MATHS PRACTICE SET – 1

9. sin⁡ 765∘ is equal to

1.Let 𝜔 be a complex number such that 2𝜔 + 1 = (a) 1 (b) 0
1 1 1 √3 1
(c) (d)
z where z = √−3 ⋅ If |1 −𝜔2 − 1 𝜔2 | = 3k, then 2 √2
1 𝜔2 𝜔7 10.The value of the expression
k is equal to sin2 ⁡ 𝑦 1+cos⁡ 𝑦 sin⁡ 𝑦
(a) z (b) -1 1 − 1+cos⁡ 𝑦 + sin⁡ 𝑦
− 1−cos⁡ 𝑦 is equal to
(c) 1 (d) -z (a) siny (b) cosy
(c) 0 (d) 1
3 𝑥 3 2
2.If | |=| | then 𝑥 is equal to
𝑥 1 4 1 11.If P = {𝜃: sin⁡ 𝜃 − cos⁡ 𝜃 = √2cos⁡ 𝜃} and Q =
(a) 8 (b) 4 {𝜃: sin⁡ 𝜃 + cos⁡ 𝜃 = √2sin⁡ 𝜃} be two sets. Then
(c) ±2√2 (d) 2 (a) P ⊂ Q and Q − P ≠ 𝜙 (b) Q ⊄ P
(c) P = Q (d) P ⊄ Q
1 2 4
3.If (1 3 5) is singular, then the value of 𝑎 is (1−sin⁡ 𝜃)(1+cos⁡ 𝜃)
12. √(1−cos⁡ 𝜃)(1+sin⁡ 𝜃) =
1 4 𝑎
(a) a = −6 (b) 𝑎 = 5
(a) (sec⁡ 𝜃 + tan⁡ 𝜃)(cosec𝜃 − cot⁡ 𝜃)
(c) a = −5 (d) 𝑎 = 6
(b) (sec⁡ 𝜃 − tan⁡ 𝜃)(cosec𝜃 + cot⁡ 𝜃)
3𝑖 −9𝑖 1 (c) (sec⁡ 𝜃 + tan⁡ 𝜃)(cosec𝜃 + cot⁡ 𝜃)
4.If | 2 9𝑖 −1| = 𝑥 + 𝑖𝑦, then (d) (sec⁡ 𝜃 − tan⁡ 𝜃)(cos⁡ 𝜃 + tan⁡ 𝜃)
10 9 𝑖 13.If 𝑥 = 5 + 2sec⁡ 𝜃 and 𝑦 = 5 + 2tan⁡ 𝜃, then (𝑥 −
(a) 𝑥 = 1, 𝑦 = 1 (b) 𝑥 = 0, 𝑦 = 1 5)2 −(𝑦 − 5)2 is equal to
(c) 𝑥 = 1, 𝑦 = 0 (d) 𝑥 = 0, 𝑦 = 0 (a) 3 (b) 1
(c) 0 (d) 4
5. The number of real values of 𝜆 for which the
system of linear equations 14.The number of principal solutions of tan⁡ 2𝜃 =
2𝑥 + 4𝑦 − 𝜆𝑧 = 0,4𝑥 + 𝜆𝑦 + 2𝑧 = 0, 𝜆𝑥 + 2𝑦 + 1 is
2𝑧 = 0 has infinitely many solutions, is (a) One (b) Two
(a) 0 (b) 1 (c) Three (d) Four
(c) 2 (d) 3 15.If 0 ≤ 𝑥 < 2π, then the number of real values of
6.Let 𝑃 be the set of all non-singular matrices of 𝑥, which satisfy the equation cos⁡ 𝑥 + cos⁡ 2𝑥 +
order 3 over R and Q be the set of all orthogonal cos⁡ 3𝑥 + cos⁡ 4𝑥 = 0, is
matrices of order 3 over R. Then (a) 3 (b) 5
(a) P is proper subset of Q c. 7 d. 9
(b) Q is proper subset of P 16.The range of sec −1 ⁡ 𝑥 is
(c) Neither P is proper subset of Q nor Q is proper 𝜋
(a) [0, 𝜋] (b) [0, 𝜋] − { 2 }
subset of P
−𝜋 𝜋 −𝜋 𝜋
(d) P ∩ Q = 𝜙, the void set (c) [ , ] (d) ( , )
2 2 2 2
7.Let 𝑃 and 𝑄 are matrices such that 𝑃𝑄 = 𝑄 and 3
𝑄𝑃 = 𝑃, then 𝑃2 + 𝑄 2 = 17.The value of cos⁡ (tan−1 ⁡ (4)) is
(a) P (b) Q (a) 5
4
(b) 5
3
(c) P + Q (d) P − Q 3 2
𝜋 𝜋
(c) 4 (d) 5
8.If 𝑥 ∈[0, 2 ] , 𝑦
∈ and sin⁡ 𝑥 + cos⁡ 𝑦 = 2,
[0, 2 ]
1 4
then the value of 𝑥 + 𝑦 is equal to 18.If 𝑥 = sin⁡(2tan−1 ⁡ 2) and 𝑦 = sin⁡ (2 tan−1 ⁡ 3),
(a) 2𝜋 (b) 𝜋 then
𝜋 𝜋
c. 4 d. 2 (a) 𝑥 > 𝑦 (b) 𝑥 = 𝑦
c. 𝑥 = 0 = 𝑦 d. x < y

