MATHS PRACTICE SET - 17

1. There are 100 students in a class. In an 9.If 𝛼 and 𝛽 are imaginary cube roots of unity,
examination, 50 of them failed in Mathematics, 1
then 𝛼 4 + 𝛽 4 + 𝛼𝛽 =
45 failed in Physics, 40 failed in Biology and 32
failed in exactly two of three subjects. Only one (a) 3 (b) 0
student passed in all the subjects. Then the (c) 1 (d) 2
number of students failing in all the three subjects 10.If 𝜔 is a complex cube root of unity, then
a. is 12 b. is 4 (1 − 𝜔)(1 − 𝜔2 ) (1 − 𝜔4 )(1 − 𝜔8 ) =
c. is 2 (a) 0 (b) 1
d. cannot determined from the given information (c) -1 (d) 9
2.If 𝑛(𝐴) = 8 and 𝑛(𝐴 ∩ 𝐵) = 2 then 𝑛((𝐴 ∩ 𝐵)′ ∩ 11.If 𝜔 is a cube root of unity, then the value of
𝐴) is equal to (1 − 𝜔 + 𝜔2 )5 + (1 + 𝜔 − 𝜔2 )5 =
a. 2 b. 4 (a) 16 (b) 32
c. 6 d. 8 (c) 48 (d) -32
3.25 people for programme A, 50 people for 12.If 𝑥 = 𝑎, 𝑦 = 𝑏𝜔, 𝑧 = 𝑐𝜔2 , where 𝜔 is a
𝑥 𝑦 𝑧
programme B, 10 people for both. So number of complex cube root of unity, then + + =
𝑎 𝑏 𝑐
employee employed for only A is (a) 3 (b) 1
a. 15 b. 20 (c) 0 (d) None of these
c. 35 d. 40
4.𝐴 and 𝐵 are subsets of universal set 𝑈 such that
𝑛(𝑈) = 800, n(A) = 300, n(B) = 400&n(A ∩ B) = 13. If the roots of the equation 𝑥 2 − 8𝑥 +
100. The number of elements in the set 𝐴𝑐 ∩ 𝐵𝑐 is (𝑎2 − 6𝑎) = 0 are real, then
a. 100 b. 200 (a) −2 < 𝑎 < 8 (b) 2 < 𝑎 < 8
c. 300 d. 400 (c) −2 ≤ 𝑎 ≤ 8 (d) 2 ≤ 𝑎 ≤ 8
5. A relation 𝑅 is defined on the set 𝑅 of all real
numbers such that for non-zero 𝑥, 𝑦 ∈ 𝑅, 𝑥𝑅𝑦 ⇒
14.The roots of the equation 𝑥 2 + 2√3𝑥 + 3 =
|𝑥 − 𝑦| < 1 then this relation is 0 are
a. Reflexive, symmetric but not transitive (a) Real and unequal
b. Reflexive, transitive but not symmetric (b) Rational and equal
c. Symmetric, transitive but not reflexive (c) Irrational and equal
d. An equivalence relation (d) Irrational and unequal
6. aRb ⇔ |a| ≤ b. Then R is
15.Roots of 𝑎𝑥 2 + 𝑏 = 0 are real and distinct if
a. reflexive b. symmetric
(a) 𝑎𝑏 > 0 (b) 𝑎𝑏 < 0
c. transitive d. equivalence
(c) 𝑎, 𝑏 > 0 (d) 𝑎, 𝑏 < 0
16.Roots of the equations 2𝑥 2 − 5𝑥 + 1 =
7. A relation defined on two natural numbers 𝑎
and 𝑏 is given by a Rb : a is divisible by b . The the 0, 𝑥 2 + 5𝑥 + 2 = 0 are
condition which holds true is (a) Reciprocal and of same sign
a. symmetric and transitive relation (b) Reciprocal and of opposite sign
b. reflexive, but not transitive relation (c) Equal in product
c. transitive but not symmetric relation (d) None of these
d. symmetric, reflexive and transitive relation 17. If 𝑎, 𝑏, 𝑐 ∈ 𝑄, then roots of the equation
8.Square of either of the two imaginary cube roots (𝑏 + 𝑐 − 2𝑎)𝑥 2 + (𝑐 + 𝑎 − 2𝑏)𝑥 + (𝑎 + 𝑏 −
of unity will be 2𝑐) = 0 are
(a) Real root of unity (a) Rational (b) Non-real
(b) Other imaginary cube root of unity (c) Irrational (d) Equal
(c) Sum of two imaginary roots of unity 18. In the expansion of (1 + 𝑥)3 , the sum of
(d) None of these the coefficient of the terms is
(a) 80 (b) 16

