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# NDA/NA(II) Exam 2018 Mathematics Previous Year Paper

July 3, 2019
###### NDA/NA(I) Exam 2018 General Ability Test Previous Year Paper
July 3, 2019

NDA/NA(II) Exam 2018 Mathematics

Q. 1 What is the value of log₇ log₇ {√7[√7(√7)]} equal to?

A. 3 log₂ 7

B. 1−3 log₂ 7

C. 1−3 log₇ 2

D. 7/8

Q. 2 If an infinite GP has the first term x and the sum 5, then which one of the following is correct ?

A. x < −10

B. −10 < x < 0

C. 0 < x < 10

D. x > 10

Q. 3 Consider the following expressions:

1. x + x² – 1/x

2. √[ax² + bx + x -c + (d/x) – (e/x²)]

3. 3x² − 5x + ab

4. 2/x² − ax + b³

5. 1/x − 2/x + 5

Which of the above are rational expressions?

A. 1, 4 and 5 only

B. 1, 3, 4 and 5 only

C. 2, 4 and 5 only

D. 1 and 2 only

Q. 4 A square matrix A is called orthogonal if

Note: where A′ is the transpose of A

A. A = A²

B. A′ = A⁻¹

C. A = A⁻¹

D. A = A′

Q. 5 If A, B and C are subsets of a Universal set, then which one of the following is not correct?

Note: where A′ is the complement of A

A. A ⋃ (B ∩ C) = (A ⋃ B) ∩ (A ⋃ C)

B. A′ ⋃ (A ⋃ B) = (B′ ∩ A)′ ⋃ A

C. A′ ⋃ (B ⋃ C) = (C′ ∩ B)′ ∩ A′

D. (A ∩ B) ⋃ C = (A ⋃ C) ⋂ (B ⋃ C)

Q. 6 Let x be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to

A. 2480

B. 2481

C. 2482

D. 2483

Q. 7 The sum of the series 3 – 1 + 1/3 – 1/9 + … is equal to

A. 20/9

B. 9/20

C. 9/4

D. 4/9

Questions: 8 – 9

A survey was conducted among 300 students. It was found that 125 students like to play cricket,145 students like to play football and 90 students like to play tennis. 32 students like to play exactly two games out of the three games.

Q. 8 How many students like to play all the three games?

A. 14

B. 21

C. 28

D. 35

Q. 9 How many students like to play exactly only one game?

A. 196

B. 228

C. 254

D. 268

Q. 10 If α and β(≠0) are the roots of the quadratic equation x²+αx-β=0, then the quadratic expression -x²+αx+β where x ⋲ R has

A. Least value -1/4

B. Least value -9/4

C. Greatest value 1/4

D. Greatest value 9/4

Q. 11 What is the coefficient of the middle term in the binomial expansion of (2 + 3x)⁴?

A. 6

B. 12

C. 108

D. 216

Q. 12 For a square matrix A, which of the following properties hold?

1. (A⁻¹)⁻¹ = A

2. det(A⁻¹) = 1/detA

3. (λA)⁻¹ = λA⁻¹ where λ is a scalar

A. 1 and 2 only

B. 2 and 3 only

C. 1 and 3 only

D. 1,2 and 3

Q. 13 Which one of the following factors does the expansion of the determinant shown in figure contain?

A. x − 3

B. x − y

C. y − 3

D. x − 3y

Q. 14 What is the adjoint of the matrix? Choose the correct option.

A. (a)

B. (b)

C. (c)

D. (d)

Q. 15 What is the value of (-1+i√3/2)³ⁿ, where i=√-1 ?

A. 3

B. 2

C. 1

D. 0

Q. 16 There are 17 cricket players, out of which 5 players can bowl. In how many ways can a team of 11 players be selected so as to include 3 bowlers?