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𝐶 𝐶
19.In △ 𝐴𝐵𝐶, (𝑎 − 𝑏)2 cos 2 ⁡ 2 + (𝑎 + 𝑏)2 sin2 ⁡ 2 = 28. For any two complex numbers 𝑧1 and 𝑧2 and
any real numbers 𝑎 and 𝑏; |(𝑎𝑧1 − 𝑏𝑧2 )|2 +
(a). 𝑏 2 (b). 𝑐 2
2 |(𝑏𝑧1 + 𝑎𝑧2 )|2 =
(c). a (d). 𝑎2 + 𝑏 2 + 𝑐 2
(a) (𝑎2 + 𝑏 2 )(|𝑧1 | + |𝑧2 |)
20.A man is walking towards a vertical pillar in a (b) (𝑎2 + 𝑏 2 )(|𝑧1 |2 + |𝑧2 |2 )
straight path, at a uniform speed. At a certain (c) (𝑎2 + 𝑏 2 )(|𝑧1 |2 − |𝑧2 |2 )
point A on the path, he observes that the angle of (d) None of these
elevation of the top of the pillar is 30∘ . After
walking for 10 minutes from A in the same 1 𝑖 √3
334
direction, at a point B , he observes that the angle 29. If 𝑖 = √−1, then 4 + 5 (− 2 + 2
) +
of elevation of the top of the pillar is 60∘ . Then the 1 𝑖√3
365

time taken (in minutes) by him, from B to reach 3 (− + ) is equal to

2 2
the pillar is (a) 1 − 𝑖√3 (b) −1 + 𝑖√3
(a). 6 (b). 10 (c) 𝑖√3 (d) −𝑖√3
c. 20 d. 5
𝑎+𝑏𝜔+𝑐𝜔2 𝑎+𝑏𝜔+𝑐𝜔2
21 Write the set builder form of 𝐴 = {−1,1) 30. The value of 𝑏+𝑐𝜔+𝑎𝜔2 + 𝑐+𝑎𝜔+𝑏𝜔2 will be
(a). A = {x: x is an integer } (a)1 (b)- 1
(b). A = {x: x is a root of the equation x 2 + 1 = 0} (c) 2 (d) -2
(c). A = {x: x is a real number }
d. A = {x: x is a root of the equation x 2 = 1} 31. The cube roots of unity when represented on
the Argand plane form the vertices of an
22. Which of the following set is an empty set? (a)Equilateral triangle
(a). {𝑥 ∣ 𝑥 is a real number and 𝑥 2 − 1 = 0} (b) Isosceles triangle
(b). {𝑥 ∣ 𝑥 is a real number and 𝑥 2 + 3 = 0} (c)Right angled triangle
(c). {𝑥 ∣ 𝑥 is a real number and 𝑥 2 − 9 = 0} (d) None of these
(d). {x ∣ x is a real number and x 2 = x + 2}
23.Which of the following set is empty? 32. If 𝜔 is a complex cube root of unity, then
(a). {𝑥 ∈ 𝑅 ∣ 𝑥 2 + 𝑥 + 1 = 0}
225 + (3𝜔 + 8𝜔2 )2 + (3𝜔2 + 8𝜔)2 =
(b). {𝑥 ∈ 𝑅 ∣ 𝑥 2 = 9 and 2𝑥 = 6}
(c). {𝑥 ∈ 𝑅 ∣ 𝑥 + 4 = 4} (a) 72 (b) 192
(d). {𝑥 ∈ 𝑅 ∣ 2𝑥 + 1 = 3} (c) 200 (d) 248
24. The set A = {x ∣ x is a real number and x 2 = 16
and 2x = 6} is equal to 33. If the product of the roots of the equation
(a). {4} (b). {3} 2𝑥 2 + 6𝑥 + 𝑎2 + 1 = 0 is −𝛼, then the value of 𝛼
(c). 𝜙 (d). None of these will be
(a) -1 (b) 1
25. The caretsian product A × A has 9 elements (c) 2 (d) -2
among which two elements are found (−1,0) and
34. The equation log 𝑒 ⁡ 𝑥 + log 𝑒 ⁡(1 + 𝑥) = 0 can be
(0,1), then set A ?
written as
(a). {1,0} (b). {1, −1,0}
(a) 𝑥 2 + 𝑥 − 𝑒 = 0 (b) 𝑥 2 + 𝑥 − 1 = 0
(c). {0, −1} (d). {1, −1} 2
(c) 𝑥 + 𝑥 + 1 = 0 (d) 𝑥 2 + 𝑥𝑥 − 𝑒 = 0
26. For non-empty sets A and B , if A ⊂ B, then
(𝐴 × 𝐵) ∩ (𝐵 × 𝐴) equals 35. ⁡{𝑥 ∈ 𝑅: |𝑥 − 2| − 𝑥 2 } −
(a). 𝐴 ∩ 𝐵 (b). A × A (a) {−1,2} (b) {1,2}
c. B × B d. None of these (c) {−1, −2} (d) {1, −2}

