𝑓(𝑥) = 2𝑥 3 − 21𝑥 2 + 36𝑥 − 30, then which
{𝑓(𝑥)}2 + {𝑔(𝑥)}2 , where ℎ(5) = 11. Find ℎ(10). one of the following is correct
(a) 0 (a) 𝑓(𝑥) has minimum at 𝑥 = 1
(b) 9 (b) 𝑓(𝑥) has maximum at 𝑥 = 6
(c) 11 (c) 𝑓(𝑥) has maximum at 𝑥 = 1
(d) None of these (d) 𝑓(𝑥) has no maxima or minima

71. Let 𝑓(𝑥) = 𝑥 + sin⁡ 𝑥, suppose 𝑔 denotes the 78. The maximum value of sin⁡ 𝑥(1 + cos⁡ 𝑥) will
be at the
inverse function of 𝑓. Then, find the value of 𝜋 𝜋
𝜋 1 (a) 𝑥 = 2 (b) 𝑥 = 6
𝑔′ ( 4 + 2). 𝜋
√ (c) 𝑥 = 3 (d) 𝑥 = 𝜋
(a) 2 + √2 (b) √2 − 2
(c) 2 − √2 (d) 2√2 79. The minimum value of 𝑓(𝑥) = |3 − 𝑥|+∣ 2 +
𝑥| + |5 − 𝑥 ∣ is

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(a) 0 (b) 7 5 3
(c) 8 (d) 10 (a) √ (b) √
3 5
3 5
80. Consider the function 𝑓(𝑥) = 𝑥 3 − 3𝑥 + 3 on (c) −5 (d) −3
the interval [−3,3/2]. Let 𝑀 = Max⁡ 𝑓(𝑥) and 𝑚 =
Min⁡ 𝑓(𝑥) on [−3,3/2]. Then 88. The non-zero vectors 𝐚, 𝐛 and 𝐜 are related by
(A) 𝑀 = 15, 𝑚 = 5 (B) 𝑀 = 5, 𝑚 = −15 𝐚 = 8𝐛 and 𝐜 = −7𝐛 angle between 𝐚 and 𝐜 is
𝜋 𝜋
(C) 𝑀 = 15, 𝑚 = −5 (D) 𝑀 = −5, 𝑚 = −15 (a) 4 (b) 2
81. The differential equation corresponding to the (c) 𝜋 (d) 0
family of curves 𝑦 = 𝑒 𝑥 (𝑎𝑥 + 𝑏) is
𝑑2 𝑦 𝑑𝑦 𝑑2 𝑦 𝑑𝑦 ˆ , 2𝐢ˆ + 3𝐣ˆ − 6𝐤
89. If the vectors 6𝐢ˆ − 2𝐣ˆ + 3𝐤 ˆ and
(a) 𝑑𝑥 2 + 2 𝑑𝑥 − 𝑦 = 0 (b) 𝑑𝑥 2 − 2 𝑑𝑥 + 𝑦 = 0
ˆ
3𝐢ˆ + 6𝐣ˆ − 2𝐤 form a triangle, then it is
𝑑2 𝑦 𝑑𝑦 𝑑2 𝑦 𝑑𝑦
(c) 𝑑𝑥 2 + 2 𝑑𝑥 + 𝑦 = 0 (d) 𝑑𝑥 2 − 2 𝑑𝑥 − 𝑦 = 0 (a) right angled (b) obtuse angled
(c) equilateral (d) isosceles
82. The order and degree of the differential
2 90. If position vectors of a point 𝐴 is 𝐚 + 2𝐛 and 𝐚
𝑑𝑦 3 𝑑3 𝑦
equation (1 + 3 𝑑𝑥 ) = 4 𝑑𝑥 3 are divides 𝐴𝐵 in the ratio 2: 3, then the position
2 vector of 𝐵 is
(a) (1, 3) (b) (3,1)
(a) 2𝐚 − 𝐛 (b) 𝐛 − 2𝐚
(c) (3,3) (d) (1,2) (c) 𝐚 − 3𝐛 (d) 𝐛
83. The differential equation of the family of ˆ and 𝐛 = 𝐢ˆ + 3𝐣ˆ − 4𝐤
91. If 𝐚 = 2𝐢ˆ + 𝐣ˆ − 8𝐤 ˆ , then the
curves 𝑥 2 + 𝑦 2 − 2𝑎𝑦 = 0, where 𝑎 is arbitrary magnitude of 𝐚 + 𝐛 is equal to
constant, is 13
𝑑𝑦 (a) 13 (b) 3
(a) (𝑥 2 − 𝑦 2 ) 𝑑𝑥 = 2𝑥𝑦 3 4
𝑑𝑦 (b) 13 (d) 13
(b) 2(𝑥 + 𝑦 2 ) 𝑑𝑥 = 𝑥𝑦
2