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(c) 32 (d) 64 (1) 3(3 − √2) (2) 6(3 − √2)

(3) 6(2 − √2) (4) 3(2 − √2)
19. What is the sum of the coefficients of (𝑥 2 −
𝑥 − 1)99 28. Let 𝑂 be the origin, the point 𝐴 be 𝑧1 = √3 +
(a) 1 (b) 0 2√2𝑖, the point 𝐵(𝑧2 ) be such that √3|𝑧2 | = |𝑧1 |
𝜋
(d) -1 (d) None of these and arg (𝑧2 ) = arg (𝑧1 ) + . Then
6
1 11
−2
20.If (𝑎 + 𝑏𝑥) = − 3𝑥 + ⋯, then (𝑎, 𝑏) = (1) area of triangle ABO is 3
4 √
(a) (2,12) (b) (−2,12) (2) ABO is an obtuse angled isosceles triangle
11
(c) (2, −12) (d) None of these (3) area of triangle ABO is 4
(4) ABO is a scalene triangle
𝑎 𝑎
21.The harmonic mean of 1−𝑎𝑏 and 1+𝑎𝑏 is equal to 29. Let 𝑇𝑟 be the 𝑟 th term of an A.P. If for some
𝑎 𝑎 1 1
(a) (b) 1−𝑎 2 𝑏2
𝑚, 𝑇𝑚 = 25 , 𝑇25 = 20, and 20 ∑25𝑟=1 𝑇𝑟 = 13, then
√1−𝑎2 𝑏2
1 2 m
(c) 𝑎 (d) 5 m ∑r=m Tr is equal to
1−𝑎 2 𝑏2
(1) 98 (2) 126
1 1
22.If first three terms of sequence 16 , 𝑎, 𝑏, 6 are in (3) 142 (4) 112
geometric series and the last three terms are in
30. The least value of 𝑛 for which the number of
harmonic series, then the values of 𝑎 and 𝑏 will be 3
1 1 1 integral terms in the Binomial expansion of (√7 +
(a) 𝑎 = − 4 , 𝑏 = 1 (b) a = 12 , 𝑏 = 9 12
√11)𝑛 is 183 , is :
(c) Both (a) and (b) (1) 2184 (2) 2196
(d) None of these (3) 2148 (4) 2172

31.How many numbers can be made with the help

23.If 𝑎, 𝑏, 𝑐 are in AP, 𝑝, 𝑞, 𝑟 are in HP and 𝑎𝑝, 𝑏𝑞, 𝑐𝑟 of the digits 0,1,2,3,4,5 which are greater than
𝑝 𝑟
are in GP, then + is equal to 3000 (repetition is not allowed)
𝑟 𝑝
a c a c (a) 180 (b) 360
(a) − (b) +
c a c a (c) 1380 (d) 1500
𝑏 𝑞 𝑏 𝑞
(c) 𝑞 +𝑏 (d) 𝑞
−𝑏
32.How many words can be made from the letters
24.If 𝑥 = 1 + 𝑎 + 𝑎2 + 𝑎3 + ⋯ ∞(|𝑎| < 1) and 𝑦 = of the word 'COMMITTEE'
1 + 𝑏 + 𝑏 2 + 𝑏 3 + ⋯ ∞(|𝑎| < 1), then 1 + 𝑎𝑏 + 9!
(a) (2!)2
9!
(b) (2!)3
𝑎2 𝑏 2 + 𝑎3 𝑏 3 + ⋯ ∞ is equal to 9!
𝑥𝑦
(a) 𝑥+𝑦−1
𝑥+𝑦
(b) 𝑥+𝑦+1 (c) 2! (d) 9 !
𝑥−𝑦
(c) 𝑥−𝑦+1 (d) None of these 33.The number of 4 digit number that can be
1 1 1 1
formed from the digits 0,1,2,3,4,5,6,7 so that each
25. 𝑏−𝑎 + 𝑏−𝑐 = 𝑎 + 𝑐 , then 𝑎, 𝑏, 𝑐 are in number contain digit 1 is
(a) AP (b) GP (a) 1225 (b) 1252
(c) HP (d) None of these (c) 1522 (d) 480
26. If the set of all 𝐚 ∈ 𝐑, for which the equation 34.𝑚 men and 𝑛 women are to be seated in a row,
2𝑥 2 + (𝑎 − 5)𝑥 + 15 = 3a has no real root, is the so that no two women sit together. If 𝑚 > 𝑛, then
interval (𝛼, 𝛽), and 𝑋 = {𝑥 ∈ 𝑍: 𝛼 < 𝑥 < 𝛽}, then the number of ways in which they can be seated is
∑𝑥∈𝑋 𝑥 2 is equal to : 𝑚!(𝑚+1)! 𝑚!(𝑚−1)!
(a) (𝑚−𝑛+1)! (b) (𝑚−𝑛+1)!
(1) 2109 (2) 2129
(𝑚−1)!(𝑚+1)!
(3) 2119 (4) 2139 (c) (d) None of these
(𝑚−𝑛+1)!

