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# NDA/NA(I) Exam 2016 Mathematics Previous Year Paper

July 3, 2019
###### NDA/NA(II) Exam 2016 Mathematics Previous Year Paper
July 3, 2019

NDA/NA(I) Exam 2016 Mathematics

Q. 1 Suppose ω is a cube root of unity with ω ≠ 1. Suppose P and Q are the points on the complex plane defined by ω and ω² . If O is the origin , then what is the angle between OP and OQ?

A. 60°

B. 90°

C. 120°

D. 150°

Q. 2 Suppose there is a relation * between the positive numbers x and y given x*y if and only if x ≤ y². Then which one of the following is correct?

A. * is reflexive but not transitive and symmetric

B. * is transitive but not reflexive and symmetric

C. * is symmetric and reflexive but not transitive

D. * is symmetric but not reflexive and transitive

Q. 3 If x² – px + 4 > 0 for all real values of x, then which one of the following is correct?

A. | p | < 4

B. | p | ≤ 4

C. | p | > 4

D. | p | ≥ 4

Q. 4 If z = x + iy = ( 1/√2 – i/√2 )⁻²⁵ , where i = √-1, then what is the fundamental amplitude of (z – √2)/(z – i√2) ?

A. π

B. π/2

C. π/3

D. π/4

Q. 5 If f(x₁) – f(x₂) = f( (x₁ – x₂)/(1 – x₁.x₂) ) for x₁,x₂ ∈ (-1, 1), then what is f(x) equal to?

A. ln( (1 – x)/(1 + x) )

B. ln( (2 + x)/(1 – x) )

C. tan⁻¹(1 – x)/(1 + x) )

D. tan⁻¹(1 + x)/(1 – x) )

Q. 6 What is the range of the function y = x²/(1 + x²) where x ∈ ℝ?

A. [0, 1)

B. [0, 1]

C. (0, 1)

D. (0, 1]

Q. 7 A straight line intersects x and y axes at P and Q respectively. If (3, 5) is the middle point of PQ, then what is the area of the triangle OPQ?

A. 12 square units

B. 15 square units

C. 20 square units

D. 30 square units

Q. 8 If a circle of radius b units with centre at (0, b) touches the line y = x – √2, then what is the value of b?

A. 2 + √2

B. 2 – √2

C. 2√2

D. √2

Q. 9 Consider the function f(θ) = 4( sin² θ + cos⁴ θ ). What is the maximum value of the function f(θ) ?

A. 1

B. 2

C. 3

D. 4

Q. 10 Consider the function f(θ) = 4( sin² θ + cos⁴θ ). What is the minimum value of the function f(θ)?

A. 0

B. 1

C. 2

D. 3

Q. 11 Consider the function f(θ) = 4( sin² θ + cos⁴ θ ). Consider the following statements : 1. f(θ) = 2 has no solution. 2. f(θ) = 7/2 has a solution. 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. 12 Consider the curves f(x) = x|x| – 1 and g(x) = { 3x/2, x>0 and 2x, x≤0}. Where do the curves intersect?

A. At (2, 3) only

B. At (-1, -2) only

C. At (2, 3) and (-1, -2)

D. Neither at (2, 3) nor at (-1, -2)

Q. 13 Consider the curves f(x) = x|x| – 1 and g(x) = { 3x/2, x>0 and 2x, x≤0}. What is the area bounded by the curves?

A. 17/6 square units

B. 8/3 square units

C. 2 square units

D. 1/3 square units

Q. 14 Consider the function f(x) = 27( x²/³ – x ) / 4 . How many solutions does function f(x) = 1 have?

A. One

B. Two

C. Three

D. Four

Q. 15 Consider the function f(x) = 27( x²/³ – x ) / 4 . How many solutions does the function f(x) = -1 have?

A. One

B. Two

C. Three

D. Four

Q. 16 Consider the functions f(x) = x.g(x) and g(x) = [1/x] where [.] is the greatest integer function. what is ∫1/3→1/2( g(x) ) dx equal to?

A. 1/6

B. 1/3

C. 5/18

D. 5/36

Q. 17 Consider the functions f(x) = x.g(x) and g(x) = [1/x] where [.] is the greatest integer function. What is ∫1→1/3 f(x)dx equal to?