27. If A and B have n elements in common, then 36. If the roots of the given equation (𝑚𝑥 2 +
the number of elements common to A × B and 1)𝑥 2 + 2𝑎𝑚𝑥 + 𝑎2 − 𝑏 2 − 0 be equal, then
B × A is (a) 𝑎2 + 𝑏 2 (𝑚2 + 1) = 0
(a). 0 (b). n (b) 𝑏 2 + 𝑎2 (𝑚2 + 1) − 0
(c). 2 n (d). n2 (c) 𝑎2 − 𝑏 2 (𝑚2 + 1) = 0
(d) 𝑏 2 − 𝑎2 (𝑚2 + 1) = 0

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2𝑛 𝑛
37. If 𝑃(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and 𝑄(𝑥) = −𝑎𝑥 2 + (a) 𝑛+1 (b) 𝑛+1
𝑑𝑥 + 𝑐 where 𝑎𝑐 ≁ 0, then 𝑃(𝑥) ⋅ 𝑄(𝑥) = 0 has at 𝑛+1 𝑛+1
least (c) 2𝑛
(d) 𝑛
(a) Four real roots
(b) Two real roots 46. If the sum of a certain number of terms of the
(c) Four imaginary roots A.P. 25,22,19,
(d) None of these ⁡ is 116 . then the last term is
(a) 0 (b) 2
38. The number of non-zero terms in the (c) 4 (d) 6
expansion of (1 + 3√2𝑥)𝑎 + (1 − 3√2𝑥)𝑎 is
(a) 9 (b) 0 47. The value of 𝑟 for which the coefficients of
(c) 5 (d) 10 (𝑟 − 5)th and (3r + 1)th terms in the expansion of
(1 + x)12 are equal, is
1 10
39. 6𝑎𝑡 term in expansion of (2𝑥 2 − ) is a. 4 b. 9
3𝑥 2
4580 896 c. 12 d. None of these
(a) 17
(b) − 27
5580 48. The number of all possible positive integral
(c) 17
(d) None of these values of 𝛼 for which the roots of the quadratic
40. If the ratio of the coefficient of third and equation, 6x 2 − 11x + 𝛼 = 0 are rational numbers
1 𝑥 is :
fourth term in the expansion of (𝑥 − 2𝑥) is 1: 2, (A) 4 (B) 2
then the value of 𝑛 will be (C) 5 (D) 3
(a) 18 (b) 16
(c) 12 (d) -10 49. If |𝑧 2 − 1| = |𝑧 2 | + 1, then 𝑧 lies on
(a) a circle
41. Sum of all two digit numbers which when (b) a parabola
divided by 4 yield unity as remainder is (c) an ellipse
(a) 1200 (b) 1210 (d) none of these
(c) 1250 (d) None of these.
50. Which one of the following relations on 𝑅 is
42. If 𝑛 arithmetic means are inserted between 1 an equivalence relation
and 31 such that the ratio of the first mean and
(a) 𝑎𝑅1 𝑏 ⇔ 𝑎| ≠ 𝑏|
𝑛𝑡ℎ mean is 3: 29, then the value of 𝑛 is
(b) ⁡𝑎𝑅2 𝑏 ⇔ 𝑎 ≥ 𝑏
(a) 10 (b) 12
(c) ⁡𝑎𝑅3 𝑏 ⇔ 𝑎 divides 𝑏
(c) 13 (d) 14
(d) 𝑎𝑅4 𝑏 ⇔ 𝑎 < 𝑏
43. If the sum of first 𝑛 even natural numbers is 51. The domain of the definition of the function
equal to 𝑘 times the sum of first 𝑛 odd natural 1
𝑓(𝑥) = 4−𝑥2 + log10 ⁡(𝑥 3 − 𝑥) is
numbers, then 𝑘 =
1 𝑛−1 (a) (−1,0) ∪ (1,2) ∪ (3, ∞)
(a) 𝑛 (b) 𝑛 (b) (−2, −1) ∪ (−1,0) ∪ (2, ∞)
𝑛+1 𝑛+1
(c) (d) (c) (−1,0) ∪ (1,2) ∪ (2, ∞)
2𝑛 𝑛
(d) (1,2) ∪ (2, ∞)
44. If the first, second and last term of an A.P are
𝑎, 𝑏 and 2𝑎 respectively, then its sum is 52. If 𝑓(𝑥) = (𝑎𝑥 2 + 𝑏)3 , then the function 𝑔 such
𝑎𝑏 𝑎𝑏 that 𝑓{𝑔(𝑥)} = 𝑔{𝑓(𝑥)} is given by
(a) 2(𝑏−𝑎) (b) 𝑏−𝑎 1/2
𝑏−𝑥 1/3
3𝑎𝑏 (a) 𝑔(𝑥) = ( )
(c) 2(𝑏−𝑎)
(d) None of these 𝑎
1
(b) 𝑔(𝑥) = (𝑎𝑥 2 +𝑏)3
45. If, S1 is the sum of an arithmetic progression (c) 𝑔(𝑥) = (𝑎𝑥 2 + 𝑏)1/3
of ' n ' odd number of terms and 𝑆2 the sum of the 𝑥 1/3 −𝑏
1/2
S (d) 𝑔(𝑥) = ( )
terms of the series in odd places, then S1 = 𝑎
2