27. The sum, of the squares of all the roots of the

35.In how many ways a garland can be made from
equation 𝑥 2 + |2𝑥 − 3| − 4 = 0, is
exactly 10 flowers

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(a) 10 ! (b) 9! (a) 3/8 (b)1/ 5

9!
(c) 2(9!) (d) 2
(c) ¾ (d) None of these
36.In how many ways can 5 boys and 5 girls sit in 44.A pack of playing cards was found to contain
a circle so that no boys sit together only 51 cards. If the first 13 cards which are
(a) 5! × 5! (b) 4! × 5! examined are all red, then the probability that the
5!×5!
(c) 2 (d) None of these missing cards is black, is
1 2
(a) 3 (b) 3
37.The number of ways in which 5 male and 2 1 25 𝐶
female members of a committee can be seated (c) 2 (d) 51 𝐶
13
13
around a round table so that the two female are
not seated together is 45.A random variable 𝑋 has the probability
(a) 480 (b) 600 distribution
(c) 720 (d) 840
38.If 𝑛 𝐶𝑟−1 = 36, 𝑛 𝐶𝑟 = 84 and 𝑛 𝐶𝑟+1 = 126 then 𝑋: 1 2 3 4 5 6 7 8
the value of 𝑟 is
(a) 1 (b) 2 𝑃(𝑋) 0 0. 0 0 0. 0. 0. 0.
(c) 3 (d) None of these .1 2 .1 .1 2 0 0 0
: 5 3 2 0 0 8 7 5
39.To fill 12 vacancies there are 25 candidates of
which five are from scheduled caste. If 3 of the
vacancies are reserved for scheduled caste
candidates while the rest are open to all, then the For the events 𝐸 = {𝑋 is a prime number } and
number of ways in which the selection can be 𝐹 = {𝑋 < 4}, the probability 𝑃(𝐸 ∪ 𝐹) is
made (a) 0.50 (b) 0.77
(c) 0.35 (d) 0.87
(a) 5 𝐶3 × 22 𝐶9 (b) 22 𝐶9 − 5 𝐶3
22 5
(c) 𝐶3 + 𝐶3 (d) None of these 46.One hundred identical coins each with
probability 𝑝 of showing up heads are tossed once.
40.There are 10 lamps in a hall. Each one of them
If 0 < 𝑝 < 1 and the probability of heads showing
can be switched on independently. The number of
on 50 coins is equal to that of heads showing on
ways in which the hall can be illuminated is
51 coins, then the value of 𝑝 is
1 49
(a) 102 (b) 1023 (c) 210 (d) 10 ! (a) (b)
2 101
50 51
(c) 101 (d) 101
41.If 𝐴 and 𝐵 are two events such that 𝑃(𝐴) ≠ 0
𝐴‾
and 𝑃(𝐵) ≠ 1, then 𝑃 (𝐵‾) = 47.If the mean of the numbers 27 + 𝑥, 31 + 𝑥, 89 +
𝐴 𝐴‾ 𝑥, 107 + 𝑥, 156 + 𝑥 is 82 , then the mean of 130 +
(a) 1 − 𝑃 (𝐵) (b) 1 − 𝑃 (𝐵)
1−𝑃(𝐴∪𝐵) 𝑃(𝐴‾)
𝑥, 126 + 𝑥, 68 + 𝑥, 50 + 𝑥, 1 + 𝑥 is
(c) (d) (a) 75 (b) 157
𝑃(𝐵‾) 𝑃(𝐵‾)
(c) 82 (d) 80
42.The probability of happening an event A in one
trial is 0.4. The probability that the event A 48.Consider the frequency distribution of the
happens at least once in three independent trials given numbers
is
Value: 1 2 3 4
(a) 0.936 (b) 0.784 (c) 0.904 (d) 0.216
Frequency: 5 4 6 𝑓
43.A man is known to speak the truth 3 out of 4
times. He throws a die and reports that it is a six.
The probability that it is actually a six, is