A. 37/72

B. 2/3

C. 17/72

D. 37/144

Q. 18 Consider the function f(x) = | x – 1 | + x² where x ∈ ℝ. Which one of the following statements is correct?

A. f(x) is continuous but not differentiable at x = 0

B. f(x) is continuous but not differentiable at x = 1

C. f(x) is differentiable at x = 1

D. f(x) is not differentiable at x = 0 and x = 1

Q. 19 Consider the function f(x) = | x – 1 | + x² where x ∈ ℝ. Which one of the following statements is correct?

A. f(x) is increasing in (-∞, 1/2) and decreasing in (1/2, ∞)

B. f(x) is decreasing in (-∞, 1/2) and increasing in (1/2, ∞)

C. f(x) is increasing in (-∞, 1) and decreasing in (1, ∞)

D. f(x) is decreasing in (-∞, 1) and increasing in (1, ∞)

Q. 20 Consider the function f(x) = | x – 1 | + x² where x ∈ ℝ. Which one of the following statements is correct?

A. f(x) has local minima at more than one point in (-∞, ∞)

B. f(x) has local maxima at more than one point in (-∞, ∞)

C. f(x) has local minima at one point only in (-∞, ∞)

D. f(x) has neither local minima nor maxima in (-∞, ∞)

Q. 21 Consider the function f(x) = | x – 1 | + x² where x ∈ ℝ. What is the area of region bounded by x-axis, the curve y = f(x) and the two ordinates x = 1/2 and x = 1?

A. 5/12 square unit

B. 5/6 square unit

C. 7/6 square units

D. 2 square units

Q. 22 Consider the function f(x) = | x – 1 | + x² where x ∈ ℝ. What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates x = 1 and x = 3/2?

A. 5/12 square unit

B. 7/12 square unit

C. 2/3 square unit

D. 11/12 square unit

Q. 23 Consider the following statements:

1. The sequence { a₂ₙ } is in AP with common difference zero.

2. The sequence {a₂ₙ₊₁} is in AP with common difference zero. 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. 24 What is aₙ₋₁ – aₙ₋₄ equal to?

A. -1

B. 0

C. 1

D. 2

Q. 25 Consider the equation x + |y| = 2y. Which of the following statements are not correct?

1. y as a function of x is not defined for all real x.

2. y as a function of x is not continuous at x = 0

3.y as a function of x is differentiable for all x.

Select the correct answer using the code given below

A. 1 and 2 only

B. 2 and 3 only

C. 1 and 3 only

D. 1, 2 and 3

Q. 26 Consider the equation x + |y| = 2y. Which of the following statements are not correct? What is the derivative of y as a function of x with respect to x for x < 0?

A. 2

B. 1

C. 1/2

D. 1/3

Q. 27 Consider the lines y = 3x, y = 6x and y = 9. What is the area of the triangle formed by these lines?

A. 27/4 square units

B. 27/2 square units

C. 19/4 square units

D. 19/2 square units

Q. 28 Consider the lines y = 3x, y = 6x and y = 9 The centroid of the triangle is at which one of the following points?

A. (3, 6)

B. (3/2, 6)

C. (3, 3)

D. (3/2, 9)

Q. 29 Consider the function: f(x) = (x – 1)²(x + 1)(x – 2)³ . What is the number of points of local minima of the function f( x )?

A. None

B. One

C. Two

D. Three

Q. 30 Consider the function: f(x) = (x – 1)²(x + 1)(x – 2)³ . What is the number of points of local maxima of the function f(x)?

A. none

B. One

C. Two

D. Three

Q. 31 Let f(x) and g(x) be twice differentiable functions on [0, 2] satisfying f’ ‘(x) = g’ ‘(x), f'(1) = 4, g'(1) = 6, f(2) = 3 and g(2) = 9. Then what is f(x) – g(x) at x = 4 equal to?