A. C (17, 11)

B. C (12, 8)

C. C (17, 5) X C (5, 3)

D. C (5, 3) X C (12, 8)

Q. 17 What is the value of log₉ 27 + log₈ 32 ?

A. 7/2

B. 19/6

C. 4

D. 7

Q. 18 If A and B are two invertible square matrices of same order, then what is (AB)⁻¹ equal to?

A. B⁻¹ A⁻¹

B. A⁻¹ B⁻¹

C. B⁻¹ A

D. A⁻¹ B

Q. 19 What is the solution of the of equation given if a+b+c=0 ?

A. x=a

B. x=[√3(a²+b²+c²)]/√2

C. x=[√2(a²+b²+c²)]/√3

D. x=0

Q. 20 What should be the value of x so that the matrix does not have an inverse?

A. 16

B. -16

C. 8

D. -8

Q. 21 The system of equations

2x + y – 3z = 5,

3x – 2y + 2z = 5 and

5x – 3y – z = 16

A. is inconsistent

B. is consistent, with a unique solution

C. is consistent, with infinitely many solutions

D. has its solution lying along x-axis in three-dimensional space

Q. 22 Which one of the following is correct in respect of the cube roots of unity?

A. They are collinear

B. They lie on a circle of radius √3

C. They form an equilateral triangle

D. None of the above

Q. 23 If u, v and w (all positive) are the pᵗʰ, qᵗʰ and rᵗʰ terms of a GP, then the determinant of the given matrix is?

A. 0

B. 1

C. (p-q)(q-r)(r-p)

D. ln u x ln v x ln w

Q. 24 Let the coefficient of the middle term of the binomial expansion of (1+x)²ⁿ be α and those of two middle terms of the binomial expansion of (1+x)²ⁿ⁻¹ be β and γ. Which one of the following relations is correct?

A. α > β + γ

B. α < β + γ

C. α = β + γ

D. α = βγ

Q. 25 Let A = {x ∈ R : – 1 ≤ x ≤ 1},

B = y ∈ R : – 1 ≤ y ≤ 1} and S be the

subset of A x B, defined by

S = [(x, y) ∈ A x B : x² + y² = 1]

Which one of the following is correct?

A. S is a one-one function from A into B

B. S is a many-one function from A into B

C. S is a bijective mapping from A into B

D. S is not a function

Q. 26 Let Tᵣ be the rth term of an AP for r=1, 2, 3, … . If for some distinct positive integers u and v we have Tᵤ=1/ᵥ and Tᵥ=1/ᵤ, then what is Tᵤᵥ equal to?

A. (uv)⁻¹

B. u⁻¹+v⁻¹

C. 1

D. 0

Q. 27 Suppose f(x) is such a quadratic expression that it is positive for all real x.

If g(x) = f(x) + f'(x) + f”(x), then for any real x

A. g(x) < 0

B. g(x) > 0

C. g(x) = 0

D. g(x) ≥ 0

Q. 28 Consider the following in respect of matrices A, B, and C of same order:

1. (A + B + C)’ = A’ + B’ + C’

2. (AB)’ = A’B’

3. (ABC) = C’B’A’

Where A’ is the transpose of the matrix A.

Which of the above is correct?

A. 1 and 2 only

B. 2 and 3 only

C. 1 and 3 only

D. 1, 2 and 3

Q. 29 The sum of the binary numbers (11011)₂, (10110110)₂ and (10011x0y)₂ is the binary number (101101101)₂. What are the values of x and y?

A. x = 1, y = 1

B. x = 1, y = 0

C. x = 0, y = 1

D. x = 0, y = 0

Q. 30 Let matrix B be the adjoint of a square matrix A, l be the identity matrix of same order as A. If k (≠0) is the determinant of the matrix A, then what is AB equal to?