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49.If the mean is known to be 3 , then the value of

𝑓 is
(a) 3 (b) 7
(c) 10 (d) 14

50.For a frequency distribution 7 th decile is

computed by the formula
𝑁
( −𝐶)
(a) 𝐷7 = 𝑙 + 7
𝑓
×𝑖
𝑁
( −𝐶)
(b) 𝐷7 = 𝑙 + 10
×𝑖
𝑓 (a) 40 (b) 50 (c) 60 (d) 75
7𝑁
( −𝐶)
(c) 𝐷7 = 𝑙 + 10
𝑓
×𝑖 54.For a normal distribution if the mean is 𝑀,
(
10𝑁
−𝐶) mode is 𝑀0 and median is 𝑀𝑑 , then
(d) 𝐷7 = 𝑙 + 7
×𝑖 (a) 𝑀 > 𝑀𝑑 > 𝑀0
𝑓
(b) 𝑀 < 𝑀𝑑 < 𝑀0
(c) 𝑀 = 𝑀𝑑 𝑀0
51.The following data was collected from the (d) 𝑀 = 𝑀𝑑 = 𝑀0
newspaper : (percentage distribution)
55.In the 13 cricket players 4 are bowlers, then
Countr Agricultu Indust Servic Other how many ways can form a cricket team of 11
players in which at least 2 bowlers included
y re ry es s
(a) 55 (b) 72 (c) 78 (d) None of these
India 45 19 28 8
𝑥+1 𝑥+2 𝑥+𝑎
56.If |𝑥 + 2 𝑥 + 3 𝑥 + 𝑏 | = 0, then 𝑎, 𝑏, 𝑐 are
U.K. 3 40 44 13
𝑥+3 𝑥+4 𝑥+𝑐
a. equal b. in A.P.
Japan 6 48 43 3 c. in G.P. d. in H.P
1 + sin2 𝜃 cos2 𝜃 4sin 2𝜃
U.S.A. 3 35 61 1 57.If | sin 𝜃
2
1 + cos 2 𝜃 4sin 2𝜃 | = 0
2 2
sin 𝜃 cos 𝜃 4sin 2𝜃 − 1
𝜋
and 0 < 𝜃 < , then cos 4𝜃 =
2
52.It is an example of a.
−1
b.
1
(a) Data given in text form 2 2
√3
(b)Data given in diagrammatic form c. d. 0
2
(c) Primary data
(d) Secondary data 1 2
58.If A = ( ), then the value of the
3 5
53. The mortality in a town during 4 quarters of a determinant |𝐴2009 − 5𝐴2008 | is
year due to various causes is given below : Based a. -6 b. -5
on this data, the percentage increase in mortality c. -4 d. 4
in the third quarter is 59.If 𝑥, 𝑦, 𝑧 are all positive and are the 𝑝th , 𝑞 th and
𝑟 th terms of a geometric progression respectively,
log 𝑥 𝑝 1
then the value of the determinant |log 𝑦 𝑞 1|
log 𝑧 𝑟 1
equals
a. log 𝑥𝑦𝑧 b. (𝑝 − 1)(𝑞 − 1)(𝑟 − 1)
c. pqr d. 0

