NDA Previous Year Question Paper 2021

 

1. The smallest positive integer n for which ((1 - i)/(1 + i))^n = 1 where i = sqrt(-1), is
(a) 2 (b) 4 (c) 6 (d) 8
(a) ((1 - i)/(1 + i))^(n^2) = 1, where i = sqrt(-1)
((1 - i)/(1 + i) × (1 - i)/(1 - i))^(n^2) = 1
((1 + i^2 - 2i)/(1 - i^2))^(n^2) = 1
((1 - 1 - 2i)/(1 + 1))^(n^2) = 1
=> (-i)^(n^2) = (-i)^4
=> n^2 = 4
n = 2
Hence, option (a) is correct.

2. The value of x, satisfying the equation log_{cos x} sin x = 1, where 0 < x < pi/2, is
(a) pi/12 (b) pi/3 (c) pi/4 (d) pi/6
(c) log_{cos x} sin x = 1, where 0 < x < pi/2
=> (cos x)^4 = sin x => cos x = sin x
=> tan x = 1 => tan x = tan pi/4
=> x = pi/4
Hence, option (c) is correct.

PAPER I: Mathematics

1. The smallest positive integer n for which
((1 - i)/(1 + i))^n = 1
where i = sqrt(-1), is
(a) 2 (b) 4 (c) 6 (d) 8
(a) ((1 - i)/(1 + i))^(n^2) = 1, where i = sqrt(-1)
((1 - i)/(1 + i) × (1 - i)/(1 - i))^(n^2) = 1
((1 + i^2 - 2i)/(1 - i^2))^(n^2) = 1
((1 - 1 - 2i)/(1 + 1))^(n^2) = 1
=> (-i)^(n^2) = (-i)^4
=> n^2 = 4
n = 2
Hence, option (a) is correct.

2. The value of x, satisfying the equation log_{cos x} sin x = 1, where 0 < x < pi/2, is
pi/12 (b) pi/3 (c) pi/4 (d) pi/6
(c) log_{cos x} sin x = 1, where 0 < x < pi/2
=> (cos x)^4 = sin x => cos x = sin x
=> tan x = 1 => tan x = tan pi/4
=> x = pi/4
Hence, option (c) is correct.

3. If Delta is the value of the determinant
then what is the value of the following determinant?
(p ≠ 0 or 1, q ≠ 0 or 1)
(a) pDelta (b) qDelta
(c) (p + q)Delta (d) pqDelta
(d) Given, a1 b1 c1
Hence, option (d) is correct.

4. If C0, C1, C2, ..., Cn are the coefficients in the expansion of (1 + x)^n, then what is the value of C1 + C2 + C3 + ... + Cn?
(a) 2^n (b) 2^n - 1
(c) 2^(n - 1) (d) 2^n - 2
(b) because (1 + x)^n = C0 + C1 x + C2 x^2 + ... + nCn x^n
and we know that
nC0 + nC1 + nC2 + ... + nCn = 2^n - nC0 = 2^n - 1
Hence, option (b) is correct.

5. If a + b + c = 4 and ab + bc + ca = 0, then what is the value of the following determinant?
(a) 32 (b) -64
(c) -128 (d) 64
(b) Let
(To take common a + b + c from C1
= (a + b + c)((b - c)(a - b) - (c - a)^2)
= (a + b + c)(ab - b^2 - ca + bc)

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6. The number of integer values of k, for which the equation 2 sin x = 2k + 1 has a solution, is
(a) zero (b) one (c) two (d) four
(c) Given, 2 sin x = 2k + 1
-1 <= sin x <= 1 => -2 <= 2 sin x <= 2
-2 - 1 <= 2 sin x - 1 <= 2 - 1
-3 <= 2k <= 1
-3/2 <= k <= 1/2 => -1.5 <= k <= 0.5
Integer values of k = -1, 0
Hence, option (c) is correct.

7. If a1, a2, a3, ..., a9 are in GP, then what is the value of the following determinant?
(a) 0 (b) 1 (c) 2 (d) 4
(a) Let first term and common ratio of GP be a and r respectively.
[because log mn = log m + log n]
(by C2 -> C2 - C1 and C3 -> C3 - C2)
= 0 [C2 = C3]

8. If the roots of the quadratic equation x^2 + 2x + k = 0 are real, then
(a) k < 0 (b) k <= 0 (c) k < 1 (d) k <= 1
(d) Given quadratic equation,
x^2 + 2x + k = 0
Since, roots are real
=> D >= 0 => b^2 - 4ac >= 0
(2)^2 - 4(1)(k) >= 0 => 4 >= 4k => k <= 1
Hence, option (d) is correct.

9. If n = 100!, then what is the value of the following?
1/log_2 n + 1/log_3 n + 1/log_4 n + ... + 1/log_100 n
(a) 0 (b) 1 (c) 2 (d) 3

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14. If A and B are square matrices of order 2 such that det(AB) = det(BA), then which one of the following is correct?
(a) A must be a unit matrix
(b) B must be a unit matrix
(c) Both A and B must be unit matrices
(d) A and B need not be unit matrices
(d) A_{2×2} and B_{2×2} are two matrices and |AB| = |BA| => |A||B| = |B||A|
Let A = [[1, 2], [3, 4]], B = [[-2, 1], [3/2, -1/2]]
then, |AB| = |BA|
Hence, we can say A and B need not be the unit matrices.
Hence, option (d) is correct.

15. What is cot 2x cot 4x - cot 4x cot 6x - cot 6x cot 2x equal to?
(a) -1 (b) 0 (c) 1 (d) 2

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22. What is the value of the following?
(sin 24° + cos 66°)(sin 24° - cos 66°)
(a) -1 (b) 0 (c) 1 (d) 2
(b) (sin 24° + cos 66°)(sin 24° - cos 66°)
= (sin 24° + cos 66°)(sin 24° - cos 66°)
= (sin 24° + cos 66°)(sin 24° - cos 66°)
[because sin(90° - theta) = cos theta]
= (sin 24° + cos 66°)(cos 66° - cos 66°)
= (sin 24° + cos 66°)(0) = 0
Hence, option (b) is correct.

23. A chord subtends an angle 120° at the centre of a unit circle. What is the length of the chord?
(a) sqrt(2) - 1 units (b) sqrt(3) - 1 units
(c) sqrt(2) units (d) sqrt(3) units
(d) Given, radius of the circle = 1 unit
angle AOB = 120°
By using cosine rule,
cos 120° = (OA^2 + OB^2 - AB^2)/(2·OA·OB)
Let AB = x unit, OA = 1 unit, OB = 1 unit
From Eq. (i),
-1/2 = (1 + 1 - x^2)/(2·1·1) => -1 = 2 - x^2
=> x^2 = 3 => x = sqrt(3) unit
Hence, option (d) is correct.

24. What is (1 + cot theta - cosec theta)(1 + tan theta + sec theta) equal to?
(a) 1 (b) 2 (c) 3 (d) 4
(b) (1 + cot theta - cosec theta)(1 + tan theta + sec theta)
= (1 + cos theta/sin theta - 1/sin theta)(1 + sin theta/cos theta + 1/cos theta)
= ((sin theta + cos theta - 1)/sin theta)((sin theta + cos theta + 1)/cos theta)
= ((sin theta + cos theta)^2 - 1^2)/(sin theta cos theta)
= (sin^2 theta + cos^2 theta + 2 sin theta cos theta - 1)/(sin theta cos theta)
= (1 + 2 sin theta cos theta - 1)/(sin theta cos theta) = 2
Hence, option (b) is correct.

25. What is (1 + tan^2 theta)/(1 + cot^2 theta) - ((1 - tan theta)/(1 - cot theta))^2 equal to?
(a) 0 (b) 1
(c) 2 tan theta (d) 2 cot theta
(a) (1 + tan^2 theta)/(1 + cot^2 theta) - ((1 - tan theta)/(1 - cot theta))^2
= (1 + tan^2 theta)/(1 + 1/tan^2 theta) - ((1 - tan theta)/(1 - 1/tan theta))^2
= tan^2 theta ((1 + tan^2 theta)/(tan^2 theta + 1)) - (tan theta(1 - tan theta)/(tan theta - 1))^2
= tan^2 theta - tan^2 theta = 0
Hence, option (a) is correct.

26. What is the interior angle of a regular octagon of side length 2 cm?
(a) pi/2 (b) 3pi/4 (c) 5pi/4 (d) 3pi/8
(b) Given, length of side of regular octagon = 2 cm
Sum of interior angles of octagon
= (8 - 2) × 180°
= 6 × 180°
[because sum of interior angles of polygon = (n - 2) × 180°]
Interior angle = (6 × 180°)/8
= 135° = 3pi/4
Hence, option (b) is correct.