𝑑𝑦
(c) 2(𝑥 2 − 𝑦 2 ) 𝑑𝑥 = 𝑥𝑦 92.The angle between the lines 2𝑥 = 3𝑦 = −𝑧 and
𝑑𝑦 6𝑥 = −𝑦 = −4𝑧 is
(d) (𝑥 2 + 𝑦 2 ) = 2𝑥𝑦
𝑑𝑥 (a) 30∘ (b) 45∘
∘
𝜋 𝑑𝑥
(c) 60 (d) 90∘
84. ∫0 1+2sin2 ⁡ 𝑥
is equal to
𝜋
(a) 3 (b) 3
𝜋 93. A line makes the same angle 𝜃 with 𝑋-axis and
𝜋
√3 𝑍-axis. If the angle 𝛽, which it makes with 𝑌-axis,
(c) 3 (d) 0 is such that sin2 ⁡ 𝛽 = 3sin2 ⁡ 𝜃, then the value of
√
cos2 ⁡ 𝜃 is
4𝜋 1 2
85. ∫0 |sin⁡ 𝑥|𝑑𝑥 is equal to (a) 5 (b) 5
(a) 0 (b) 2 (c) 5
3
(d) 3
2
(c) 4 (d) 8

86. The distance of the point (1,0,2) from the 94. Two sides of a parallelogram are along the
𝑥−2 𝑦+1 𝑧−2 lines, 𝑥 + 𝑦 = 3 and 𝑥 − 𝑦 + 3 = 0. If its diagonals
point of intersection of the line 3 = 4 = 12 intersect at ( 2, 4), then one of its vertex is
and the plane 𝑥 − 𝑦 + 𝑧 = 16 is (a) (3,6) (b) (2,6)
(a) 2√14 (b) 8 (c) (2,1) (d) (3,5)
(c) 3√21 (d) 13
95. A straight line through the origin 𝑂 meets the
87. If an angle between the line,
𝑥+1 𝑦−2
= 1 = −2
𝑧−3 parallel lines 4𝑥 + 2𝑦 = 9 and 2𝑥 + 𝑦 + 6 = 0 at
2
2√2
points 𝑃 and 𝑄 respectively. Then, the point 𝑂
and the plane, 𝑥 − 2𝑦 − 𝑘𝑧 = 3 is cos−1 ⁡ ( 3 ), divides the segment 𝑃𝑄 in the ratio
then value of 𝑘 is (a) 1: 2 (b) 3: 4
(c) 2: 1 (d) 4: 3

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96. Orthocentre of triangle with vertices a. unique solution