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60.The value of 𝜆 and 𝜇 for which the 69.A value of 𝜃 satisfying sin 5𝜃 − sin 3𝜃 +
simultaneous equation 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑦 + sin 𝜃 = 0 such that 0 < 𝜃 < 𝜋/2 is
3𝑧 = 10 and 𝑥 + 2𝑦 + 2𝑧 = 𝜇 have a unique a. 𝜋/12 b. 𝜋/6
solution are c. 𝜋/4 d. 𝜋/2
a. 𝜆 = 3 only
70.The equation 𝑘sin 𝑥 + cos 2𝑥 = 2𝑘 − 7 has a
b. 𝜇 = 3 only
solution, if
c. 𝜆 = 3 and 𝜇 = 3
a. k > 6 b. 2 ≤ k ≤ 6
d. 𝜆 ≠ 3 and 𝜇 can take any value
c. k < 2 d. −6 ≤ k ≤ −2 (e)
0 3 0 4a
61.If A = [ ] and kA = [ ] then value of 2𝑥
71.If 𝑓(𝑥) = 2tan−1 𝑥 + sin−1 (1+𝑥2 ) , 𝑥 > 1, then
4 5 3 b 60
k, a and b are respectively 𝑓(5) is equal to
a. 12,19,16 b. 9,12,16 a. 𝜋/2 b. 𝜋
c. 12,9,16 d. 16,9,12 −1 65
c. 4tan (5) d. tan−1 (156)
62.For two 3 × 3 matrices A and B , let A + B =
2 B ′ and 3 A + 2 B = I3 , where B ′ is the transpose 2
72.Value of cos (3sin−1 (5)) is
of B and I3 is 3 × 3 identity matrix. Then 17 31
a. 10 A + 5 B = 3I3 b. 5 A + 10 B = 2I3 a. 25 b. − 125
c. 3 A + 6 B = 2I3 d. B + 2 A = I3 9√21 9√21
c. − 125
d. 125
63.Let f: (−1,1) → R be such that f(cos 4𝜃) =
2 𝜋 𝜋 𝜋 73.The value of tan2 (sec −1 2) + cot 2 (cosec −1 3) is
for 𝜃 ∈ (0, ) ∪ ( , ). Then the values of
2
2−sec 𝜃 4 4 2 a. 13 b. 15
1
f (3) is (are) c. 11 d. None of these
1 1 3
a. 1 − √2
3
b. 1 + √2
3 74.In a triangle ABC, a+c + b+c = a+b+c then ∠ABC
in degrees equals
2 2
c. 1 − √3 d. 1 + √3 a. 60∘ b. 30∘
c. 45∘ d. 90∘
64.If 𝜃 lies in the second quadrant and 3tan 𝜃 + 75.The angle of elevation of the top of a tower at
4 = 0, then the value of sin 𝜃 + cos 𝜃 is equal to point on the ground is 30∘ . If on walking 20
1 2
a. 5 b. 5 metres toward the tower, the angle of elevation
3
c. 5 d. 5
4 become 60∘ , then the height of the tower is
10
(a) 10 metre (b) metre
1 √3
65.If sin 𝑥cos 𝑦 = 4 and 3tan 𝑥 = 4 tany, then (c) 10√3 metre (d) None of these
sin (𝑥 − 𝑦) equals to
1 1
a. 16 b. 8
3 3 76. If position vectors of four points 𝐴, 𝐵, 𝐶 and 𝐷
c. 16 d. 4 ˆ , 2𝐢ˆ + 3𝐣ˆ, 3𝐢ˆ + 5𝐣ˆ − 2𝐤
ˆ and 𝐤
ˆ − 𝐣ˆ
are𝐢ˆ + 𝐣ˆ + 𝐤
𝜋 𝜋 respectively, then 𝐀𝐁 and 𝐂𝐃 are related as
66.The value of cot ( 4 − 𝜃) ⋅ cot ( 4 + 𝜃) is
(a) perpendicular
a. -1 b. 0 (b) parallel
c. 1 d. 2 (c) independent
sin 70∘ +cos 40∘ (d) None of these
67. cos 70∘ +sin 40∘ =
1 77. If (3𝐚 − 𝐛) × (𝐚 + 3𝐛) = 𝑘𝐚 × 𝐛, then what is
a. 1 b. the value of 𝑘 ?
√3
1 (a) 10 (b) 5
c. √3 d. 2
(c) 8 (d) -8
68.The value of √2(cos 15∘ − sin 15∘ ) is equal to
a. √3 b. √2 78. Point 𝐴 is 𝑎 + 2𝑏, 𝑃 is 𝑎 and 𝑃 divides 𝐴𝐵 in
c. 1 d. 2 the ratio 2:3. The position vector of 𝐵 is

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(a) 2𝑎 − 𝑏 (b) 𝑏 − 2𝑎 88. The area of a triangle is 5 and two of its