27. If 7 sin theta + 24 cos theta = 25, then what is the value of (sin theta + cos theta)?
(a) 1 (b) 26/25 (c) 6/5 (d) 31/25
(d) Given, 7 sin theta + 24 cos theta = 25
Since, we know that if
a sin theta + b cos theta = c
then b sin theta - a cos theta = sqrt(a^2 + b^2 - c^2)
7 sin theta + 24 cos theta = 25 ... (i)
24 sin theta - 7 cos theta = sqrt(7^2 + 24^2 - 25^2)
= sqrt(49 + 576 - 625) = 0 ... (ii)
Eq. (i) × 7 + Eq. (ii) × 24
49 sin theta + 168 cos theta = 175
576 sin theta - 168 cos theta = 0
625 sin theta = 175
sin theta = 175/625 = 7/25
cos theta = sqrt(1 - (7/25)^2) = 24/25
sin theta + cos theta = 7/25 + 24/25 = 31/25
Hence, option (d) is correct.

28. A ladder 6 m long reaches a point 6 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is 75°. What is the height of the flagstaff?
(a) 12 m (b) 9 m
(c) (6 + sqrt(3)) m (d) (6 + 3sqrt(3)) m
(d) Let AC be a vertical flagstaff.
CD = 6 m, BD = 6 m
angle CBD = 75°
Let AD = h meter
In triangle ABC
90 + 75 + angle C = 180° [because sum of interior angle of triangle is 180°]
angle C = 15°
In triangle BCD,
BD = CD => angle BCD = angle CBD = 15°
angle ABD = 75° - 15° = 60°
In triangle ABD, sin 60° = h/6 => sqrt(3)/2 = h/6
h = 3sqrt(3) m
Height of the flagstaff = (h + 6) m = (3sqrt(3) + 6) m
Hence, option (d) is correct.

29. The shadow of a tower is found to be x metre longer, when the angle of elevation of the sun changes from 60° to 45°. If the height of the tower is 5(sqrt(3) + 1) m, then what is x equal to?
(a) 8 m (b) 10 m
(c) 12 m (d) 15 m
(b) In the given diagram,
AB represents the position of tower, where h = 5(sqrt(3) + 1) m
CD = x m
In triangle ABC,
tan 60° = (5(sqrt(3) + 1))/BC => sqrt(3) = (5(sqrt(3) + 1))/BC

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BC = 5(sqrt(3) + 1) m
In triangle ABD
tan 45° = (5(sqrt(3) + 1))/BD
=> 1 = (5(sqrt(3) + 1))/BD
therefore BD = 5(3 + sqrt(3)) m
Since, x = BD - BC
x = 5(3 + sqrt(3)) - 5(sqrt(3) + 1)
x = 5(3 + sqrt(3) - sqrt(3) - 1)
x = 10 m
Hence, option (b) is correct.

30. If 3 cos theta = 4 sin theta then what is the value of tan(45° + theta)?
(a) 10 (b) 7 (c) 7/2 (d) 7/4
(b) If 3 cos theta = 4 sin theta
Hence, option (b) is correct.

31. tan^{-1} x + cot^{-1} x = pi/2 holds, when
(a) x in R
(b) x in R - (-1, 1) only
(c) x in R - {0} only
(d) x in R - [-1, 1] only
(a) Since, tan^{-1} x + cot^{-1} x = pi/2 for all x in R
Hence, option (a) is correct.

32. If tan A = 1/7 then what is cos 2A equal to?
(a) 24/25 (b) 18/25 (c) 12/25 (d) 6/25
(a) tan A = 1/7
therefore cos 2A = (1 - tan^2 A)/(1 + tan^2 A) = (1 - (1/7)^2)/(1 + (1/7)^2)
= (49 - 1)/(49 + 1) = 48/50
cos 2A = 24/25
Hence, option (a) is correct.

33. The sides of a triangle are m, n and sqrt(m^2 + n^2 + mn). What is the sum of the acute angles of the triangle?
(a) 45° (b) 60° (c) 75° (d) 90°
(b) Let AB = m, AC = n
BC = sqrt(m^2 + n^2 + mn)
By using cosine rule,
cos A = (AB^2 + AC^2 - BC^2)/(2 AB·AC)
=> cos A = (m^2 + n^2 - m^2 - n^2 - mn)/(2mn)
=> cos A = -1/2 => A = 120°
therefore angle B + angle C = 180 - angle A
[because sum of interior angle is 180°]
= 180° - 120°
angle B + angle C = 60°
Hence, option (b) is correct.

34. What is the area of the triangle ABC with sides a = 10 cm, c = 4 cm and angle B = 30°?
(a) 16 cm^2 (b) 12 cm^2 (c) 10 cm^2 (d) 8 cm^2
(c) Given, a = 10 cm
c = 4 cm
angle B = 30°
Area of triangle = 1/2 ac sin(angle B)
= 1/2 × 10 × 4 × sin 30° = 1/2 × 40 × 1/2
= 10 sq cm
Hence, option (c) is correct.

35. Consider the following statements
1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets.
2. A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets
Which of the above statements is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
(c) A = {1, 3, 5} and B = {2, 4, 7}
Since, number of elements are same in both the sets.
=> A and B are equivalent sets.
If A = {1, 5, 9}, B = {1, 5, 5, 9, 9}
Which is nothing but B = {1, 5, 9}
Since, elements are same in A and B
=> A and B are equal sets
Hence, option (c) is correct.

36. Consider the following statements
1. The null set is a subset of every set.
2. Every set is a subset of itself.
3. If a set has 10 elements, then its power set will have 1024 elements.
Which of the above statements are correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
(d) Since we know that null set is a subset of every set and every set is a subset of itself.
If n(A) = 10
therefore n(P(A)) = 2^10 = 1024
all the given statements are true.
Hence, option (d) is correct.

37. Let R be a relation defined as xRy if and only if 2x + 3y = 20, where x, y in N. How many elements of the form (x, y) are there in R?
(a) 2 (b) 3 (c) 4 (d) 6
(b) because xRy <=> 2x + 3y = 20 where, x, y in N
therefore y = (20 - 2x)/3
All ordered pair which satisfies the given relations are (1, 6), (4, 4), (7, 2).
therefore R = {(1, 6), (4, 4), (7, 2)}
n(R) = 3
Hence, option (b) is correct.

38. Consider the following statements
1. A function f: Z -> Z defined by f(x) = x + 1 is one-one as well as onto.
2. A function f: N -> N defined by f(x) = x + 1 is one-one but not onto.
Which of the above statement(s) is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2

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(c) Statement I
f: Z -> Z
f(x) = x + 1
Let f(x1) = f(x2)
=> x1 + 1 = x2 + 1
=> x1 = x2
=> f is one-one in Z and every element of co-domain has its pre-image in domain.
=> f is onto.
Statement II
f: N -> N
f(x) = x + 1
Let f(x1) = f(x2)
x1 + 1 = x2 + 1
=> x1 = x2
=> f is one-one in N
But there is no element in N such that f(x) = 1
Hence, f is not onto on N
Given statements are correct.
Hence, option (c) is correct.

39. Consider the following in respect of a complex number z.
1. (z^{-1}) = (z)^{-1}
2. z z^{-1} = |z|^2
Which of the above is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
(a) Let z = x + iy
z bar = x - iy
(z bar)^{-1} = 1/(x - iy) = (x + iy)/(x^2 + y^2)
Also, z^{-1} = 1/(x + iy) = (x - iy)/(x^2 + y^2)
(z^{-1}) bar = (x + iy)/(x^2 + y^2) = (z bar)^{-1}
Statement 1 is correct.
|z| = sqrt(x^2 + y^2)
=> |z|^2 = x^2 + y^2
But z z^{-1} = (x + iy)(x - iy)/(x^2 + y^2)
= (x^2 + y^2)/(x^2 + y^2) = 1 ≠ |z|^2
Statement 2 is wrong.
Hence, option (a) is correct.

40. Consider the following statements in respect of an arbitrary complex number z.
1. The difference of z and its conjugate is an imaginary number.
2. The sum of z and its conjugate is a real number.
Which of the above statement(s) is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2
z = x + iy
z bar = x - iy
therefore z - z bar = x + iy - x + iy = 2iy which is an imaginary number.
=> Statement-1 is correct.
Also, z + z bar = x + iy + x - iy = 2x which is real.
=> Statement-2 is correct.
Hence, option (c) is correct.