A. l

B. kl

C. k²l

D. (1/k)l

Q. 31 If (0.2)ⁿ = 2 and log₁₀2 = 0.3010, then what is the value of n to the nearest tenth?

A. -10.0

B. -0.5

C. -0.4

D. -0.2

Q. 32 The total number of 5-digit numbers that can be composed of distinct digits from 0 to 9 is

A. 45360

B. 30240

C. 27216

D. 15120

Q. 33 What is the determinant of the matrix?

A. (x – y) (y – z) (z – x)

B. (x – y) (y – z)

C. (y – z) (z – x)

D. (z – x)² (x + y + z)

Q. 34 As given in the picture, if A, B and C are angles of a triangle then which one of the following is correct?

A. The triangle ABC is isosceles

B. The triangle ABC is equilateral

C. The triangle ABC is scalene

D. No conclusion can be drawn with regard to the nature of the triangle

Q. 35 Consider the following in respect of matrices A and B of same order:

1. A² – B² = (A + B)(A – B)

2. (A – I)(I + A) = O ⇔ A² = 1

where I is the identity matrix and O is the null matrix.

Which of the above is/are correct?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

Q. 36 What is 2 tanθ/1 + tan²θ equal to?

A. cos 2θ

B. tan 2θ

C. sin 2θ

D. cosec 2θ

Q. 37 If sec (θ – a), sec θ and sec (θ + a) are in AP, where cos a ≠ 1, then what is the value of sin²θ + cos a?

A. 0

B. 1

C. -1

D. 1/2

Q. 38 If A + B + C = 180°, then what is sin 2A – sin 2C equal to?

A. -4 sin A sin B sin C

B. -4 cos A sin B cos C

C. -4 cos A cos B sin C

D. -4 sin A cos B cos C

Q. 39 A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the

balloon is 48°. If the height of the balloon above the bridge is 122 m, then what is the length of the bridge?

A. 122 sin 48° m

B. 122 tan 42° m

C. 122 cos 48° m

D. 122 tan 48° m

Q. 40 A is an angle in the fourth quadrant. It satisfies the trigonometric equation 3 (3 – tan²A – cot A)² = 1. Which one of the following is the value of A?

A. 300°

B. 315°

C. 330°

D. 345°

Q. 41 The top of a hill observed from the top and bottom of a building of height h is at angles of elevation π/6 and π/3 respectively. What is the height of the hill?

A. 2h

B. 3h/2

C. h

D. h/2

Q. 42 What is/are the solution(s) of the trigonometric equation cosec x + cot x = √3, where 0 < x < 2π ?

A. 5π/3 only

B. π/3 only

C. π only

D. π, π/3, 5π/3

Q. 43 If θ=π/8, then what is the value of

(2cosθ+1)¹⁰(2cos2θ-1)¹⁰(2cosθ-1)¹⁰(2cos4θ-1)¹⁰?

A. 0

B. 1

C. 2

D. 4

Q. 44 If cos α and cos β (0<α<β<π) are the roots of the quadratic equation 4x²-3=0, then what is the value of sec α and sec β?

A. -4/3

B. 4/3

C. 3/4

D. -3/4

Q. 45 Consider the following values of x:

1. 8

2. -4

3. 1/6

4. -1/4

Which of the above values of x is/are the solution(s) of the equation tan⁻¹(2x)+tan⁻¹(3x)=π/4 ?

A. 3 only

B. 2 and 3 only

C. 1 and 4 only

D. 4 only

Q. 46 If the second term of a GP is 2 and the sum of its infinite terms is 8, then the GP is

A. 8, 2, 1/2, 1/8

B. 10, 2, 2/5, 2/25

C. 4, 2, 1, 1/2, 1/2²

D. 6, 3, 3/2, 3/4

Q. 47 If a, b, c are in AP or GP or HP, then a -b/b – c is equal to

A. b/a or 1 or b/c

B. c/a or c/b or 1

C. 1 or a/b or a/c

D. 1 or a/b or c/a

Q. 48 What is the sum of all three-digit numbers that can be formed using all the digits 3, 4 and 5, when repetition of digits is not allowed?