A. -10

B. -6

C. -4

D. 2

Q. 32 Consider the curves y =| x – 1 | and |x| = 2. What is/are the point(s) of intersection of the curves?

A. (-2, 3) only

B. (2, 1) only

C. (-2, 3) and (2, 1)

D. neither (-2, 3) nor (2, 1)

Q. 33 Consider the curves y =| x – 1 | and |x| = 2. What is the area of the region bounded by the curves and x-axis?

A. 3 square units

B. 4 square units

C. 5 square units

D. 6 square units

Q. 34 What is the value of f'(0)?

A. p³

B. 3 p³

C. 6 p³

D. -6 p³

Q. 35 What is the value of p for which f’ ‘(0) = 0?

A. -1/6 or 0

B. -1 or 0

C. -1/6 or 1

D. -1 or 1

Q. 36 Consider a triangle ABC which cos A + cos B + cos C = √3 sin π/3. What is the value of sin(A/2).sin(B/2).sin(C/2)?

A. 1/2

B. 1/4

C. 1/8

D. 1/16

Q. 37 Consider a triangle ABC which cos A + cos B + cos C = √3 sin π/3. What is the value of cos((A+B)/2).cos((B+C)/2).cos((C+A)/2)?

A. 1/4

B. 1/2

C. 1/16

D. None of the above

Q. 38 Given that tanα and tanβ are the roots of the equation x² + bx + c = 0 with b≠0. What is tan(α + β) equal to?

A. b(c – 1)

B. c(b – 1)

C. c(b – 1)^(-1)

D. b(c – 1)^(-1)

Q. 39 Given that tanα and tanβ are the roots of the equation x² + bx + c = 0 with b≠0. What is sin(α + β) secα secβ equal to?

A. b

B. -b

C. c

D. -c

Q. 40 Consider the two circles (x – 1)² + (y – 3)² = r² and x² + y² – 8x + 2y + 8 =0. What is the distance between the centres of the two circles?

A. 5 units

B. 6 units

C. 8 units

D. 10 units

Q. 41 Consider the two circles (x – 1)² + (y – 3)² = r² and x² + y² – 8x + 2y + 8 =0. If the circles intersect at two distinct points, then which one of the following is correct?

A. r = 1

B. 1 < r < 2

C. r = 2

D. 2 < r < 8

Q. 42 Consider the two lines x + y + 1 = 0 and 3x + 2y + 1 = 0. What is the equation of the line passing through the point of intersection of the given lines and parallel to x-axis?

A. y + 1 = 0

B. y – 1 = 0

C. y – 2 = 0

D. y + 2 = 0

Q. 43 Consider the two lines x + y + 1 = 0 and 3x + 2y + 1 = 0. What is the equation of the line passing through the point of intersection of the given lines and parallel to y-axis?

A. x + 1 = 0

B. x – 1 = 0

C. x – 2 = 0

D. x + 2 = 0

Q. 44 Consider the equation k sin x + cos2x = 2k – 7. If the equation possesses a solution, then what is the minimum value of k?

A. 1

B. 2

C. 4

D. 6

Q. 45 Consider the equation k sin x + cos2x = 2k – 7. If the equation possesses solution, then what is the maximum value of k?

A. 1

B. 2

C. 4

D. 6

Q. 46 Consider the function f( x ) = (a |ˣ| ⁺ ˣ – 1)/( [x] + x ) where [ . ] denotes the greatest integer function. What is lim x→ 0+ f( x ) equal to?

A. 1

B. ln a

C. 1 = a⁻¹

D. Love does not exist

Q. 47 Consider the function f( x ) = (a |ˣ| ⁺ ˣ – 1)/( [x] + x ) where [ . ] denotes the greatest integer function. What is lim x→ 0- f( x ) equal to?

A. 0

B. ln a

C. 1 – a⁻¹

D. Limit does not exist

Q. 48 Let z₁, z₂ and z₃ be the non-zero complex numbers satisfying z² = iz̅, where i = √-1. What is z₁ + z₂ + z₃ equal to?

A. i

B. -i

C. 0

D. 1

Q. 49 Let z₁, z₂ and z₃ be the non-zero complex numbers satisfying z² = iz̅, where i = √-1. Consider the following statements:

1. z₁.z₂.z₃ is purely imaginary.

2. z₁.z₂ + z₂.z₃ + z₃.z₁ is purely real.