41. What is the modulus of the complex number i^{2n} + (-i)^{2n - 1} where n in N and i = sqrt(-1)?
(a) -1 (b) 1 (c) sqrt(2) (d) 2
(b) Let z = i^{2n} + (-i)^{2n - 1} where n in N
= (i)^{2n} + ((-i)^2)^n (-i)^{-1}
= (i^{2n}) + (-1)^{2n} (i^{2n}) (i/(-i))
= (i^{4n})(-1) = (i^4)^n (-1)
= -1 = -1 + 0i
therefore |z| = 1
Hence, option (b) is correct.

42. If alpha and beta are the roots of the equation 4x^2 + 2x - 1 = 0 then which one of the following is correct?
(a) beta = -2alpha^2 - 2alpha
(b) beta = 4alpha^2 - 3alpha
(c) beta = alpha^2 - 3alpha
(d) beta = -2alpha^2 + 2alpha
(a) Given quadratic equation
4x^2 + 2x - 1 = 0 ... (i)
If alpha, beta are the roots of Eq. (i), then these value will satisfy the given equation.
4alpha^2 + 2alpha - 1 = 0 ... (ii)
and 4beta^2 + 2beta - 1 = 0 ... (iii)
From Eq. (i), Sum of roots = -2/4
alpha + beta = -1/2
beta = -1/2 - alpha
On putting the value of beta in Eq. (iii),
4(-1/2 - alpha)^2 + 2beta - 1 = 0
4(1/4 + alpha^2 + alpha) - 1 = -2beta
1 + 4alpha^2 + 4alpha - 1 = -2beta
=> beta = (4(alpha^2 + alpha))/(-2)
beta = -2alpha^2 - 2alpha
Hence, option (a) is correct.

43. If one root of 5x^2 + 26x + k = 0 is reciprocal of the other, then what is the value of k?
(a) 2 (b) 3 (c) 5 (d) 8
(c) Given quadratic equation
5x^2 + 26x + k = 0
Let alpha and beta be the roots.
According to question, beta = 1/alpha
Product of roots = k/5
alpha·beta = k/5
=> alpha·(1/alpha) = k/5 => 1 = k/5
=> k = 5
Hence, option (c) is correct.

44. In how many ways can a team of 5 players be selected from 8 players so as not to include a particular player?
(a) 42 (b) 35 (c) 21 (d) 20
(c) Given that there are 8 players among which one particular player is there.
Hence, number of ways to select 5 players = 8^{-1} · C5
= 7C5 = (7 × 6)/(1 × 2) = 21
Hence, option (c) is correct.

45. What is the coefficient of the middle term in the expansion of (1 + 4x + 4x^2)^5?
(a) 8064 (b) 4032 (c) 2016 (d) 1008
(a) (1 + 4x + 4x^2)^5
= ((1 + 2x)^2)^5 = (1 + 2x)^10
Total number of term in the expansion of (1 + 2x)^10 = 10 + 1 = 11
Middle term = ((11 + 1)/2) th term = 6th term
T6 = T5 + 1 = 10C5 (2x)^5
= 10C5 × 2^5 × x^5
Coefficient of middle term = 10C5 · 2^5
= (10 × 9 × 8 × 7 × 6)/(1 × 2 × 3 × 4 × 5) × 2^5 = 8064
Hence, option (a) is correct.

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46. What is C(n, 1) + C(n, 2) + ... + C(n, n) equal to?
(a) 2 + 2^2 + 2^3 + ... + 2^n
(b) 1 + 2 + 2^2 + 2^3 + ... + 2^2
(c) 1 + 2 + 2^2 + 2^3 + 2^4 + 2^{n - 1}
(d) 2 + 2^2 + 2^3 + ... + 2^{n - 1}
(c) C(n, 1) + C(n, 2) + ... + C(n, n)
= nC1 + nC2 + nC3 + ... + nCn
because nC0 + nC1 + nC2 + ... + nCn = 2^n
= 2^n - nC0 = 2^n - 1
Now, we shall solve the option to check whether sum is 2^n - 1 or not.
Let's take S = 1 + 2 + 2^2 + 2^3 + ... + 2^{n - 1}
which forms a GP.
where a = 1
r = 2/1 = 2 > 1
therefore S = a(r^n - 1)/(r - 1)
therefore S = 1(2^n - 1)/(2 - 1) = 2^n - 1
Hence, 2^n - 1 = nC1 + nC2 + ... + nCn.
Option (c) is correct.

47. What is the sum of the coefficients of first and last terms in the expansion of (1 + x)^{2n}, where n is a natural number?
(a) 1 (b) 2 (c) n (d) 2n
(b) Expand (1 + x)^{2n} by using binomial expansion
= 2nC0 x^0 + 2nC1 x^1 + 2nC2 x^2 + ...
therefore the coefficient of first and last term of the expansion
= 2nC0 + 2nC2n
= 1 + 1 = 2
Hence, option (b) is correct.

48. If the first term of an AP is 2 and the sum of the first five terms is equal to one-fourth of the sum of the next five terms, then what is the sum of the first ten terms?
(a) -500 (b) -250 (c) 500 (d) 250
(b) Given, first term of an AP (a) = 2
and a1 + a2 + a3 + a4 + a5 = 1/4 (a6 + a7 + a8 + a9 + a10)
where an = a + (n - 1)d
=> 5/2 [2a + (5 - 1)d]
= 1/4 × 5/2 [2a6 + (5 - 1)d]
[because sum of n terms of AP, Sn = n/2 [2a + (n - 1)d]]
4(2 × 2 + 4d) = 2a6 + 4d
16 + 16d = 2a6 + 4d
16 + 16d = 2(a + 5d) + 4d
16 + 16d = 2a + 14d
16 + 16d = 2 × 2 + 14d
2d = -12 => d = -6
therefore S10 = 10/2 [2a + (10 - 1)d]
= 5[2 × 2 + 9(-6)]
= 5[4 - 54]
S10 = -250
Hence, option (b) is correct.

49. Consider the following statements
1. If each term of a GP is multiplied by same non-zero number, then the resulting sequence is also a GP.
2. If each term of a GP is divided by same non-zero number, then the resulting sequence is also a GP.
Which of the above statements is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
(c) Let us take a GP.
a, ar, ar^2, ... is in GP.
=> ak, akr, akr^2, ... will also be in GP
where, k is non-zero number.
=> a/m, ar/m, ar^2/m, ... will also be in GP
Hence, both statements are correct.

50. How many 5-digit prime numbers can be formed using the digits 1, 2, 3, 4, 5 if the repetition of digits is not allowed?
(a) 5 (b) 4 (c) 3 (d) 0
(d) Given digits are 1, 2, 3, 4, 5
Since, the sum of digits = 1 + 2 + 3 + 4 + 5 = 15 is divisible by 3.
=> Every 5 digit number formed by the given digits will be divisible by 3.
=> There is no prime number.
Hence, option (d) is correct.
Hence, option (d) is correct.

51. If f(x + 1) = x^2 - 3x + 2 then what is f(x) equal to?
(a) x^2 - 5x + 4 (b) x^2 - 5x + 6
(c) x^2 + 3x + 3 (d) x^2 - 3x + 1
(b) If f(x + 1) = x^2 - 3x + 2
Let x + 1 = y => x = y - 1 or x -> x - 1
therefore f(x) = (x - 1)^2 - 3(x - 1) + 2
= x^2 + 1 - 2x - 3x + 3 + 2
= x^2 - 5x + 6
Hence, option (b) is correct.
Hence, option (b) is correct.

52. If x^2, x, -8 are in AP, then which one of the following is correct?
(a) x in (-2) (b) x in (4)
(c) x in (-2, 4) (d) x in (-4, 2)
(c) If x^2, x, -8 are in AP, then
=> x^2 - 2x - 8 = 0
=> (x - 4)(x + 2) = 0
x in (-2, 4)
Hence, option (c) is correct.

53. The third term of a GP is 3. What is the product of its first five terms?
(a) 81 (b) 243
(c) 729 (d) Cannot be determined due to insufficient data
(b) Given
therefore a3 = 3
therefore a3 = ar^2 in GP [because an = ar^{n - 1} in GP]
ar^2 = 3
To find a1·a2·a3·a4·a5
= a(ar)(ar^2)(ar^3)(ar^4)
= a^5 r^10 = (ar^2)^5 = 3^5 = 243
Hence, option (b) is correct.