A. 2664

B. 3882

C. 4044

D. 4444

Q. 49 The ratio of roots of the equations ax²+bx+c=0 and px²+qx+r=0 are equal. If D₁ and D₂ are respective discriminants, then what is D₁/D₂ equal to?

A. a²/p²

B. b²/q²

C. c²/r²

D. None of the above

Q. 50 If A=sin²θ+cos⁴θ, then for all real θ, which one of the following is correct?

A. 1≤ A ≤ 2

B. 3/4 ≤ A ≤ 1

C. 13/16 ≤ A ≤ 1

D. 3/4 ≤ A ≤ 13/16

Q. 51 The equation of a circle whose end points of a diameter are (x₁,y₁) and (x₂,y₂) is

A. (x-x₁)(x-x₂)+(y-y₁)(y-y₂)=x²+y²

B. (x-x₁)²+(y-y₁)²=x₂y₂

C. x²+y²+2x₁x₂+2y₁y₂=0

D. (x-x₁)(x-x₂)+(y-y₁)(y-y₂)=0

Q. 52 The second degree equation x² + 4y² – 2x – 4y + 2 = 0 represents

A. A point

B. An ellipse of semi-major axis 1

C. An ellipse with eccentricity √3/2

D. None of the above

Q. 53 The angle between the two lines lx+my+n=0 and l’x+m’y+n’=0 is given by tan⁻¹θ. What is θ equal to?

A. |lm’-l’m/ll’-mm’|

B. |lm’+l’m/ll’+mm’|

C. |lm’-l’m/ll’+mm’|

D. |lm’+l’m/ll’+mm’|

Q. 54 Consider the following statements:

1. The distance between the lines y=mx+c₁ and y=mx+c₂ is |c₁-c₂|/[√1+m²]

2. The distance between the lines ax+by+c₁=0 and ax+by+c₂=0 is |c₁-c₂|/[√a²+b²]

3. The distance between the lines x=c₁ and x=c₂ is |c₁-c₂|

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

Q. 55 What is the equation of straight line passing through the point of intersection of the lines x/2 + y/3 = 1 and x/3 + y/2 = 1, and parallel to the line 4x + 5y – 6 = 0 ?

A. 20x + 25y – 54 = 0

B. 25x + 20y – 54 = 0

C. 4x + 5y – 54 = 0

D. 4x + 5y – 45 = 0

Q. 56 What is the distance of the point (2, 3, 4) from the plane 3x – 6y + 2z + 11 = 0?

A. 1 unit

B. 2 units

C. 3 units

D. 4 units

Q. 57 Coordinates of the points O, P ,Q and R are respectively (0, 0, 0), (4, 6, 2m), (2, 0, 2n) and (2, 4, 6). Let L, M, N and K be points on the sides OR, OP, PQ and QR respectively such that LMNK is a parallelogram who two adjacent sides LK and LM are each of length √2. What are the values of m and n respectively?

A. 6, 2

B. 1, 3

C. 3, 1

D. None of the above

Q. 58 The line x-1/2 = y-2/3 = z-3/4 is given by

A. x+y+z=6, x+2y-3z=-4

B. x+2y-2z=-1, 4x+4y-5z-3=0

C. 3x+2y-3z=0, 3x-6y+3z=-2

D. 3x+2y-3z=-2, 3x-6y+3z=0

Q. 59 Consider the following statements:

1. The angle between the planes

2x-y+z=1 and x+y+2z=3 is π/3.

2. The distance between the planes

6x-3y+6z+2=0 and

2x-y+2z+4=0 is 10/9

Which of the above statements is/are correct?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

Q. 60 Consider the following statements:

Statement I: If the line segment joining the points P(m, n) and Q(r, s) subtends an angle α at the origin, then cos α = ms-nr/[√(m²+n²)(r²+s²)]

Statement II: In any triangle ABC, it is true that a²=b²+c²-2bc cos A

Which one of the following is correct in respect of the above two statements?

A. Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I

B. Both Statement I and Statement II are true, but Statement II is not the correct

explanation of Statement I

C. Statement I is true, but Statement II is false

D. Statement I is false, but Statement II is true

Q. 61 What is the area of the triangle with vertices (x₁, 1/x₁), (x₂, 1/x₂), (x₃, 1/x₃)?

A. |(x₁-x₂)(x₂-x₃)(x₃-x₁)|

B. 0

C. |(x₁-x₂)(x₂-x₃)(x₃-x₁)/x₁x₂x₃|

D. |(x₁-x₂)(x₂-x₃)(x₃-x₁)/2x₁x₂x₃|

Q. 62 If y-axis touches the circle

x²+y²+gx+fy+e/4=0, then the normal at this point intersects the circle at the point

A. (-g/2, -f/2)

B. (-g, -f/2)

C. (-g/2, f)

D. (-g, -f)

Q. 63 Let |a̅|≠0, |b̅|≠0

(a̅+b̅).(a̅+b̅)=|a̅|²+|b̅|²

holds if and only if

Note: a̅ and b̅ represent vector sign

A. a̅ and b̅ are perpendicular

B. a̅ and b̅ are parallel

C. a̅ and b̅ are inclined at an angle of 45°

D. a̅ and b̅ are anti-parallel

Q. 64 if r̅=xî+yĵ+zk̂, then what is r̅.(î+ĵ+k̂) equal to?

Note: r̅ represents the vector sign.

A. x

B. x+y

C. -(x+y+z)

D. (x+y+z)

Q. 65 A unit vector perpendicular to each of the vectors 2î-ĵ+k̂ and 3î-4ĵ-k̂ is

A. (1/√3)î+(1/√3)ĵ-(1/√3)k̂

B. (1/√2)î+(1/2)ĵ-(1/2)k̂

C. (1/√3)î-(1/√3)ĵ-(1/√3)k̂

D. (1/√3)î+(1/√3)ĵ+(1/√3)k̂

Q. 66 if |a̅|=3, |b̅|=4, and |a̅-b̅|=5, then what is the value of |a̅+b̅|?

Note: a̅, b̅ represent the vector sign.

A. 8

B. 6

C. 5√2

D. 5

Q. 67 Let a̅, b̅ and c̅ be three mutually perpendicular vectors each of unit magnitude. If A̅=a̅+b̅+c̅, B̅=a̅-b̅+c̅ and C̅=a̅-b̅-c̅, then which one of the following is correct?

Note: A̅, B̅, C̅, a̅, b̅ and c̅ represent the vector sign.

A. |A̅|>|B̅|>|C̅|

B. |A̅|=|B̅|≠|C̅|

C. |A̅|=|B̅|=|C̅|

D. |A̅|≠|B̅|≠|C̅|

Q. 68 What is (a̅-b̅)×(a̅+b̅) equal to?

Note: a̅ and b̅ represent the vector sign.

A. 0̅

B. a̅×b̅

C. 2(a̅×b̅)

D. |a̅|²-|b̅|²

Q. 69 A spacecraft located at î+2ĵ+3k̂ is subjected to a force λk̂ by firing a rocket. The spacecraft is subjected to a moment of magnitude