(c) 𝑎 − 3𝑏 (d) 𝑏 vertices are 𝐴(2,1), 𝐵(3, −2). Then, the third
vertex, in Ist quadrant which lies on the line 𝑦 =
79. If 𝐚 + 𝐛 + 𝐜 = 𝑝𝐝, 𝐛 + 𝐜 + 𝐝 = 𝑞𝐚 and 𝐚, 𝐛, 𝐜 are 𝑥 + 3 is
non-coplanar, then 𝐚 + 𝐛 + 𝐜 + 𝐝 is equal to 7 13
(a) (2 , 2 )
5 5
(b) (2 , 2)
(a) 0 (b) 𝑝𝑎 3 3
(c) 𝑞𝐛 (d) (𝑝 + 𝑞)𝐜 (c) (2 , 2) (d) (0,0)

80. If 𝐚 is a non-zero vector of modulus 𝑎 and 𝜆 is 89. The diagonals of a quadrilateral 𝐴𝐵𝐶𝐷 are
a non-zero scalar and 𝜆, 𝐚 is a unit vector, then along the lines 𝑥 + 3𝑦 = 4 and 6𝑥 − 2𝑦 = 7. Then,
(a) 𝜆 = ±1 (b) a = |𝜆| 𝐴𝐵𝐶𝐷 must be a
1 1
(c) 𝐚 = |𝜆| (d) a = 𝜆 (a) rectangle (b) parallelogram
(c) cyclic quadrilateral (d) rhombus
81. The ratio in which the line joining (2,4,5), (
90. The equation of straight line passing through
3,5, −4 ) is divided by the 𝑌𝑍-plane is
the point of intersection of the straight line 3𝑥 −
(a) 2: 3 (b) 3: 2
𝑦 + 2 = 0 and 5𝑥 − 2𝑦 + 7 = 0 and having infinite
(c) −2: 3 (d) 4: −3
slope is
82. A straight line which makes an angle of 60∘ (a) 𝑥 = 2 (b) 𝑥 + 𝑦 = 3
with each of 𝑌 and 𝑍-axes, is inclined with 𝑋-axis (c) 𝑥 = 3 (d) 𝑥 = 4
at an angle
Directions (Q. Nos. 91-92) In a circle of radius ' 𝑟
(a) 45∘ (b) 30∘
', a right circular cone is drawn.
(c) 75∘ (d) 60∘
91. What will be the maximum height of cone
83. The foot of the perpendicular from (0,2,3) to having maximum valume?
𝑥+3 𝑦−1 𝑧+4
the line = = is 4𝑟
(a) 3
3𝑟
(b) 4
5 2 3
(a) (−2,3,4) (b) (2, −1,3) √3 2
(c) (2,3, −1) (d) (3,2, −1) (c) 𝑟 (d) 𝑟
3 3

𝑥−2 𝑦−3 𝑧−4

84. The line 3 = 4 = 5 is parallel to the 92. What will be the radius of cone having
maximum volume?
plane
2√3 2√3
(a) 2𝑥 + 𝑦 − 2𝑧 = 0 (b) 3𝑥 + 4𝑦 + 5𝑧 = 7 (a) 9
𝑟 (b) 3
𝑟
(c) 𝑥 + 𝑦 + 𝑧 = 2 (d) 2𝑥 + 3𝑦 + 4𝑧 = 0 √3
(c) 2
𝑟 (d) √3𝑟
85. The equation of line through the point (1,2,3)
𝑥−4 𝑦+1 𝑧+10
parallel to line 2 = −3 = 8 is 93. What is the value of 𝑘 for which the following
𝑥−1 𝑦−2 𝑧−3 𝑥−1 𝑦−2 𝑧−3
function 𝑓(𝑥) is continuous for all 𝑥 ?
(a) 2
= −3
= 8
(b) 1
= 2
= 3 𝑥 3 − 3𝑥 + 2
𝑥−4 𝑦+1 𝑧+10 , for 𝑥 ≠ 1
(c) = = (d) None of these 𝑓(𝑥) = { (𝑥 − 1)2
1 2 3
𝑘, for 𝑥 = 1
86. If (0,4) and (0,2) are respectively, the vertex (a) 3 (b) 2
and focus of a parabola, then its equation is (c) 1 (d) -1
(a) 𝑥 2 + 8𝑦 = 32 (b) 𝑦 2 + 8𝑥 = 32
2 𝑥+6 𝑥+4
(c) 𝑥 − 8𝑦 = 32 (d) 𝑦 2 − 8𝑥 = 32 94. What is the value of lim𝑥→∞ (𝑥+1) ?
2
87. The coordinates of a point on the parabola (a) e (b) 𝑒
𝑦 2 = 8𝑥, whose focal distance is 4 , is (c) 𝑒 4 (d) 𝑒 5
(a) (2,4) (b) (4,2)
95. The growth of a quantity 𝑁(𝑡) at any instant 𝑡
(c) (−2, −4) (d) (4, −2) 𝑑𝑁(𝑡)
is given by 𝑑𝑡 = 𝛼𝑁(𝑡). If 𝑁(𝑡) = 𝑐𝑒 𝑘𝑡 , 𝑐 is a
constant, then what is the value of 𝛼 ?