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53. Find lim 𝑓(𝑥) and lim 𝑓(𝑥), where 𝑓(𝑥) = 62. If the function 𝑓(𝑥) = 2𝑥 3 − 9𝑎𝑥 2 + 12𝑎2 𝑥 +
𝑥→0 𝑥→1
2𝑥 + 3,𝑥 ≤ 0 1, where 𝑎 > 0, attains its maximum and
{ . minimum at 𝑝 and 𝑞 respectively, such that 𝑝2 =
3(𝑥 + 1),𝑥 > 0
(a) 3,5 (b) 3,6 𝑞, then 𝑎 is equal to
(c) 4,7 (d) 3, −6 (a) 3 (b) 1
(c) 2 (d) 1/2
54. The abscissa of the points, where the tangent
𝑑2 𝑦
to the curve 𝑦 = 𝑥 3 − 3𝑥 2 − 9𝑥 + 5 is parallel to 𝑋- 63. If 𝑦 = 𝐴𝑒 𝑚𝑥 + 𝐵𝑒 𝑛𝑥 , then the value of 𝑑𝑥 2 −
axis, are 𝑑𝑦
(a) 𝑥 = 0 and 0 (b) 𝑥 = 1 and -1 (𝑚 + 𝑛) 𝑑𝑥 + 𝑚𝑛𝑦 is equal to
(c) 𝑥 = 1 and -3 (d) 𝑥 = −1 and 3 (a) −𝑦 (b) 0
(c) 𝑚𝑛𝑦 (d) −𝑚𝑛𝑦
𝑥 3 +1
55. If lim𝑥→∞ [ 2 − (𝑎𝑥 + 𝑏)] = 2, then
𝑥 +1 𝑥 𝑥2 𝑥3
(a) 𝑎 = 1 and 𝑏 = 1 (b) 𝑎 = 1 and 𝑏 = −1 64. If 𝑓(𝑥) = |1 2𝑥 3𝑥 2 |, then 𝑓 ′ (𝑥) is equal to
(c) 𝑎 = 1 and 𝑏 = −2 (d) 𝑎 = 1 and 𝑏 = 2 0 2 6𝑥
(a) 6𝑥 2 (b) 6𝑥
56. The function 𝑥 𝑥 is increasing, when (c) 0 (d) None of these
1 1
(a) 𝑥 > 𝑒 (b) 𝑥 < 𝑒
(c) 𝑥 < 0 (d) for all real 𝑥 65. If 𝑓(𝑥) = cos⁡ 𝑥 ⋅ cos⁡ 2𝑥 ⋅ cos⁡ 4𝑥 ⋅ cos⁡ 8𝑥 ⋅
𝜋
cos⁡ 16𝑥, then the value of 𝑓 ′ ( 4 ) is
1+sin⁡ 𝑥
57. ∫ e𝑥 (1+cos⁡ 𝑥) 𝑑𝑥 is equal to (a) 1 (b) √2
𝑥 𝑒𝑥 1
𝑥