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. 50 Given that logₓ y, log􀀁 x, logᵧ z are in GP, xyz = 64 and x³, y³, z³ are in AP. Which one of the following is correct?

x,y and z are

A. in AP only

B. in GP only

C. in both AP and GP

D. neither in AP nor in GP

Q. 51 Given that logₓ y, log􀀁 x, logᵧ z are in GP, xyz = 64 and x³, y³, z³ are in AP. Which one of the following is correct?

xy, yz and zx are

A. in AP only

B. in GP only

C. in both AP and GP

D. neither in AP nor in GP

Q. 52 Let z be a complex number satisfying | (z – 4)/(z – 8)| = 1 and | z/(z – 2)| = 3/2. What is |z| equal to?

A. 6

B. 12

C. 18

D. 36

Q. 53 Let z be a complex number satisfying | (z – 4)/(z – 8)| = 1 and | z/(z – 2)| = 3/2. What is |(z – 6)/(z + 6)| equal to?

A. 3

B. 2

C. 1

D. 0

Q. 54 A function f(x) is defined as follows: f(x) = { x + π for x∈[ -π, 0); n.cosx for x∈[ 0, π/2]; (x – π)² for x∈( π/2, π] }.

Consider the following statements:

1. The function f(x) is continuous at x = 0.

2. The function f(x) is continuous at x = π/2.

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. 55 A function f(x) is defined as follows: f(x) = { x + π for x∈[ -π, 0); n.cosx for x∈[ 0, π/2]; (x – π)² for x∈( π/2, π] }.

Consider the following statements:

1. The function f(x) is differentiable at x = 0.

2. The function f(x) is differentiable at x = π/2.

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. 56 Let α and β ( α < β ) be the roots of the equation s² + bx + c = 0, where b > 0 and c > 0. Consider the following :

1. β < -α

2. β < |α|.

Which of the above is/are correct?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

Q. 57 Let α and β ( α < β ) be the roots of the equation s² + bx + c = 0, where b > 0 and c > 0. Consider the following :

1. α + β + βα > 0

2. α²β + β²α > 0.

Which of the above is/are correct?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

Q. 58 Consider a parallelogram whose vertices are A(1,2), B(4,y), C(x,6) and D(3,5) taken in order. What is the value of AC² – BD²?

A. 25

B. 30

C. 36

D. 40

Q. 59 Consider a parallelogram whose vertices are A(1,2), B(4,y), C(x,6) and D(3,5) taken in order. What is the point of intersection of the diagonals?

A. (7/2, 4)

B. (3, 4)

C. (7/2, 5)

D. (3, 5)

Q. 60 Consider a parallelogram whose vertices are A(1,2), B(4,y), C(x,6) and D(3,5) taken in order. What is the area of parallelogram?

A. 7/2 square units

B. 4 square units

C. 11/2 square units

D. 7 square units

Q. 61 Let f: ℝ->ℝ be a function such that f(x) = x³ + x²f'(1) + xf”(2) + f”'(3) for x∈ℝ. What is f(1) equal to?

A. -2

B. -1

C. 0

D. 4

Q. 62 Let f: ℝ->ℝ be a function such that f(x) = x³ + x²f'(1) + xf”(2) + f”'(3) for x∈ℝ. What is f'(1) equal to?

A. -6

B. -5

C. 1

D. 0

Q. 63 Let f: ℝ->ℝ be a function such that f(x) = x³ + x²f'(1) + xf’ ‘(2) + f’ ‘ ‘(3) for x∈ℝ. What is f’ ‘(10) equal to?

A. 1

B. 5

C. 6

D. 8

Q. 64 Let f: ℝ->ℝ be a function such that f(x) = x³ + x²f'(1) + xf”(2) + f”'(3) for x∈ℝ.

Consider the following:

1. f(2) = f(1) – f(0)

2. f’ ‘(2) – f'(1) = 12 .

Which of the above is/are correct?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

Q. 65 A plane P passes through the line of intersection of the planes 2x – y + 3z = 2, x + y – z =1 and the point (1, 0, 1). What are the direction ratios of the line of intersection of the given planes?

A. <2, -5, -3>

B. <-1, -5, -3>

C. <2, 5, 3>

D. <1, 3, 5>

Q. 66 A plane P passes through the line of intersection of the planes 2x – y + 3z = 2, x + y – z =1 and the point (1, 0, 1). What is the equation of the plane P?

A. 2x + 5y – 2 = 0

B. 5x + 2y – 5 = 0

C. x + z – 2 = 0

D. 2x – y – 2z = 0

Q. 67 A plane P passes through the line of intersection of the planes 2x – y + 3z = 2, x + y – z =1 and the point (1, 0, 1). If the plane P touches the sphere x² + y² + z² = r², then what is r equal to?