A. λ

B. √3λ

C. √5λ

D. None of the above

Q. 70 In a triangle ABC, if taken in order, consider the given statements.

How many of the statements given in the image are correct

A. One

B. Two

C. Three

D. Four

Q. 71 Let the slope of the curve y=cos⁻¹(sin x) be tan θ. Then the value of θ in the interval (0, π) is

A. π/6

B. 3π/4

C. π/4

D. π/2

Q. 72 If f(x)=(√x-1)/x-4 defines a function on ℝ, then what is its domain?

A. (-∞, 4)⋃(4, ∞)

B. [4, ∞)

C. (1, 4)⋃(4, ∞)

D. [1, 4)⋃(4, ∞)

Q. 73 Which one of the following is correct in respect of the function?

A. It is not continuous at x=0

B. It is continuous at every x

C. It is not continuous at x=π

D. It is continuous at x=0

Q. 74 For the function f(x) = |x-3|, which one of the following is not correct ?

A. The function is not continuous at x = -3

B. The function is continuous at x = 3

C. The function is differentiable at x = 0

D. The function is differentiable at x = -3

Q. 75 If the function f(x)=2x-sin⁻¹x/2x+tan⁻¹x is continuous at each point in its domain, then what is the value of f(0) ?

A. -1/3

B. 1/3

C. 2/3

D. 2

Q. 76 If f(x)=(√25-x²), then what is the given matrix equal to?

A. -1/√24

B. 1/√24

C. -1/4√3

D. 1/4√3

Q. 77 If y=tan⁻¹(5-2tan√x/2+5tan√x), then what is dy/dx equal to?

A. -1/2√x

B. 1

C. -1

D. 1/2√x

Q. 78 Which one of the following is correct in respect of the function f(x)=x sin x + cos x + ½ cos²x?

A. It is increasing in the interval (0, π/2)

B. It remains constant in the interval (0, π/2)

C. It is decreasing in the interval (0, π/2)

D. It is decreasing in the interval (π/4, π/2)

Q. 79 What is the given matrix equal to?

A. √2

B. 2√2

C. 1/√2

D. -1/2√2

Q. 80 A function f : A→R is defined by the equation f(x)=x²-4x+5 where A=(1, 4). What is the range of the function?

A. (2, 5)

B. (1, 5)

C. [1, 5)

D. [1, 5]

Q. 81 What is the given matrix equal to, where [.] is the greatest integer function?

A. b-a

B. a-b

C. 0

D. 2(b-a)

Q. 82 What is the given matrix equal to?

A. 2

B. 3

C. 4

D. 9

Q. 83 What is ∫ sin³ x cos x dx equal to?

Note: where c is the constant of integration.

A. cos⁴x+c

B. sin⁴x+c

C. [(1-sin²x)²/4] + c

D. [(1-cos²x)²/4] + c

Q. 84 What is the given matrix equal to?

A. ln|tan x|+c

B. ln|sec x|+c

C. tan x+c

D. eᵗᵃⁿ ˣ+c

Q. 85 What is the given matrix equal to?

A. 0

B. -π/4

C. -π/2

D. π/2

Q. 86 In which one of the following intervals is the function f(x)=x²-5x+6 decreasing?

A. (∼∞, 2]

B. [3, ∞)

C. (∼∞, ∞)

D. (2, 3)

Q. 87 The differential equation of the family of curves y=p cos (ax)+ q sin (ax), where p, q are arbitrary constants is

A. (d²y/dx²) – a²y = 0

B. (d²y/dx²) – ay = 0

C. (d²y/dx²) + ay = 0

D. (d²y/dx²) + a²y = 0

Q. 88 The equation of the curve passing through the point (-1, -2) which satisfies dy/dx=-x²-1/x³, is

A. 17x²y-6x²+3x⁵-2=0

B. 6x²y+17x²+2x⁵-3=0

C. 6xy-2x²+17x⁵+3=0

D. 17x²y+6xy-3x⁵+5=0

Q. 89 What is the order of the differential equation whose solution is y=a cos x + b sin x + ce⁻ⁿ + d, where a, b, c and d are arbitrary constants?