(a) 𝑒 sec⁡ + 𝐶 (b) +𝐶 (c) 2 (d) 0
2 1+cos⁡ 𝑥 √
𝑥
(c) e𝑥 tan⁡ 2 + 𝐶 (d) None of these
1 𝑥 3 +|𝑥|+1
66. ∫−1 𝑥 2 +2|𝑥|+1 𝑑𝑥 is equal to
58. If 𝑓(𝑥) = 𝑎𝑥 + 𝑏, where 𝑎 and 𝑏 are integers, (a) log⁡ 2 (b) 2log⁡ 2
𝑓(−1) = −5 and 𝑓(3) = 3, then 𝑎 and 𝑏 are equal 1
(c) log⁡ 2 (d) 4log⁡ 2
to 2
(a) 𝑎 = −3, 𝑏 = −1 (b) 𝑎 = 2, 𝑏 = −3
(c) 𝑎 = 0, 𝑏 = 2 (d) 𝑎 = 2, 𝑏 = 3 67. The maximum and minimum values of 𝑥 +
sin⁡ 2𝑥 on [0,2𝜋] is
1
59. The value of lim
𝑥 4 −4
is (a) 2𝜋 and 0 (b) 𝜋 and
2
𝑥→√2 𝑥 2 +3√2𝑥−8 𝜋
8 7 (c) 2 and -1 (d) None of these
(a) − 5 (b) 5
8 1 𝑑𝑥
(c) 5 (d) None of these 68. ∫0 𝑑𝑥 is equal to
𝑒 𝑥 +𝑒 −𝑥
𝜋
1/𝑥
(a) 𝜋/4 (b) tan−1 ⁡ 𝑒 −
𝜋 4
𝜋
60. Let 𝑓(𝑥) = {[tan⁡ ( 4 + 𝑥)]
, 𝑥 ≠ 0. For (c) tan−1 ⁡ 𝑒 (d) 4 tan−1 ⁡ 𝑒
𝑘, 𝑥=0
what value of 𝑘 is 𝑓(𝑥) continuous at 𝑥 = 0 ? 4
69. ∫0 |𝑥 − 1|𝑑𝑥 is equal to
(a) 1 (b) e 5 3
1 (a) 2 (b) 2
(c) (d) e2
𝑒 1
(c) (d) 5
2
𝑥+𝑦
61. If 𝑓(𝑥) + 𝑓(𝑦) = 𝑓 (1−𝑥𝑦) for ail 𝑥, 𝑦 ∈ (−1,1)
70. The value of ' 𝑎 ' for which the function 𝑓(𝑥) =
𝑓(𝑥) 1
and lim = 2, then 𝑓 ( ) and 𝑓 ′ (1) are 1 𝜋
𝑎sin⁡ 𝑥 + 3 sin⁡ 3𝑥 has an extremum at 𝑥 = 3 is
𝑥→0 𝑥 √3
𝜋 𝜋 1
(a) and 1 (b) and (a) 1 (b) -1
3 6 2
𝜋 𝜋 (c) 2 (d) 0
(c) 2 and 0 (d) 4 and √3