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(a) c (b) 𝑘 104. The function 𝑓(𝑥) = 𝑘sin 𝑥 + 3 sin 3𝑥 has

1

(c) 𝑐 + 𝑘 (d) 𝑐 − 𝑘 𝜋
maximum value at 𝑥 = 3 , what is the value of 𝑘 ?
96. If 𝑓(𝑥) = 3𝑥 2 + 6𝑥 − 9, then (a) 3 (b) 1/3
(a) 𝑓(𝑥) is increasing in (−1,3) (c) 2 (d) 1/2
(b) 𝑓(𝑥) is decreasing in (3, ∞)
(c) 𝑓(𝑥) is increasing in (−∞, −1) Directions (Q. Nos. 105-107)
(d) 𝑓(𝑥) is decreasing in (−∞, −1) If ∫ 𝑥 2 𝑒 −2𝑥 𝑑𝑥 = 𝑒 −2𝑥 (𝑎𝑥 2 + 𝑏𝑥 + 𝑐) + 𝐷

97. The function 𝑓(𝑥) = 105. The value of 𝑎 is

𝑥 2 /𝑎, 0≤𝑥<1 (a) 1 (b) 2
(c) 4 (d) -2
{ 𝑎, 1 ≤ 𝑥 < √2 is continuous for
2 2
(2𝑏 − 4𝑏)/𝑥 , √2 ≤ 𝑥 < ∞ 106. The value of 𝑐 is
0 ≤ 𝑥 < ∞, then the most suitable values of 𝑎 and (a) 0 (b)
1
𝑏 are 1
2
1
(a) 𝑎 = 1, 𝑏 = −1 (b) 𝑎 = −1, 𝑏 = 1 + √2 (c) 4 (d) 8
(c) 𝑎 = −1, 𝑏 = 1 (d) None of these
107. The value of 𝑏 is
4
98. If for a continuous function 𝑓, 𝑓(0) = 𝑓(1) = (a) 4 (b) 3
0, 𝑓 ′ (1) = 2 and 𝑔(𝑥) = 𝑓(𝑒 𝑥 )𝑒 𝑓(𝑥) , then 𝑔′ (0) is (c) 1 (d) None of these
equal to
(a) 1 (b) 2 𝜋/2 𝑑𝑥
108. The value of ∫−𝜋/2 𝑒 sin 𝑥 +1 is equal to
(c) 0 (d) 3
(a) 0 (b) 1
𝜋 𝜋
99. If 𝑥 = cos (2𝑡) and 𝑦 = sin2 𝑡, then what is 𝑑𝑥 2
𝑑2 𝑦 (c) − (d)
2 2
equal to? 1
(a) 0 (b) sin (2𝑡) 109. What is the value of the integral ∫−1 |𝑥|𝑑𝑥 ?
(c) −cos (2𝑡)
1
(d) − 2 (a) 1 (b) 0
(c) 2 (d) -1
𝑑2 𝑦 𝑑2 𝑥
100. If 𝑦 = 𝑒 2𝑥 , then ⋅ is equal to 110. What is ∫0
𝜋/2 sin3 𝑥
𝑑𝑥 ?
𝑑𝑥 2 𝑑𝑦 2
−2𝑥 −2𝑥 sin3𝑥+cos3 𝑥
(a) 𝑒 (b) −2𝑒 (a) 𝜋
𝜋
(b) 2
(c) 2𝑒 −2𝑥 (d) 1 𝜋
(c) (d) 0
4
2𝑥 −1
101. lim (1+𝑥)1/2 −1 is equal to 𝜋/4
𝑥→0
111. If 𝐼𝑛 = ∫0 tan𝑛 𝑥𝑑𝑥, then what is 𝐼𝑛 + 𝐼𝑛−2
(a) log 2 (b) 2log 2
1 equal to?
(c) 2 log 2 (d) 0 1 1
(a) 𝑛 (b) (𝑛−1)
(𝑥−3) 𝑛 1
102. lim is equal to (c) (𝑛−1) (d) (𝑛−2)
𝑥→3 |𝑥−3|
(a) 0 (b) 1
(c) -1 (d) does not exist 112.The differential equation of the family of
curves 𝑦 = 𝐴𝑒 3𝑥 + 𝐵𝑒 5𝑥 , where 𝐴 and 𝐵 are
103. What is the derivative of 𝑥√𝑎2 − 𝑥 2 + arbitrary constants, is
𝑥 𝑑2 𝑦 𝑑𝑦
𝑎2 sin−1 ( ) ? (a) 𝑑𝑥 2 + 8 𝑑𝑥 + 15𝑦 = 0
𝑎
𝑑2 𝑦 𝑑𝑦
(a) √𝑎2 − 𝑥 2 (b) 2√𝑎2 − 𝑥 2 (b) 𝑑𝑥 2 − 8 𝑑𝑥 + 15𝑦 = 0
(c) √𝑥 2 − 𝑎2 (d) 2√𝑥 2 − 𝑎2 𝑑2 𝑦 𝑑𝑦
(c) 𝑑𝑥 2 − 𝑑𝑥 + 𝑦 = 0
(d) None of these