A. 1

B. 2

C. 3

D. 4

Q. 90 What is the solution of the differential equation ln(dy/dx)=ax+by?

Note: where c is an arbitrary constant

A. aeᵃˣ + beᵇʸ = c

B. (1/a)eᵃˣ +(1/b)eᵇʸ = c

C. aeᵃˣ + be⁻ᵇʸ = c

D. (1/a)eᵃˣ + (1/b)e⁻ᵇʸ = c

Q. 91 If u=eⁱ ⁿ sin bx and v=eⁱ ⁿ cos bx, then what is u(du/dx)+v(dv/dx) equal to?

A. a e²ⁱ ⁿ

B. (a²+b²)eⁱ ⁿ

C. ab e²ⁱ ⁿ

D. (a+b)eⁱ ⁿ

Q. 92 If y=sin(ln x), then which one of the following is correct?

A. (d²y/dx²) + y = 0

B. d²y/dx²=0

C. x²(d²y/dx²)+x(dy/dx)=0

D. x²(d²y/dx²)-x(dy/dx)=0

Q. 93 A flower-bed in the form of a sector has been fenced by a wire of 40 m length. If the flowerbed has the greatest possible area, then what is the radius of the sector?

A. 25 m

B. 20 m

C. 10 m

D. 5 m

Q. 94 What is the minimum value of [x(x-1)+1]¹⁄³, where 0≤x≤1 ?

A. (3/4)¹⁄³

B. 1

C. 1/2

D. (3/8)¹⁄³

Q. 95 If y=|sin x||ˣ|, then what is the value of dy/dx at x=-π/6 ?

A. [2⁻π/⁶ (6 ln 2 – √3π)] / 6

B. [2π/⁶ (6 ln 2 + √3π)] / 6

C. [2⁻π/⁶ (6 ln 2 + √3π)] / 6

D. [2π/⁶ (6 ln 2 – √3π)] / 6

Q. 96 What is d√(1-sin2x)/dx equal to, where π/4

A. cos x + sin x

B. -(cos x+ sin x)

C. ±(cos x + sin x)

D. None of the above

Q. 97 What is ∫ dx/a²sin²x+b²cos²x equal to?

Note: where c is the constant of integration

A. c+1/ab tan⁻¹(a tan x/b)

B. c-1/ab tan⁻¹(b tan x/a)

C. c+1/ab tan⁻¹(b tan x/a)

D. None of the above

Q. 98 According to the matrix given, let f(x+y)=f(x)f(y) and f(x)=1+xg(x)φ(x). Then what is f'(x) equal to?

A. 1+abf(x)

B. 1+ab

C. ab

D. abf(x)

Q. 99 What is solution of the differential equation dx/dy=x+y+1/x+y-1 ?

Note: where c is an arbitrary constant.

A. y-x+4ln(x+y)=c

B. y+x+2ln(x+y)=c

C. y-x+ln(x+y)=c

D. y+x+2ln(x+y)=c

Q. 100 What is the matrix given equal to?

A. -1/2

B. -1/3

C. -2

D. -3

Q. 101 If two dices are thrown and at least one of the dice shows 5, then the probability that the sum is 10 or more is

A. 1/6

B. 4/11

C. 3/11

D. 2/11

Q. 102 The correlation coefficient computed from a set of 30 observations is 0.8. Then the percentage of variation not explained by linear regression is

A. 80%

B. 20%

C. 64%

D. 36%

Q. 103 The average age of a combined group of men and women is 25 years. If the average age of the group of men is 26 years and that of the group of women is 21 years, then the percentage of men and women in the group is respectively

A. 20, 80

B. 40, 60

C. 60, 40

D. 80, 20

Q. 104 If sin β is the harmonic mean of sin α and cos α, and sin θ is the arithmetic mean of sin α and cos α, then which of the following is/are correct?

1. √2 sin (α+π/4)sinβ = sin 2α

2. √2 sin θ = cos (α-π/4)

Select the correct answer using the code below:

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

Q. 105 Let A, B and C be three mutually exclusive and exhaustive events associated with a random experiment. If P(B) = 1.5 P(A) and P(C) = 0.5 P(B), then P(A) is equal to

A. 3/4

B. 4/13

C. 2/3

D. 1/2

Q. 106 In a bolt factory, machines X, Y, Z manufacture bolts that are respectively 25%, 35% and 40% of the factory’s total output. The machines X, Y, Z respectively produce 2%, 4% and 5% defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine X?