54. The element in the ith row and the jth column of a determinant of third order is equal to 2(i + j). What is the value of the determinant?
(a) 0 (b) 2 (c) 4 (d) 6
(a) Given,
therefore aij = 2(i + j)
therefore a11 = 2(1 + 1) = 4, a21 = 2(2 + 1) = 6
a12 = 2(1 + 2) = 6, a22 = 2(2 + 2) = 8
a13 = 2(1 + 3) = 8

===== Page 8 =====

55. With the numbers 2, 4, 6, 8, all the possible determinants with these four different elements are constructed. What is the sum of the values of all such determinants?
(a) 128 (b) 64 (c) 32 (d) 0
(d) Given numbers are 2, 4, 6, 8.
We can form determinant of order 2.
Number of determinants = 4 × 3 × 2 × 1 = 24
Let's observe some determinants
|2 6| = 8 - 48 = -40, |6 2| = 40
|4 8| |4 8|
|2 8| = 8 - 48 = -40, |6 4| = 40
|4 6| |2 8|
|4 8| = 8 - 48 = -40, |8 4| = 40
|6 2| |2 6|
|4 6| = 8 - 48 = -40, |8 2| = 40
|8 2| |4 6|
Hence, we can see that we are getting a pattern where each determinant value will be neutralised by other value.
Hence, sum of the values of all determinants = 0
Hence, option (d) is correct.

56. What is the radius of the circle 4x^2 + 4y^2 - 20x + 12y - 15 = 0?
(a) 14 units (b) 10.5 units
(c) 7 units (d) 3.5 units
(d) Given equation of circle
4x^2 + 4y^2 - 20x + 12y - 15 = 0
=> x^2 + y^2 - 5x + 3y - 15/4 = 0
On comparing with
x^2 + y^2 + 2gx + 2fy + c = 0
g = -5/2, f = 3/2, c = -15/4
therefore Radius = sqrt(g^2 + f^2 - c)
= sqrt(25/4 + 9/4 + 15/4) = 7/2 = 3.5 unit
Hence, option (d) is correct.

57. A parallelogram has three consecutive vertices (-3, 4), (0, -4) and (5, 2). The fourth vertex is
(a) (2, 10) (b) (2, 9)
(c) (3, 9) (d) (4, 10)
(a)
Let the fourth vertex be D(x, y).
Diagonals of a parallelogram bisect each other.
O is mid-point of AC.
=> Coordinate of O is ((-3 + 5)/2, (4 + 2)/2) or (1, 3)
O is mid-point of BD.
=> Coordinate of O is ((x + 0)/2, (y - 4)/2) or (x/2, (y - 4)/2)
Therefore, compare the coordinate of O
=> x/2 = 1 => x = 2
and (y - 4)/2 = 3 => y = 10
Hence, the fourth vertex is (2, 10)

58. If the lines y + px = 1 and y - qx = 2 are perpendicular, then which one of the following is correct?
(a) pq + 1 = 0 (b) p + q + 1 = 0
(c) pq - 1 = 0 (d) p - q + 1 = 0
(c) Given y + px = 1 ... (i)
y - qx = 2 ... (ii)
Eqs. (i) and (ii) are perpendicular
=> m1 m2 = -1 where m1 and m2 are the slope of Eqs. (i) and (ii)
and m = - coefficient of x / coefficient of y
=> -p/1 × (-q)/1 = -1
=> -pq = -1
=> pq = 1
Hence, option (c) is correct.

59. If A, B and C are in AP, then the straight line Ax + 2By + C = 0 will always pass through a fixed point. The fixed point is
(a) (0, 0) (b) (-1, 1)
(c) (1, -2) (d) (1, -1)
(d) Given A, B, C are in AP.
=> 2B = A + C
=> A - 2B + C = 0 ... (i)
On comparing A - 2B + C = 0 with the given line Ax + 2By + C = 0, we get x = 1, y = -1
Hence, line Ax + 2By + C = 0 will pass through (1, -1)
Hence, option (d) is correct.

60. If the image of the point (-4, 2) by a line mirror is (4, -2), then what is the equation of the line mirror?
(a) y = x (b) y = 2x
(c) 4y = x (d) y = 4x
(b) Let A = (-4, 2)
image point A' = (4, -2)
therefore Mid-point of AA' = ((-4 + 4)/2, (2 + (-2))/2) = (0, 0)
Slope of AA' = (-2 - 2)/(4 - (-4))
= -4/8 = -1/2
Since, AA' and mirror line are perpendicular.
Slope of line mirror = -1 / Slope of AA' = -1 / (-1/2) = 2
Equation of line is y - y1 = m(x - x1)
therefore Equation of a line mirror is y - 0 = 2(x - 0)
=> y = 2x
Hence, option (b) is correct.

61. Consider the following statements in respect of the points (p, p - 3), (q + 3, q) and (6, 3)
1. The points lie on a straight line.
2. The points always lie in the first quadrant only for any value of p and q.
Which of the above statement(s) is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
(a) Given points are A(p, p - 3), B(q + 3, q) and C(6, 3)
As, Points lies on a straight line,
so slope of AB = slope of BC
(q - p + 3)/(q + 3 - p) = (3 - q)/(6 - q - 3)
[because slope of a line = (y2 - y1)/(x2 - x1)]
1 = 1
=> Statement 1 is correct.
But it's not necessary that the collinear points lie in the first quadrant only.
=> Statement 2 is wrong.
Hence, option (a) is correct.

===== Page 9 =====

62. What is the acute angle between the lines x - 2 = 0 and sqrt(3)x - y - 2 = 0?
(a) 0° (b) 30° (c) 45° (d) 60°
(b) l1: x - 2 = 0
l2: sqrt(3)x - y - 2 = 0
Slope of line l1, m1 = - coefficient of x / coefficient of y
= -1/0 = infinity
The line l1 is parallel to Y-axis or perpendicular to X-axis.
Slope of line l2, m2 = -sqrt(3)/(-1) = sqrt(3)
The line l2 makes an angle 60° from positive X-axis.
Angle between l1 and l2 = 90° - 60° = 30°
Hence, option (b) is correct.

63. The point of intersection of diagonals of a square ABCD is at the origin and one of its vertices is at A(4, 2). What is the equation of the diagonal BD?
(a) 2x + y = 0 (b) 2x - y = 0
(c) x + 2y = 0 (d) x - 2y = 0
(a) Since, diagonal BD passes through the origin O(0, 0).
Slope of OA = (0 - 2)/(0 - 4) = 1/2
OA and OB are perpendicular to each other
slope of OB = -1 / slope of OA = -1 / (1/2) = -2
Eqs. of BD having slope -2 and passes through (0, 0)
y - 0 = -2[x - 0] [Equation of a line => y - y1 = m(x - x1)]
2x + y = 0
Hence, option (a) is correct.

64. If any point on a hyperbola is (3 tan theta, 2 sec theta), then what is the eccentricity of the hyperbola?
(a) 3/2 (b) 5/2 (c) sqrt(11)/2 (d) sqrt(13)/2
(d) Given point is (3 tan theta, 2 sec theta)
=> x = 3 tan theta, y = 2 sec theta
x/3 = tan theta, y/2 = sec theta
therefore sec^2 theta - tan^2 theta = 1
y^2/4 - x^2/9 = 1
which represents conjugate Hyperbola.
=> a^2 = 9, b^2 = 4
therefore e = sqrt(1 + a^2/b^2) = sqrt(1 + 9/4) = sqrt(13/4)
e = sqrt(13)/2
Hence, option (d) is correct.

65. Consider the following with regard to eccentricity (e) of a conic section
1. e = 0 for circle
2. e = 1 for parabola
3. e < 1 for ellipse
Which of the above are correct?
(a) 1 and 2 (b) 2 and 3
(c) 1 and 3 (d) 1, 2 and 3
(d) Since, we know that circle has eccentricity 0 and parabola has eccentricity 1 and ellipse has eccentricity < 1 and hyperbola has eccentricity > 1
Hence, option (d) is correct.

66. What is the angle between the two lines having direction ratios (6, 3, 6) and (3, 3, 0)?
(a) pi/6 (b) pi/4 (c) pi/3 (d) pi/2
(b) Direction ratios of line l1 = <6, 3, 6>
a1 = 6, b1 = 3, c1 = 6
Direction ratios of line l2 = <3, 3, 0>
=> a2 = 3, b2 = 3, c2 = 0
therefore cos theta = (a1a2 + b1b2 + c1c2)/(sqrt(a1^2 + b1^2 + c1^2) sqrt(a2^2 + b2^2 + c2^2))
= (6×3 + 3×3 + 6×0)/(sqrt(6^2 + 3^2 + 6^2) sqrt(3^2 + 3^2 + 0))
=> cos theta = 27/(9 × 3sqrt(2))
=> cos theta = 1/sqrt(2) = cos pi/4
therefore theta = pi/4
Hence, option (b) is correct.