A. 2 / √29

B. 4 / √29

C. 5 / √29

D. 1

Q. 68 Consider the function f(x) = | x² – 5x + 6 |. What is tf'(4) equal to?

A. -4

B. -3

C. 3

D. 2

Q. 69 Consider the function f(x) = | x² – 5x + 6 |. What is f’ ‘(2.5) equal to?

A. -3

B. -2

C. 0

D. 2

Q. 70 Let f(x) be the greatest integer functions and g(x) be the modulus function. What is (g ∘ f) (-5/3) – (f ∘ g)(-5/3) equal to?

A. -1

B. 0

C. 1

D. 2

Q. 71 Let f(x) be the greatest integer functions and g(x) be the modulus function. What is (f ∘ f) (-9/5) + (g ∘ g)(-2) equal to?

A. -1

B. 0

C. 1

D. 2

Q. 72 Consider a circle passing through the origin and the points (a, b) and (-b, -a). On which line does the centre of the circle lie?

A. x + y = 0

B. x – y = 0

C. x + y = a + b

D. x – y = a² – b²

Q. 73 Consider a circle passing through the origin and the points (a, b) and (-b, -a). What is the sum of the squares of the intercepts cut off by the circle on the axes?

A. ( a² + b²)/(a² – b²)²

B. 2( a² + b²)/(a – b)²

C. 4( a² + b²)/(a – b)²

D. None of the above

Q. 74 Let â, b̂ be two unit vectors and θ be the angle between them. What is cos(θ/2) equal to?

A. | â – b̂|/2

B. | â + b̂|/2

C. | â – b̂|/4

D. | â + b̂|/4

Q. 75 Let â, b̂ be two unit vectors and θ be the angle between them. What is sin(θ/2) equal to?

A. | â – b̂|/2

B. | â + b̂|/2

C. | â – b̂|/4

D. | â + b̂|/4

Q. 76 Consider the following statements :

1.There exists θ ∈ ( -π/2, π/2) for which tan⁻¹ ( tan θ ) ≠ θ.

2. sin⁻¹ (1/3) – sin⁻¹ (1/5) = sin⁻¹ ( 2√2( √3 – 1)/15) .

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. 77 Consider the following statements :

1. tan⁻¹ ( x ) + tan⁻¹ (1/x) = π

2. There exist x, y∈[-1, 1], where x≠y such that sin⁻¹ ( x ) + cos⁻¹ ( y ) = π/2.

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. 78 What are the order and degree respectively of the differential equation whose solution is y = cx + c² – 3c³/² + 2, where c is a parameter?

A. 1, 2

B. 2, 2

C. 1, 3

D. 1, 4

Q. 79 What is ∫ (-2→2) x dx – ∫ (-2→2) [ x ] dx equal to, where [ . ] is the greatest integer function?

A. 0

B. 1

C. 2

D. 4

Q. 80 If ∫ (-2→5) f(x)dx = 4 and ∫ (0→5) { 1 + f(x) }dx = 7 then what is ∫ (-2→0) f(x)dx equal to?

A. -3

B. 2

C. 3

D. 5

Q. 81 If lim x→0 Φ( x ) = a^2, where a ≠ 0, then what is lim x→0 Φ( x/a ) equal to?

A. a²

B. a⁻²

C. -a²

D. -a

Q. 82 What is lim x→0 e⁻¹/ˣ² equal to?

A. 0

B. 1

C. -1

D. Limit does not exist

Q. 83 If A is square matrix, then what is adj(A⁻¹) – (adj A)⁻¹ equal to?

A. 2 |A|

B. Null matrix

C. Unit matrix

D. None of the above

Q. 84 What is the binary equivalent of the decimal number 0.3125?

A. 0.0111

B. 0.1010

C. 0.0101

D. 0.1101

Q. 85 Let R be a relation on the set N of natural numbers defined by ‘nRm ⇔ n is a factor of m’. Then which one of the following is correct?

A. R is reflexive, symmetric but not transitive

B. R is transitive, symmetric but not reflexive

C. R is reflexive, transitive but not symmetric

D. R is an equivalence relation

Q. 86 What is ∫ 0→4π ( | cos x | )dx equal to?

A. 0

B. 2

C. 4

D. 8

Q. 87 What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?

A. 860

B. 854

C. 840

D. 824

Q. 88 (a, 2b) is the mid-point of the line segment joining the points (10, 06) and (k, 4). If a – 2b = 7, then what is the value of k?