A. 5/39

B. 14/39

C. 20/39

D. 34/39

Q. 107 8 coins are tossed simultaneously. The probability of getting at least 6 heads is

A. 7/64

B. 57/64

C. 37/256

D. 229/256

Q. 108 Three groups of children contain 3 girls and 1 boy; 2 girls and 2 boys; 1 girl and 3 boys. One child is selected at random from each group. The probability that the three selected consist of 1 girl and 2 boys is

A. 13/32

B. 9/32

C. 3/32

D. 1/32

Q. 109 Consider the following statements:

1. If 10 is added to each entry on a list, then the average increases by 10.

2. If 10 is added to each entry on a list, then the standard deviation increases by 10.

3. If each entry on a list is doubled, then the average doubles.

Which of the above statements are correct?

A. 1, 2 and 3

B. 1 and 2 only

C. 1 and 3 only

D. 2 and 3 only

Q. 110 The variance of 25 observations is 4. If 2 is added to each observation, then the new variance of the resulting observations is

A. 2

B. 4

C. 6

D. 8

Q. 111 If xᵢ>0, yᵢ>0 (i=1, 2, 3 … n) are the values of two variables X and Y with geometric means P and Q respectively, then the geometric mean of X/Y is

A. P/Q

B. antilog (P/Q)

C. n(logP-logQ)

D. n(logP+logQ)

Q. 112 If the probability of simultaneous occurrence of two events A and B is p and the probability that exactly one of A, B occurs is q, then which of the following is/are correct?

1. P(A̅)+P(B̅)=2-2p-q

2. P(A̅∩B̅)=1-p-q

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

Q. 113 If the regression coefficient of Y on X is -6, and the correlation coefficient between X and Y is -1/2, then the regression coefficient of X on Y would be

A. 1/24

B. -1/24

C. -1/6

D. 1/6

Q. 114 The set of bivariate observations (x₁, y₁), (x₂, y₂) …, (xᵣ, yᵣ) are such that all the values are distinct and all the observations fall on a straight line with non-zero slope. Then the possible values of the correlation coefficient between x and y are

A. 0 and 1 only

B. 0 and -1 only

C. 0, 1 and -1

D. -1 and 1 only

Q. 115 Two integers x and y are chosen with replacement from the set {0, 1, 2, …, 10}. The probability that |x – y| > 5 is

A. 6/11

B. 35/121

C. 30/121

D. 25/121

Q. 116 An analysis of monthly wages paid to the workers in two firms A and B belonging to the same industry gives the result as shown in the table. The average of monthly wage and variance of distribution of wages of all the workers in the firms A and B taken together are

A. Rs 1860, 100

B. Rs. 1750, 100

C. Rs. 1800, 81

D. None of the above

Q. 117 Three dice having digits, 1, 2, 3, 4, 5 and 6 on their faces are marked I, II and III and rolled. Let x, y and z represent the number on die-I, die-II and die-III respectively. What is the number of possible outcomes such that x > y > z ?

A. 14

B. 16

C. 18

D. 20

Q. 118 Which one of the following can be obtained from an ogive?

A. Mean

B. Median

C. Geometric mean

D. Mode

Q. 119 In any discrete series (when all values are not same), if x represents mean deviation about mean and y represents standard deviation, then which one of the following is correct ?

A. y ≥ x

B. y ≤ x

C. x = y

D. x < y

Q. 120 In which one of the following cases would you expect to get a negative correlation?

A. The ages of husbands and wives

B. Shoe size and intelligence

C. Insurance companies’ profits and the number of claims they have to pay

D. Amount of rainfall and yield of crop