67. If l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z then what is l^4 + m^4 + n^4 equal to?
(a) 1 (b) 11/27 (c) 13/27 (d) 4
(b) Given line is x - 1 = 2(y + 3) = 1 - z
=> (x - 1)/2 = (y + 3)/1 = (1 - z)/2
=> (x - 1)/2 = (y - (-3))/1 = (z - 1)/(-2)
Direction ratios are <2, 1, -2>
Direction cosines are 2/sqrt(2^2 + 1^2 + (-2)^2), 1/sqrt(2^2 + 1^2 + (-2)^2), -2/sqrt(2^2 + 1^2 + (-2)^2)
therefore l = 2/3, m = 1/3, n = -2/3
therefore l^4 + m^4 + n^4 = (2/3)^4 + (1/3)^4 + (-2/3)^4
= (16 + 1 + 16)/81 = 33/81 = 11/27
Hence, option (b) is correct.

68. What is the projection of the line segment joining A(1, 7, -5) and B(-3, 4, -2) on Y-axis?
(a) 5 (b) 4 (c) 3 (d) 2
(c) A = (1, 7, -5) and B = (-3, 4, -2)
Direction ratios of AB = <(-3 - 1), (4 - 7), (-2 + 5)>
= <-4, -3, 3>
=> a = -4, b = -3, c = 3
Direction cosines of Y-axis = <0, 1, 0>
l = 0, m = 1, n = 0
therefore Projection of AB on Y-axis = |al + bm + cn|
= |-4 × 0 + (-3) × 1 + 3 × 0| = 3
Hence, option (c) is correct.

69. What is the number of possible values of k for which the line joining the points (k, 1, 3) and (1, -2, k + 1) also passes through the point (15, 2, -4)?
(a) Zero (b) One (c) Two (d) Infinite
(c) Let A = (k, 1, 3), B = (1, -2, k + 1) and C = (15, 2, -4)
Since, line AB passes through C also.
Hence, points A, B and C are collinear.
therefore |k 1 3; 1 -2 k + 1; 15 2 -4| = 0
k(8 - 2k - 2) - 1(-4 - 15k - 15) + 3(2 + 30) = 0

===== Page 10 =====

6k - 2k^2 + 19 + 15k + 96 = 0
2k^2 - 21k - 115 = 0 which is quadratic equation.
=> k has two values.
Hence, option (c) is correct.

70. The foot of the perpendicular drawn from the origin to the plane x + y + z = 3 is
(a) (0, 1, 2) (b) (0, 0, 3)
(c) (1, 1, 1) (d) (-1, 1, 3)
(c) Let the foot of the perpendicular drawn from the origin to the plane x + y + z = 3 be (a, b, c).
Direction ratios of the plane = <1, 1, 1>
Direction ratios of OA and normal will be in the same ratio.
therefore (a - 0)/1 = (b - 0)/1 = (c - 0)/1
=> a = 1, b = 1, c = 1
A = (1, 1, 1)
Hence, option (c) is correct.

71. A vector r = a i + b j is equally inclined to both x and y axes. If the magnitude of the vector is 2 units, then what are the values of a and b respectively?
(a) 1/2, 1/2 (b) 1/sqrt(2), 1/sqrt(2)
(c) sqrt(2), sqrt(2) (d) 2, 2
(c) r = a i + b j
|r| = sqrt(a^2 + b^2) = 2
Since, r is equally inclined from X-axis and Y-axis.
Hence, r makes 45° from the X-axis.
therefore a = |r| cos 45° and b = |r| sin 45°
a = 2 × 1/sqrt(2), and b = 2 × 1/sqrt(2)
a = sqrt(2) and b = sqrt(2)
Hence, option (c) is correct.

72. Consider the following statements in respect of a vector c = a + b where |a| = |b| ≠ 0
1. c is perpendicular to (a - b)
2. c is perpendicular to (a × b)
Which of the above statements is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
(c) c = a + b where |a| = |b| ≠ 0
Consider, c · (a - b) = (a + b) · (a - b)
= |a|^2 - |b|^2 = |b|^2 - |b|^2 = 0
=> c is perpendicular to (a - b).
Also, c · (a × b) = (a + b) · (a × b)
= a · (a × b) + b · (a × b)
= 0 + 0 = 0
=> c is perpendicular to (a × b)
Hence, option (c) is correct.

73. If a and b are two vectors such that |a + b| = |a - b| = 4, then one of the following is correct?
(a) a and b must be unit vectors
(b) a must be parallel to b
(c) a must be perpendicular to b
(d) a must be equal to b
(c) Given, |a + b| = |a - b| = 4
=> |a + b|^2 = |a - b|^2
|a|^2 + |b|^2 + 2a·b = |a|^2 + |b|^2 - 2a·b
=> 4a·b = 0 => a·b = 0
=> a must be perpendicular to b.
Hence, option (c) is correct.

74. If a, b and c are coplanar, then what is (2a × 3b) · 4c + (5b × 3c) · 6a equal to?
(a) 114 (b) 66 (c) 0 (d) -66
(c) Given that, a, b and c are coplanar
=> [a b c] = 0 ... (i)
therefore (2a × 3b) · 4c + (5b × 3c) · 6a
= 2·3·4 [a b c] + 5·3·6 [b c a]
= 24[a b c] + 90[a b c] [because [a b c] = [b c a]]
= 24 × 0 + 90 × 0 = 0
Hence, option (c) is correct.

75. Consider the following statements
1. The cross product of two unit vectors is always a unit vector.
2. The dot product of two unit vectors is always unity.
3. The magnitude of sum of two unit vectors is always greater than the magnitude of their difference.
Which of the above statements are not correct?
(a) 1 and 2 (b) 2 and 3
(c) 1 and 3 (d) 1, 2 and 3
(d) Statement I
Let a and b are unit vectors
i.e. |a| = |b| = 1
a × b = |a||b| sin theta n hat
|a × b| = |a||b| sin theta
= sin theta in [-1, 1]
Therefore, statement I is incorrect.
Statement II
Let a and b are unit vectors
i.e. a · b = |a||b| cos theta
= cos theta in [-1, 1]
Therefore, statement II is incorrect.
Statement III
|a + b| = sqrt(|a|^2 + |b|^2 + 2|a||b| cos theta)
= sqrt(2 + 2 cos theta)
|a - b| = sqrt(2 - 2 cos theta)
For theta = 0, |a + b| = 2, |a - b| = 0
For theta = pi, |a + b| = 0, |a - b| = 2
So, statement III is incorrect.
Hence, option (d) is correct.

===== Page 11 =====

On integrating both sides
∫ dx/(x + 1) = ∫ dt
ln(x + 1) = t + c ... (i)
Since, at t = 0, distance (x) = 0
therefore ln(0 + 1) = 0 + c
0 = c
therefore ln(x + 1) = t
At x = 24 m
t = ln(24 + 1) = ln 25 = ln 5^2
t = 2 ln 5
Hence, option (c) is correct.

78. What is ∫_0^a f(a - x)/(f(x) + f(a - x)) dx equal to?
(a) a (b) 2a
(c) 0 (d) a/2
(d) Let I = ∫_0^a f(a - x)/(f(x) + f(a - x)) dx ... (i)
x -> a - x
I = ∫_0^a f(a - (a - x))/(f(a - x) + f(a - (a - x))) dx
I = ∫_0^a f(x)/(f(a - x) + f(x)) dx ... (ii)
Adding Eqs. (i) and (ii), we get
2I = ∫_0^a (f(a - x) + f(x))/(f(a - x) + f(x)) dx
2I = ∫_0^a 1 dx
2I = [x]_0^a
2I = a - 0
I = a/2
Hence, option (d) is correct.

79. What is lim_{x -> 1} (x^3 + x^2)/(x^2 + 3x + 2) equal to?
(a) 0 (b) 1 (c) 2 (d) 3
(b) Given, lim_{x -> 1} (x^3 + x^2)/(x^2 + 3x + 2) (0/0 form)
By using L'Hospital rule
lim_{x -> 1} (3x^2 + 2x)/(2x + 3)
= (3(-1)^2 + 2(-1))/(-2 + 3) = (3 - 2)/1 = 1
Hence, option (b) is correct.