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𝑑𝑥 𝜋 𝜋
71. ∫ ⁡ 9𝑥 2 +6𝑥+5 is equal to (a) 2 (b) 6
𝜋 𝜋
(a) tan−1 ⁡ (
3𝑥+1
)+𝐶 (b) 6tan−1 ⁡ (
3𝑥+1
)+𝐶 (c) 4 (d) 3
2 2
3𝑥+1 1 3𝑥+1
(c) cot −1 ⁡ ( )+𝐶 (d) tan−1 ⁡ ( )+𝐶 1
2 6 2 80. If 5𝑓(𝑥) + 3𝑓 (𝑥) = 𝑥 + 2 and 𝑦 = 𝑥𝑓(𝑥), then
𝜋−𝑎 𝑑𝑦
72. Let 𝐼1 = ∫𝑎 𝑥𝑓(sin⁡ 𝑥)𝑑𝑥, 𝐼2 = (𝑑𝑥 ) is equal to
𝑥=1
𝜋−𝑎
∫𝑎 𝑓(sin⁡ 𝑥)𝑑𝑥, then 𝐼2 is equal to (a) 14 (b) 7/8
𝜋
(a) 2 𝐼1 (b) 𝜋𝐼1 (c) 1 (d) None of these
2
(c) 𝜋 𝐼1 (d) 2𝐼1 81. If the points (0,0), (2,2√3) and (𝑎, 𝑏) are the
vertices of an equilateral triangle, then (𝑎, 𝑏) is
73. The area of the region bounded by 𝑦 2 = (a) (0, −4) (b) (0,4)
9𝑥, 𝑥 = 2, 𝑥 = 4 and the 𝑋-axis in the first (c) (4,0) (d) (−4,0)
quadrant is
(a) 16 sq units (b) 4√2 sq units 82. Suppose 𝐴(1,1) and 𝐵(2, −3) are two points
(c) 4(4 − √2) sq units (d) 4(4 + √2) sq units and 𝐷 is a point on 𝐴𝐵 produced such that 𝐴𝐷 =
3𝐴𝐵. Then, coordinates of 𝐷 are
74. Order and degree of the differential equation (a) (3, −12) (b) (4, −11)
𝑑4 𝑦 𝑑3 𝑦 (c) (11,4) (d) (−11,7)
𝑑𝑥 4
+ sin⁡ (𝑑𝑥 3 ) = 0 are
(a) order = 4, degree = 1 83. The distance between the lines 5𝑥 − 12𝑦 +
(b) order = 3, degree = 1 65 = 0 and 10𝑥 − 24𝑦 − 39 = 0 is
(c) order = 4, degree = 0 √169 √169
(a) (b)
(d) order = 4, degree = not defined 5 2
(c) √169 (d) None of these
75. The differential equation of the family of
curves 𝑦 2 = 4𝑎(𝑥 + 𝑎) is 84. If the point (3, −4) divides the intercept of a
𝑑𝑦 𝑑𝑦
(a) 𝑦 2 = 4 (𝑥 + ) (b) 2𝑦 = 4𝑎
𝑑𝑦 line between the coordinate axes in the ratio 2: 3.
𝑑𝑥 𝑑𝑥 𝑑𝑥 Then, its equation is
𝑑2 𝑦 𝑑𝑦 2 𝑑𝑦 𝑑𝑦 2
(c) 𝑦 2 +( ) = 0 (d) 2𝑥 +𝑦( ) −𝑦 = 0 (a) 2𝑥 + 3𝑦 + 1 = 0 (b) 3𝑥 − 4𝑦 + 1 = 0
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
(c) 2𝑥 − 3𝑦 = 5 (d) 2𝑥 − 𝑦 = 10
𝑑𝑦
76. The differential equation 𝑦 𝑑𝑥 + 𝑥 = 𝐶 85. If one end of a diameter of the circle 𝑥 2 + 𝑦 2 −
represents family of 4𝑥 − 6𝑦 + 11 = 0 is (3,4), then find the coordinate
(a) hyperbolas (b) parabolas of the other end of the diameter.
(c) ellipses (d) circles (a) (2,1) (b) (1,2)
3
(c) (1,1) (d) None of these
𝑥
77. The value of ∫−1 [tan−1 ⁡ (𝑥 2 +1) +
𝑥 2 +1 86. Find the coordinates of a point on 𝑌-axis
tan−1 ⁡ ( 𝑥
)] 𝑑𝑥 is which are at a distance of 5√2 from the point
(a) 2𝜋 (b) 𝜋 𝑃(3, −2,5).
(c) 𝜋/2 (d) 𝜋/4 (a) (0, −6,0) or (0,2,0) (b) (0,6,0) or (0, −2,0)
(c) (0, −6,0) or (1,2,0) (d) None of these
78. Function 𝑓(𝑥) = 2𝑥 2 − log⁡ |𝑥|, 𝑥 ≠ 0
monotonically increases in 87. A point 𝑅 with 𝑥-coordinate 4 lies on the line
1 1 1 1
(a) (−∞, − ) ∪ (0, ) (b) (− , 0) ∪ ( , ∞) segment joining the points 𝑃(2, −3,4) and
2 2 2 2
1 𝑄(8,0,10). Find the coordinates of the point 𝑅.
(c) (−∞, 0) ∪ (2 , ∞) (d) None of these (a) (4, −2, −6) (b) (4,2,6)
(c) (4, −2,6) (d) None of these
79. Angle between the tangents to the curve 𝑦 =
𝑥 2 − 5𝑥 + 6 at the points (2,0) and (3,0) is 88. The angle between the lines whose direction
cosines satisfy the equation