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113. The order of the differential equation whose

solution is 𝑦 = 𝑎cos 𝑥 + 𝑏sin 𝑥 + 𝑐𝑒 −𝑥 is
(a) 3 (b) 2
(c) 1 (d) None of these

114. The differential equation of the family of

curves represented by the equation 𝑥 2 𝑦 = 𝑎, is
𝑑𝑦 2𝑦 𝑑𝑦 2𝑥
(a) 𝑑𝑥 + 𝑥 = 0 (b) 𝑑𝑥 + 𝑦 = 0
𝑑𝑦 2𝑦 𝑑𝑦 2𝑥
(c) − =0 (d) − =0
𝑑𝑥 𝑥 𝑑𝑥 𝑦

115. If 𝑓(𝑥) satisfies the relation 2𝑓(𝑥) + 𝑓(1 −

𝑥) = 𝑥 2 for all real 𝑥, then 𝑓(𝑥) is
𝑥 2 +2𝑥−1 𝑥 2 +2𝑥−1
(a) 6
(b) 3
𝑥 2 +4𝑥−1 𝑥 2 +4𝑥−1
(c) 3
(d) 6

116. The function 𝑓(𝑥) = log (𝑥 + √𝑥 2 + 1) is

(a) an even function (b) an odd function
(c) periodic function (d) None of these

117. The values of 𝑏 and 𝑐 for which the identity

𝑓(𝑥 + 1) − 𝑓(𝑥) = 8𝑥 + 3 is satisfied, where
𝑓(𝑥) = 𝑏𝑥 2 + 𝑐𝑥 + 𝑑, are
(a) 𝑏 = 2, 𝑐 = 1
(b) 𝑏 = 4, 𝑐 = −1
(c) 𝑏 = −1, 𝑐 = 4
(d) None of these

118. Let 𝑓 be a function with domain [−3,5] and

let 𝑔(𝑥) = |3𝑥 + 4|, then the domain of 𝑓 ∘ 𝑔(𝑥) is
1 1
(a) (−3, 3) (b) [−3, 3]
1
(c) [−3, 3) (d) None of these

𝛼𝑥
119.Let 𝑓(𝑥) = 𝑥+1 , 𝑥 ≠ −1. Then, for what value
of 𝛼 is 𝑓[𝑓(𝑥)] = 𝑥 ?
(a) √2 (b) −√2
(c) 1 (d) -1

120. Which one of the following functions, 𝑓: 𝑅 →

𝑅 is injective?
(a) 𝑓(𝑥) = |𝑥|, ∀𝑥 ∈ 𝑅 (b) 𝑓(𝑥) = 𝑥 2 , ∀𝑥 ∈ 𝑅
(c) 𝑓(𝑥) = 11, ∀𝑥 ∈ 𝑅 (d) 𝑓(𝑥) = −𝑥, ∀𝑥 ∈ 𝑅

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