80. If ∫_a^b [f(x) + f(-x)] dx = ∫_a^b g(x) dx then what is g(x) equal to?
(a) f(x) (b) f(-x) + f(x)
(c) -f(x) (d) None of these
(a) Given that,
∫_a^b [f(x) + f(-x)] dx = ∫_a^b g(x) dx
If g(x) = f(x)
R.H.S. Let I = ∫_a^b g(x) dx
=> I = ∫_a^b f(x) dx ... (i)
[because ∫_a^b f(x) dx = ∫_a^b f(a + b - x) dx]
I = ∫_a^b f(-x) dx ... (ii)
Adding Eqs. (i) and (ii), we get
2I = ∫_a^b [f(x) + f(-x)] dx (even function)
=> 2I = 2∫_0^b [f(x) + f(-x)] dx
I = ∫_0^b [f(x) + f(-x)] dx = L.H.S
=> g(x) = f(x) and
Hence, option (a) is correct.

81. What is the area bounded by y = sqrt(16 - x^2), y >= 0 and the X-axis?
(a) 16π sq. units (b) 8π sq. units
(c) 4π sq. units (d) 2π sq. units
(b) Shaded portion in the diagram represents the area bounded by y = sqrt(16 - x^2), y >= 0 and X-axis.
Put y = 0, then 16 - x^2 = 0
=> x = ±4
therefore Required area = ∫_{-4}^4 sqrt(16 - x^2) dx
= 2∫_0^4 sqrt(16 - x^2) dx
= 2[ x/2 sqrt(16 - x^2) + 16/2 sin^{-1}(x/4)]_0^4
= 2[0 + 8 sin^{-1}1] = 2 × 8π/2
= 8π sq units
Hence, option (b) is correct.

82. The curve y = -x^3 + 3x^2 + 2x - 27 has the maximum slope at
(a) x = -1 (b) x = 0
(c) x = 1 (d) x = 2
(c) Given that, y = -x^3 + 3x^2 + 2x - 27
Slope = dy/dx = -3x^2 + 6x + 2
therefore f'(x) = -3x^2 + 6x + 2
For maxima/minima of f'(x).
d/dx f'(x) = -6x + 6 = 0
=> 6x = 6 => x = 1
At x = 1, d^2/dx^2 f'(x) = -6 < 0
therefore At x = 1, f'(x) is maximum.
Hence, option (c) is correct.

83. A 24 cm long wire is bent to form a triangle with one of the angles as 60°. What is the altitude of the triangle having the greatest possible area?
(a) 4sqrt(3) cm (b) 2sqrt(3) cm
(c) 6 cm (d) 3 cm
(a)
Given, a + b + c = 24
=> c = 24 - (a + b)
Again cos C = (a^2 + b^2 - c^2)/(2ab)
=> cos 60° = (a^2 + b^2 - c^2)/(2ab)
=> 1/2 = (a^2 + b^2 - c^2)/(2ab)
=> ab = a^2 + b^2 - c^2
=> ab = a^2 + b^2 - [24 - (a + b)]^2
=> ab = a^2 + b^2 - 576 + 48(a + b) - (a + b)^2
=> ab = a^2 + b^2 - 576 + 48a + 48b - a^2 - b^2 - 2ab
=> 3ab - 48(a + b) = -576
=> ab - 16(a + b) = -192
=> ab - 16a - 16b = -192
=> a(b - 16) = 16b - 192
=> a = 16(b - 12)/(b - 16)
Again ar(ΔABC) = 1/2 ab sin C
= 1/2 × 16(b - 12)b/(b - 16) × sin 60°
= 1/2 × 16(b - 12)b/(b - 16) × sqrt(3)/2
= 4sqrt(3)(b^2 - 12b)/(b - 16)
dA/db = 4sqrt(3) [(2b - 12)(b - 16) - (b^2 - 12b)]/(b - 16)^2 = 0
Maximum value for A
dA/db = 0
=> 4sqrt(3)/(b - 16)^2 [2b^2 - 32b - 12b + 192 - b^2 + 12b] = 0

===== Page 12 =====

=> b^2 - 32b + 192 = 0
=> (b - 24)(b - 8) = 0
=> b = 24, 8
when, b = 24
a = 16(24 - 12)/(24 - 16) = (16 × 12)/8 = 24
and c = 24 - (24 + 24) = -24 It is impossible,
when, b = 8
a = 16(8 - 12)/(8 - 16) = 16(-4)/(-8) = 8
and c = 24 - (8 + 8) = 8
So triangle will be equilateral.
Height = sqrt(3)/2 (side) = sqrt(3)/2 × 8 = 4sqrt(3) cm

84. If f(x) = e^{|x|}, then which one of the following is correct?
(a) f'(0) = 1 (b) f'(0) = -1
(c) f'(0) = 0 (d) f'(0) does not exist
(d) Given that, f(x) = e^{|x|}
=> f(x) = { e^x ; x >= 0
           e^{-x} ; x < 0 }
LHD at x = 0
f'(0^-) = lim_{h -> 0^-} (f(0 + h) - f(0))/h
= lim_{h -> 0^-} (e^{-h} - e^0)/h
(b) using L'Hospital rule lim_{h -> 0^-} e^{-h}/1 = -1
RHD at x = 0
f'(0^+) = lim_{h -> 0^+} (f(0 + h) - f(0))/h
= lim_{h -> 0^+} (e^h - e^0)/h
= lim_{h -> 0^+} e^h/1 = e^0 = 1
LHD ≠ RHD
f'(x) does not exist at x = 0
Hence, option (d) is correct.

85. What is ∫ dx/(sec x + tan x) equal to?
(a) ln(sec x) + ln|sec x + tan x| + c
(b) ln(sec x) - ln|sec x + tan x| + c
(c) sec x tan x - ln|sec x - tan x| + c
(d) ln|sec x + tan x| - ln|sec x| + c
(d) Let I = ∫ dx/(sec x + tan x)
I = ∫ 1/(sec x + tan x) × (sec x - tan x)/(sec x - tan x) dx

86. What is ∫ dx/(sec^2(tan^{-1} x)) equal to?
(a) sin^{-1} x + c (b) tan^{-1} x + c
(c) sec^{-1} x + c (d) cos^{-1} x + c
(b) Let I = ∫ dx/(sec^2(tan^{-1} x))
= ∫ dx/(1 + x^2)
= ∫ dx/(1 + x^2)
I = tan^{-1} x + C
Hence, option (b) is correct.

87. If x + y = 20 and P = xy, then what is the maximum value of P?
(a) 100 (b) 96 (c) 84 (d) 50
(a) Given, x + y = 20
=> y = 20 - x
P = xy
P = x(20 - x)
P = 20x - x^2
therefore dP/dx = 20 - 2x
For maxima/minima, dP/dx = 0
20 - 2x = 0
=> x = 10
=> x = 10
therefore P
=> y = 20 - 10
therefore Maximum value of P = xy
= 10 × 10 = 100
Hence, option (a) is correct.

88. What is the derivative of sin(ln x) + cos(ln x) with respect to x at x = e?
(a) (cos 1 - sin 1)/e
(b) (cos 1 + sin 1)/e
(c) e
(d) 1/e
(a) Let y = sin(ln x) + cos(ln x)
therefore dy/dx = cos(ln x)·1/x + (-sin(ln x)·1/x)
= 1/x [cos(ln x) - sin(ln x)]
At x = e
dy/dx = 1/e [cos(ln e) - sin(ln e)]
= 1/e [cos 1 - sin 1]
[because ln e = 1]
Hence, option (a) is correct.

89. If x = e^t cos t and y = e^t sin t, then what is dx/dy at t = 0 equal to?
(a) 0 (b) 1 (c) -1 (d) 2
(b) Given that, x = e^t cos t, y = e^t sin t
therefore dx/dt = e^t d/dt cos t + cos t d/dt e^t
= e^t(-sin t) + cos t·e^t
dy/dt = e^t d/dt sin t + sin t·d/dt e^t
dy/dt = e^t cos t + e^t sin t
therefore dx/dy = (dx/dt)/(dy/dt)
therefore dx/dy = (dx/dt)/(dy/dt)
dx/dy = e^t(cos t - sin t)/(e^t(cos t + sin t))
therefore dx/dy = (cos 0 - sin 0)/(cos 0 + sin 0) = (1 - 0)/(1 + 0)
(dx/dy)_{t = 0} = 1
Hence, option (b) is correct.

90. What is the maximum value of sin 2x·cos 2x?
(a) 1/2 (b) 1 (c) 2 (d) 1/4

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92. If a differentiable function f(x) satisfies lim_{x -> -1} (f(x) + 1)/(x^2 - 1) = -3/2, then what is lim_{x -> -1} f(x) equal to?
(a) -3/2 (b) -1 (c) 0 (d) 1
(b) Given, lim_{x -> -1} (f(x) + 1)/(x^2 - 1) = -3/2
lim_{x -> -1} (f(x) + 1) = 0
=> lim_{x -> -1} f(x) + 1 = 0
=> lim_{x -> -1} f(x) = -1
Hence, option (b) is correct.