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𝑙 + 𝑚 + 𝑛 = 0 and 𝑙 2 = 𝑚2 + 𝑛2 is (a)1/5 (b)1/4

𝜋 𝜋
(a) 3 (b) 4 (c)1/ 3 (d) 1/12
𝜋 𝜋
(c) 6 (d) 2 97. Three boys P, Q, R and three girls S, T, U
are to be arranged in a row for a group
89. The equation of a plane containing the line of photograph.
intersection of the planes 2𝑥 − 𝑦 − 4 = 0 and 𝑦 +
2𝑧 − 4 = 0 and passing through the point (1,1,0) What is the probability that boys and girls sit
is alternatively ?
(a) 𝑥 − 3𝑦 − 2𝑧 = −2 (b) 2𝑥 − 𝑧 = 2 (a)4/5 (b) 1/10
(c) 𝑥 − 𝑦 − 𝑧 = 0 (d) 𝑥 + 3𝑦 + 𝑧 = 4 (c)5/6 (d)1/7
90. The lines
𝑥+3
=
𝑦−1
=
𝑧−5
and
𝑥+1
=
𝑦−2
=
𝑧−5 98. Three boys P, Q, R and three girls S, T, U
−3 1 5 −1 2 5
are are to be arranged in a row for a group
(a) parallel (b) perpendicular photograph.
(c) coplanar (d) None of the above What is the probability that no two girls sit
together ?
91. The projection of the vector 𝐢ˆ − 𝐣ˆ on the vector
𝐢ˆ + 𝐣ˆ is (a)2/5 (b) 3/5
1
(a) 0 (b) 2 (c)1/18 (d)1/5
√
1
(c) 2 (d) None of these 99. Three boys P, Q, R and three girls S, T, U
are to be arranged in a row for a group
92. If position vector of a point 𝐴 is 𝐚 + 2𝐛 and photograph.
any point 𝑃(𝐚) divides 𝐀𝐁 in the ratio of 2: 3, then What is the probability that P and Q take the
position vector of 𝐵 is
two end positions ?
(a) 2𝐚 − 𝐛 (b) 𝐛 − 2𝐚
(c) 𝐚 − 3𝐛 (d) 𝐛 (a)1/15 (b) 7/15
(c)14/15 (d)11/45
93. The angle between two vectors 𝐚 and 𝐛 with
magnitudes √3 and 2 respectively, having 𝐚 ⋅ 𝐛 = 100. Three boys P, Q, R and three girls S, T, U
are to be arranged in a row for a group
√6 is
𝜋 𝜋 photograph.
(a) 4 (b) 2
(c)
𝜋
(d)
𝜋 What is the probability that Q and U sit
6 3 together ?
ˆ , 𝐛 = −𝐢ˆ + 2𝐣ˆ + 𝐤
94. If 𝐚 = 2𝐢ˆ + 2𝐣ˆ + 3𝐤 ˆ and 𝐜 = (a)2/3 (b) 1/4
3𝐢ˆ + 𝐣ˆ such that 𝐚 + 𝜆𝐛 is perpendicular to 𝐜, then (c)5/6 (d)1/3
the value of 𝜆 is
(a) 2 (b) 4 101. How many 4-letter words(with or
(c) 6 (d) 8 without meaning) containing two vowels can
be constructed using only the letters(without
95. If |𝐚| = 8, |𝐛| = 3 and |𝐚 × 𝐛| = 12, then value repetition) of the word 'LUCKNOW' ?
of 𝐚 ⋅ 𝐛 is
(a) 240 (b) 200
(a) 6√3 (b) 8√3
(c) 150 (d) 120
(c) 12√3 (d) None of these
102. Five letters word, having distinct letters
96. Three boys P, Q, R and three girls S, T, U are to be constructed using the letters of the
are to be arranged in a row for a group word 'EQUATION' so that each word contains
photograph. exactly three vowels and two consonants.
What is the probability that all three boys sit How many of them have all the vowels
together ? together ?