(0,0), (3,4) and (4,0) is b. infinitely many solutions
5
(a) (3, 4) (b) (3,12) c. no solutions
3
d. only finite number of solutions
(c) (3, 4) (d) (3,9) 1 0 0
106.Let [ 4 1 0] and 𝐼 be the identity matrix of
97. The locus of the mid-point of a chord of the 16 4 1
circle 𝑥 2 + 𝑦 2 = 4 which subtends a right angle at order 3. If 𝑄 = [𝑞𝑖𝑗 ] is a matrix such that 𝑃50 −
𝑞31 +𝑞32
the origin, is 𝑄 = 1, then equals
𝑞21
(a) 𝑥 + 𝑦 = 2 (b) 𝑥 2 + 𝑦 2 = 1 a. 52 b. 103
(c) 𝑥 2 + 𝑦 2 = 2 (d) 𝑥 + 𝑦 = 1 c. 201 d. 205
98. If the line 𝑥 − 1 = 0 is the directrix of the 3 1
107.If 𝐴 = [ ] then 𝐴2 − 5𝐴 is equal to
parabola 𝑦 2 − 𝑘𝑥 + 8 = 0, then one of the values −1 2
of 𝑘 is a. I b. -I
1 c. 7I d. -7 I
(a) 8 (b) 8
1 cosec2𝜃−sec2 ⁡ 𝜃
(c) 4 (d) 4
1 108.If tan⁡ 𝜃 = , then (cosec2𝜃+sec2 ⁡ 𝜃) =
√7
a. ½ b. 3/4
2
99. If 𝑥 + 𝑦 = 𝑘 is normal to 𝑦 = 12𝑥, then 𝑘 is c. 5/4 d. 2
(a) 3 (b) 9
(c) -9 (d) -3 109.If sin⁡ 𝐴 − √6cos⁡ 𝐴 = √7cos⁡ 𝐴, then cos⁡ 𝐴 +
√6sin⁡ 𝐴 is equal to
100. If 5𝑥 + 9 = 0 is the directrix of the hyperbola a. √6sin⁡ A b. √7sin⁡ A
16𝑥 2 − 9𝑦 2 = 144, then its corresponding focus is c. √6cos⁡ A d. √7cos⁡ A
5
(a) (− , 0) (b) (−5,0) 110.If 𝑆𝑛 = cos𝑛 ⁡ 𝜃 + sin𝑛 ⁡ 𝜃, then the value of
3
(c)
5
(3 , 0) (d) (5,0) 3𝑆4 − 2𝑆6 is given by
a. 4 b. 0
c. 1 d. 7
101.If 𝛼, 𝛽, 𝛾 are the roots of 𝑥 3 + 𝑎2 𝑥 + 𝑏 = 0 then
𝛼 𝛽 𝛾 111.If sin⁡ 𝑥 + cosec𝑥 = 2, then sin𝑛 ⁡ 𝑥 + cosec 𝑛 𝑥 is
the value of |𝛽 𝛾 𝛼 | is equal to
𝛾 𝛼 𝛽 a. 2 b. 2n
n−1
a. −a3 b. 𝑎3 − 3𝑏 c. 2 d. 2n−2
3
c. a d. 0 112.If 8sin⁡ 𝜃 = 4 + cos⁡ 𝜃, then one of the values of
a2 2ab b2 sin⁡ 𝜃 is
102.The value of | b2 a2 2ab| = ? 5
a. 11
5
b. 13
2ab b2 a2 5 5
2 2
a. (𝑎 + 𝑏 ) 3
b. (𝑎 + 𝑏 3 )2
3 c. 7 d. 9
4 4 2
c. (𝑎 + 𝑏 ) d. (𝑎2 + 𝑏 2 )4 2𝑧tan⁡ 𝜃
113.If 𝑥sin⁡ 𝜃 = 𝑦cos⁡ 𝜃 = , then 4𝑧 2 (𝑥 2 +
1 log b ⁡ a 1−tan2 ⁡ 𝜃
103.If A = [ ] then |A| is equal to 𝑦2) =
log a ⁡ b a
a. 0 b. log a ⁡ b a. (𝑥 2 + 𝑦 2 )3 b. (𝑥 2 − 𝑦 2 )3
2 2 2
c. -1 d. log b ⁡ a c. (𝑥 − 𝑦 ) d. (𝑥 2 + 𝑦 2 )2
1
104.If the entries in a 3 × 3 determinant are either 114.If tan⁡ 𝜃 ⋅ tan⁡(120∘ − 𝜃) ⋅ tan⁡(120∘ + 𝜃) = ,
√3
0 or 1 , then the greatest value of this determinant then 𝜃 =
is n𝜋 𝜋
a. 3 + 18 , n ∈ Z b.
n𝜋 𝜋
+ 12 , n ∈ Z
a. 1 b. 2 n𝜋 𝜋
3
n𝜋 𝜋
c. 3 d. 9 c. 12 + 12 , n ∈ Z d. 3
+ 6,n ∈ Z

105. The system of equations 𝑥 + 4𝑦 − 3𝑧 = 3, 𝑥 −

𝑦 + 7𝑧 = 11, 2𝑥 + 8𝑦 − 6𝑧 = 7 have

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 MATHS PRACTICE SET - 5

115.The number of values of 𝛼 ∈ [−𝜋, 𝜋] for which

𝜋 𝜋 1
sin2 ⁡ ( + 𝛼) − sin2 ⁡ ( − 𝛼) = is
8 8 2√2
a. 1 b. 2
c. 3 d. 4
3𝑥+1 𝐴 𝐵 𝐴
116. (𝑥−1)(𝑥+3) = 𝑥−1 + 𝑥+3, then sin−1 ⁡ 𝐵 =
𝜋 𝜋
a. 4 b. 2
𝜋 𝜋
c. 3 d. 6
2
117.The value of cos⁡ (sin−1 ⁡ (3)) is equal to
√3 5
a. 5
b. 3
5 √5
c. d.
√3 3

118.The domain of the function 𝑓(𝑥) =

𝑥+5
sin−1 ⁡ ( ) is
2
a. [−1,1] b. [2,3]
c. [3,7] d. [−7, −3]
119.If the angled of a triangle are in the ratio
1: 1: 4, then the ratio of the perimeter of the
triangle to its largest side is
a. 3: 2 b. √3 + 2: √2
c. √3 + 2: √3 d. √2 + 2: √3
120.From an aeroplane flying, vertically above a
horizontal road, the angles of depression of two
consecutive stones on the same side of the
aeroplane are observed to be 30∘ and 60∘
respectively. The height at which the aeroplane
is flying in km is
4 √3
a. b.
√3 2
2
c. d. 2
√3

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