93. If the function
is continuous, then what is the value of (a + b)?
(a) 5 (b) 10 (c) 15 (d) 20
(a) Given that, f(x) = { a + bx ; x < 1
                        5 ; x = 1
                        b - ax ; x > 1 }
therefore f(x) is continuous.
=> f(x) will be continuous at x = 1
lim_{x -> 1^-} f(x) = f(1) = lim_{x -> 1^+} f(x)
lim_{x -> 1^-} (a + bx) = 5 = lim_{x -> 1^+} (b - ax)
a + b = 5 = b - a
=> a + b = 5
Hence, option (a) is correct.

94. Consider the following statements in respect of the function f(x) = sin x
1. f(x) increases in the interval (0, pi)
2. f(x) decreases in the interval (5pi/2, 3pi)
Which of the above statement is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
(b) Given, f(x) = sin x
From the graph of sin x
We can see that f(x) increases in [0, pi/2] and decreases in [pi/2, pi] and (5pi/2, 3pi)
=> Statement-1 is wrong and Statement-2 is correct.
Hence, option (b) is correct.

95. What is the domain of the function f(x) = 3^x?
(a) (-infinity, infinity) (b) (0, infinity)
(c) [0, infinity)
(a) Given, f(x) = 3^x
We know that, domain of exponential function is (-infinity, infinity)
Domain of 3^x = (-infinity, infinity)
Hence, option (a) is correct.

96. If the general solution of a differential equation is y^2 + 2cy - cx + c^2 = 0, where c is an arbitrary constant, then what is the order of the differential equation?
(a) 1 (b) 2 (c) 3 (d) 4
(a) Given that, y^2 + 2cy - cx + c^2 = 0
Since, the above equation contains only one variable constant.
Hence, order of the differential equation = 1
Hence, option (a) is correct.

97. What is the degree of the following differential equation?
x = sqrt(1 + d^2y/dx^2)
(a) 1 (b) 2 (c) 3 (d) Degree is not defined
(a) Let x = sqrt(1 + d^2y/dx^2)
=> x^2 = 1 + d^2y/dx^2 => (d^2y/dx^2)^1 = x^2 - 1
Degree = exponent of highest order derivative = 1
Hence, option (a) is correct.

98. Which one of the following differential equations has the general solution y = ae^x + be^{-x}?
(a) d^2y/dx^2 + y = 0 (b) d^2y/dx^2 - y = 0
(c) d^2y/dx^2 + y = 1 (d) dy/dx - y = 0
(b) Given, y = ae^x + be^{-x}
therefore dy/dx = ae^x - be^{-x}
d^2y/dx^2 = ae^x + be^{-x} = y
=> d^2y/dx^2 - y = 0
Hence, option (b) is correct.

99. What is the solution of the following differential equation?
ln(dy/dx) + y = x
(a) e^x + e^y = c (b) e^x + y = c
(c) e^x - e^y = c (d) e^x - y = c
(c) Given, ln(dy/dx) + y = x
=> ln(dy/dx) = x - y
=> dy/dx = e^{x - y}
=> dy/dx = e^x/e^y
=> e^y dy = e^x dx
On integrating both sides,
∫ e^y dy = ∫ e^x dx
e^y + c = e^x => e^x - e^y = c
Hence, option (c) is correct.

100. What is ∫ e^{(2 ln x + ln x^2)} dx equal to?
(a) x^4/4 + C (b) x^3/3 + C
(c) 2x^5/5 + C (d) x^5/5 + C
(d) Let I = ∫ e^{(2 ln x + ln x^2)} dx
= ∫ e^{(ln x^2 + ln x^2)} dx
= ∫ e^{2 ln x^2} dx = ∫ e^{ln (x^2)^2} dx = ∫ x^4 dx
I = x^5/5 + C
Hence, option (d) is correct.

101. Consider the following measures of central tendency for a set of N numbers
1. Arithmetic mean
2. Geometric mean

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(c) Since, we know that the measures of central tendency are Mean, Median and Mode. Where Arithmetic Mean and Geometric mean are the type of mean.
Hence, option (c) is correct.

102. The numbers of Science, Arts and Commerce graduates working in a company are 30, 70 and 50 respectively. If these figures are represented by a pie chart, then what is the angle corresponding to Science graduates?
(a) 36° (b) 72° (c) 120° (d) 168°
(b) The ratio of Science, Arts and Commerce graduates
= 30:70:50 = 3:7:5
Angle corresponding to Science graduates = 3/(3 + 7 + 5) × 360°
= 3/15 × 360° = 72°
Hence, option (b) is correct.

103. For a histogram based on a frequency distribution with unequal class intervals, the frequency of a class should be proportional to
(a) the height of the rectangle
(b) the area of the rectangle
(c) the width of the rectangle
(d) the perimeter of the rectangle
(b) Since, we know that for a histogram, based on a frequency distribution with equal intervals, the frequency of a class is proportional to height of the rectangle and for a histogram based on frequency distribution with unequal intervals, the frequency of a class is proportional to Area of the rectangle.
Hence, option (b) is correct.

104. The coefficient of correlation is independent of
(a) change of scale only
(b) change of origin only
(c) both change of scale and change of origin
(d) neither change of scale nor change of origin
(c) Since, we know that coefficient of correlation is independent of both change of scale and change of origin.
Hence, option (c) is correct.

105. The following table gives the frequency distribution of number of peas per pea pod of 198 pods

Number of peas: 1 2 3 4 5 6 7
Frequency: 4 3 3 7 6 5 0 2 6 8 1

What is the median of this distribution?
(a) 3 (b) 4 (c) 5 (d) 6
(a)
Number of peas | Frequency | Cumulative frequency
1 | 4 | 4
2 | 3 | 7
3 | 3 | 10
4 | 7 | 17
5 | 6 | 23
6 | 5 | 28
7 | 0 | 28? (table OCR unclear)
Σf = 198
therefore N = 198, N/2 = 198/2 = 99
therefore Median = (N/2)th term + (N/2 + 1)th term / 2
Median = (99th term + 100th term)/2
= (3 + 3)/2 = 3
therefore Median = 3
Hence, option (a) is correct.

106. If M is the mean of n observations x1 - k, x2 - k, x3 - k, ..., xn - k, where k is any real number, then what is the mean of x1, x2, x3, ..., xn?
(a) M (b) M + k (c) M - k (d) kM
(b) Given that, Mean of x1 - k, x2 - k, x3 - k, ..., xn - k
therefore M = ((x1 - k) + (x2 - k) + ... + (xn - k))/n
M = (x1 + x2 + ... + xn)/n - nk/n
M + k = (x1 + x2 + ... + xn)/n
therefore Mean of x1, x2, ..., xn = M + k
Hence, option (b) is correct.

107. What is the sum of deviations of the variate values 73, 85, 92, 105, 120 from their mean?
(a) -2 (b) -1 (c) 0 (d) 5
(c) Mean of 73, 85, 92, 105, 120
x bar = (73 + 85 + 92 + 105 + 120)/5
= 475/5
x bar = 95
Sum of deviations from their mean
= (73 - 95) + (85 - 95) + (92 - 95) + (105 - 95) + (120 - 95)
= -22 - 10 - 3 + 10 + 25 = 0
Hence, option (c) is correct.

108. Let x be the HM and y be the GM of two positive numbers m and n. If 5x = 4y, then which one of the following is correct?
(a) 5m = 4n (b) 2m = n
(c) 4m = 5n (d) m = 4n
(d) Given, two positive numbers are m and n.
therefore H.M. of m and n = 2mn/(m + n)
therefore x = 2mn/(m + n)
G.M. of m and n = sqrt(mn)
y = sqrt(mn)
therefore 5x = 4y
Squaring both sides, we get
(5mn/(m + n))^2 = (2sqrt(mn))^2
=> 25m^2n^2/(m^2 + n^2 + 2mn) = 4mn
=> 25mn = 4m^2 + 4n^2 + 8mn
[because m ≠ 0, n ≠ 0]
=> 4m^2 + 4n^2 - 17mn = 0
=> 4m^2 - 16mn - mn + 4n^2 = 0
=> 4m(m - 4n) - n(m - 4n) = 0
=> (m - 4n)(4m - n) = 0
=> m = 4n or n = 4m
Hence, option (d) is correct.