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 MATHS PRACTICE SET – 1

(a) 3600 (b) 1800 (a)5/14 (b) 5/16

(c) 1080 (d) 900 (c)5/18 (d)25/52
103. How many 5-letter words can be made 110. Six boys and six girls sit in a row. What is
from the letters of the word 'EQUATION' with the probability that the boys and girls sit
3 vowels and 2 consonants such that the two alternatively
consonants are never together ? (a) 1/462 (b) 1/924
(a) 1400 (b) 2160 (c) 1/2 (d) None of these
(c) 3600 (d) 7200
111. If ∑9𝑖=1 (𝑥𝑖 − 5) = 9 and ∑9𝑖=1 (𝑥𝑖 − 5)2 =
104. In how many ways can we arrange 6 45, then the standard deviation of the 9 items
different flowers in a circle? In how many 𝑥1 , 𝑥2 , … , 𝑥9 is
ways we can form a garland (a) 4 (b) 2
using these flowers? (c) 3 (d) 9
(a)50 (b) 60 112.The mean of a set of numbers is 𝑥‾. If each
(c) 70 (d) 80 number is multiplied by 𝜆, then the mean of
new set is
105. In how many ways can 5 boys and 5 girls (a) 𝑥‾ (b) 𝜆 + 𝑥‾
be seated at a round table so that no two girls (c) 𝜆𝑥‾ (d) None of these
are together?
113.The mean of discrete observations
(a) 5! x 5! (b) 5! × 4!
1 𝑦1 , 𝑦2 , … … , 𝑦𝑛 is given by
(c) 4! × 4! (d) 2 (5! ⁡ × ⁡4!⁡⁡) ∑𝑛
𝑖=1 𝑦𝑖 ∑𝑛
𝑖=1 𝑦𝑖
(a) (b) ∑𝑛
𝑛 𝑖=1 𝑖
106. A bag x contains 3 white balls and 2 ∑𝑛 𝑦𝑖 𝑓 𝑖 ∑𝑛
𝑖=1 𝑦𝑖 𝑓𝑖
black balls and another bag y contains 2 white (c) 𝑖=1
(d) ∑𝑛
𝑛 𝑖=1 𝑓𝑖
balls and 4 black balls. A bag and a ball out of
it are picked at random. The probability that 114.⁡𝑑𝑖 is the deviation of a class mark 𝑦𝑖 from
the ball is white, is ' 𝑎 ' the assumed mean and 𝑓𝑖 is the
1
frequency, if 𝑀𝑔 = 𝑥 + ∑ ⁡𝑓 (∑ ⁡ 𝑓𝑖 𝑑𝑖 ), then 𝑥 is
(a) 3/5 (b) 7/15 𝑖
(c) 1/2 (d) None of these (a) Lower limit
(b) Assumed mean
107. Two dice are thrown. The probability
(c) Number of observations
that the sum of the points on two dice will be
(d) Class size
7, is
115.The mean of a set of observation is 𝑥‾. If
(a)5/36 (b) 6/ 36
each observation is divided by 𝛼, 𝛼 ≠ 0 and
(c) 7/36 (d)8/36
then is increased by 10 , then the mean of the
108. The probability that A speaks truth is new set is
4/5, while this probability for B is 3/4. The 𝑥‾
(a) 𝛼
𝑥‾+10
(b) 𝛼
probability that they contradict each other 𝑥‾+10𝛼
when asked to speak on a fact (c) (d) 𝛼𝑥‾ + 10
𝛼
(a)4/5 (b)1/5 116.If the arithmetic mean of the numbers
(c) 7/20 (d) 3/20 𝑥1 , 𝑥2 , 𝑥3 , … … , 𝑥𝑛 is 𝑥‾, then the arithmetic
109. A bag 'A' contains 2 white and 3 red balls mean of numbers 𝑎𝑥1 + 𝑏, 𝑎𝑥2 + 𝑏, 𝑎𝑥3 +
and bag 'B' contains 4white and 5 red balls. 𝑏, … … . , 𝑎𝑥𝑛 + 𝑏, where 𝑎, 𝑏 are two constants
One ball is drawn at random from a randomly would be
chosen bag and is found to be red. The (a) 𝑥‾ (b) 𝑛𝑎𝑥‾ + 𝑛𝑏
probability that it was drawn from bag 'B' was (c) 𝑎𝑥‾ (d) 𝑎𝑥‾ + 𝑏

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117. If a variable takes the discrete values 𝛼 +

7 5 1 1
4, 𝛼 − 2 , 𝛼 − 2 , 𝛼 − 3, 𝛼 − 2, 𝛼 + 2 , 𝛼 − 2 , 𝛼 +
5(𝛼 > 0), then the median is
5 1
(a) 𝛼 − 4 (b) ⁡𝛼 − 2
5
(c) 𝛼 − 2 (d) 𝛼 + 4
118. The mode of the distribution

Marks 4 5 6 7 8

No. of students 6 7 10 8 3

(a) 5 (b) 6
(c) 8 (d) 10

119. If in a moderately asymmetrical

distribution mode and mean of the data are
6𝜆 and 9𝜆 respectively, then median is
(a) 8𝜆 (b) 7𝜆
(c) 6𝜆 (d) 5𝜆

120. What is the standard deviation of the

following series

Measurements 0-10 10- 20- 30-

20 30 40

Frequency 1 3 4 2

(a) 81 (b) 7.6

(c) 9 (d) 2.26

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