109. If the mean of a frequency distribution is 100 and the coefficient of variation is 45%, then what is the value of the variance?
(a) 2025 (b) 450 (c) 45 (d) 4.5
(a) Since, we know that Coefficient of variation (CV) = sigma/x bar × 100
Where sigma is standard deviation and x bar is mean.

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111. Let two events A and B be such that P(A) = L and P(B) = M. Which one of the following is correct?
(a) P(A|B) < (L + M - 1)/M
(b) P(A|B) > (L + M - 1)/M
(c) P(A|B) >= (L + M - 1)/M
(d) P(A|B) = (L + M - 1)/M
(c) Given, P(A) = L, P(B) = M
therefore P(A|B) = P(A ∩ B)/P(B)
P(A|B) = (P(A) + P(B) - P(A ∪ B))/P(B)
P(A|B) = (L + M - P(A ∪ B))/P(B)
therefore P(A ∪ B) = L + M - P(B)P(A|B)
therefore 0 <= P(A ∪ B) <= 1
=> L + M - P(B)·P(A|B) <= 1
=> P(B)·P(A|B) >= L + M - 1
P(A|B) >= (L + M - 1)/M [P(B) = M]
Hence, option (c) is correct.

112. For which of the following sets of numbers do the mean, median and mode have the same value?
(a) 12, 12, 12, 24
(b) 6, 18, 18, 18, 30
(c) 6, 6, 12, 30, 36
(d) 6, 6, 6, 12, 30
(b) For option (a),
Mean = (12 + 12 + 12 + 24)/5 = 14.4 ≠ mode (12)
For option (b),
Mean = (6 + 18 + 18 + 18 + 30)/5 = 18
Mode = 18
Median = 18
Hence, for the data 6, 18, 18, 18, 30, Mean = Mode = Median = 18
Hence, option (b) is correct.

113. The mean of 12 observations is 75. If two observations are discarded, then the mean of the remaining observations is 65. What is the mean of the discarded observations?
(a) 250 (b) 125 (c) 120 (d) Cannot be determined due to insufficient data
(b) Given, mean of 12 observations = 75
x bar = (Σ_{i=1}^{12} xi)/12 => 75 = (Σ_{i=1}^{12} xi)/12
=> Σ_{i=1}^{12} xi = 900 ... (i)
Let observations x11 and x12 is discarded then mean = (Σ_{i=1}^{10} xi)/10 = 65
therefore Σ_{i=1}^{10} xi = 10 × 65 = 650
From Eq. (i) Σ_{i=1}^{12} xi = 900
=> Σ_{i=1}^{10} xi + x11 + x12 = 900
=> 650 + x11 + x12 = 900
=> x11 + x12 = 250
Mean of x11 and x12 = 250/2 = 125
Hence, option (b) is correct.

114. If k is one of the roots of the equation x(x + 1) + 1 = 0 then what is its other root?
(a) 1 (b) -k (c) k^2 (d) -k^2
(c) Given, quadratic equation
x(x + 1) + 1 = 0
x^2 + x + 1 = 0 ... (i)
Since, we know that omega, omega^2 are the roots of Equation when
omega = (-1 + sqrt(3)i)/2 and omega^2 = (-1 - sqrt(3)i)/2
If one of the roots of Eqs. (i) is k then other root will be k^2
Hence, option (c) is correct.

115. The geometric mean of a set of observations is computed as 10. The geometric mean obtained when each observation xi is replaced by 3xi^4 is
(a) 810 (b) 900 (c) 30000 (d) 81000
(c) Given that, geometric mean of a set of observations = 10
Since, we know that if Geometric mean of x1, x2, x3, ..., xn is G
=> Geometric mean of x1^2, x2^2, x3^2, ..., xn^2 is G^2
Geometric mean of k1^2, k2^2, ..., kn^2 is G^2
Required geometric mean = 3(10)^4
= 3 × 10000 = 30000
Hence, option (c) is correct.

116. If P(A ∪ B) = 5/6, P(A ∩ B) = 1/3 and P(A bar) = 1/2 then which of the following is/are correct?
1. A and B are independent events.
2. A and B are mutually exclusive events.
Select the correct answer using the code given below.
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
(a) Given, P(A ∪ B) = 5/6, P(A ∩ B) = 1/3
P(A bar) = 1/2
=> P(A) = 1 - 1/2 = 1/2
P(B) = P(A ∪ B) - P(A) + P(A ∩ B)
= 5/6 - 1/2 + 1/3 = 4/6 = 2/3
If A and B are independents, then P(A ∩ B) = P(A)·P(B)
therefore P(A ∩ B) = 1/3 and P(A)·P(B) = 1/2 × 2/3 = 1/3
=> Statement-1 is correct.
If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B)
therefore P(A ∪ B) = 5/6
P(A) + P(B) = 1/2 + 2/3 = 7/6
=> Statement-2 is wrong.
Hence, option (a) is correct.

117. The average of a set of 15 observations is recorded, but later it is found that for one observation, the digit in the tens place was wrongly recorded as 8 instead of 3. After correcting the observation, the average is
(a) reduced by 1/3 (b) increased by 10/3
(c) reduced by 10/3 (d) reduced by 50
(c) Let unit digit for wrongly recorded observation = b
When tens digit is 8, then number = 10 × 8 + b = 80 + b
When tens digit is 3, then number = 10 × 3 + b = 30 + b
=> One observation is recorded ((80 + b) - (30 + b)) more while calculating average.
Hence, after correcting the observation, the average will be reduced by
= ((80 + b) - (30 + b))/15
= 50/15 = 10/3
Hence, option (c) is correct.

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118. A coin is tossed twice. If E and F denote occurrence of head on first toss and second toss respectively, then what is P(E ∪ F) equal to?
(a) 1/4 (b) 1/2
(c) 3/4 (d) 1/3
(c) Given that, a coin is tossed twice.
therefore S = {HH, HT, TH, TT}
Given, E be the event of occurrence of head on first toss and F be the event of occurrence of head on second toss.
therefore E = {HH, HT}
therefore P(E) = n(E)/n(S) = 2/4 = 1/2
F = {TH, HH}, P(F) = 2/4 = 1/2
therefore E ∩ F = {HH}
therefore P(E ∩ F) = 1/4
therefore P(E ∪ F) = P(E) + P(F) - P(E ∩ F)
= 1/2 + 1/2 - 1/4
= 1 - 1/4 = 3/4
Hence, option (c) is correct.

119. In a binomial distribution, the mean is 2/3 and variance is 5/9. What is the probability that random variable X = 2?
(a) 5/36 (b) 25/36
(c) 25/54 (d) 25/216
(d) For a binomial distribution mean = np and variance = npq
Where p is probability of success and q is the probability of unsuccess and n is number of observations.
therefore Given, np = 2/3, npq = 5/9
=> 2/3 q = 5/9
q = 5/6
therefore p = 1 - q = 1 - 5/6 = 1/6
therefore np = 2/3
=> n(1/6) = 2/3
=> n = 4
therefore P(X = x) = nCx p^x q^{n - x}
therefore P(X = 2) = 4C2 (1/6)^2 (5/6)^2 = 25/216
Hence, option (d) is correct.

119. If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
(a) 10 (b) 12
(c) 13 (d) 15
(d) Given, observations are
10, 12, 13, 15, 15, 13, 12, 10, x
therefore Mode = 15
Scores | Frequency
10 | 2
12 | 2
13 | 2
15 | 2
Since, frequency of all other numbers is same as frequency of 15.
But mode is the number of highest frequency.
therefore x should be 15
Hence, option (d) is correct.

120. If A and B are two events such that P(A) = 3/4 and P(B) = 5/8, then consider the following statements
1. The minimum value of P(A ∪ B) is 3/4.
2. The maximum value of P(A ∩ B) is 5/8.
Which of the above statements is/are correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
(c) Given, P(A) = 3/4
P(B) = 5/8
because P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
and P(A ∩ B) = P(A) + P(B) - P(A ∪ B)
Here P(A ∪ B) will be minimum if P(A ∩ B) is maximum and vice-versa.
Since, minimum value of P(A ∩ B) is zero and maximum value of P(A ∩ B) is minimum (P(A), P(B))
=> maximum P(A ∩ B) = minimum (3/4, 5/8) = 5/8
=> Statement-2 is correct.
Also, minimum value of P(A ∪ B) is maximum (P(A), P(B))
therefore Minimum value of P(A ∪ B) = maximum (3/4, 5/8) = 3/4
=> Statement-1 is correct.
Hence, correct option is (c).

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