A. 2

B. 3

C. 4

D. 5

Q. 89 Consider the following statements:

1. If ABC is an equilateral triangle then 3.tan( A + B ).tan C = 1.

2. If ABC is a triangle in which A = 78°, B = 66°, then tan(A/2 + C) < tan A.

3. If ABC is any triangle, then tan( A+B/2 ). sin(C/2) < cos(C/2).

Which of the above statements is/are correct?

A. 1 only

B. 2 only

C. 1 and 2

D. 2 and 3

Q. 90 If A = ( cos 12° – cos 36° )( sin 96° + sin 24° ) and B = (sin 60° – sin 12° )( cos 48° – cos 72° ), then what is A/B equal to?

A. -1

B. 0

C. 1

D. 2

Q. 91 What is the mean deviation from the mean of the numbers 10, 9, 21, 16, 24?

A. 5.2

B. 5.0

C. 4.5

D. 4.0

Q. 92 Three dice are thrown simultaneously. What is the probability that the sum on the three faces is at least 5?

A. 17/18

B. 53/54

C. 103/108

D. 215/216

Q. 93 Two independent events A and B have P(A) = 1/3 and P(B) = 3/4. What is the probability that exactly one of the two events A or B occurs?

A. 1/4

B. 5/6

C. 5/12

D. 7/12

Q. 94 A coin is tossed three times. What is the probability of getting head and tail alternatively?

A. 1/8

B. 1/4

C. 1/2

D. 3/4

Q. 95 if the total number of observations is 20, Σ(xi) = 1000 and Σ (xi²) = 84000, then what is the variance of the distribution?

A. 1500

B. 16000

C. 1700

D. 1800

Q. 96 A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is queen of spade?

A. 1/52

B. 1/13

C. 1/4

D. 1/8

Q. 97 If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to 4?

A. 13/18

B. 5/6

C. 11/12

D. 35/36

Q. 98 A certain type of missile hits the target with probability p = 0.3. What is the least number of missiles should be fired so that there is at least 80% probability that target is hit?

A. 5

B. 6

C. 7

D. None of the above

Q. 99 For two mutually exclusive events A and B, P(A) = 0.2 and P(A’∩B) 0.3. What is P(A| (A U B)) equal to?

A. 1/2

B. 2/5

C. 2/7

D. 2/3

Q. 100 What is the probability of 5 Sundays in the month of December?

A. 1/7

B. 2/7

C. 3/7

D. None of the above

Q. 101 if m is the geometric mean of (y/z)ˡᵒᵍ ⁽ʸᶻ⁾ , (z/x)ˡᵒᵍ ⁽ᶻˣ⁾ and (x/y)ˡᵒᵍ ⁽ˣʸ⁾ then what is the value of m?

A. 1

B. 3

C. 6

D. 9

Q. 102 A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?

A. 2/3

B. 1/3

C. 1/4

D. 2/5

Q. 103 The mean of the series x₁, x₂, …, xₙ is X’. If x₂ is replaced by λᵢ then what is the new mean?

A. X’ – x₂ + λ

B. (X’ – x₂ – λ)/n

C. (X’ – x₂ + λ)/n

D. (nX’ – x₂ + λ)/n

Q. 104 For the data 3,5,1,6,5,9,2,8,6 the mean, median and mode are x, y and z respectively. Which one of the following is correct?

A. x = y ≠ z

B. x ≠ y = z

C. x ≠ y ≠ z

D. x = y = z

Q. 105 Consider the following statements in respect of a histogram :

1. The total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequency polygon and the x-axis.

2. When class intervals are unequal in a frequency distribution, the area of the rectangle is proportional to the frequency.

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. 106 A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?

A. 1/2

B. 3/8

C. 1/4

D. 1/8

Q. 107 What is the number of ways in which 3 holiday travel tickets are to be given to 10 employees of an organization, if each employee is eligible for any one or more of the tickets?

A. 60

B. 120

C. 500

D. 1000

Q. 108 If one root of the equation (l – m)x² + l.x + 1 = 0 is double the other and l is real, then what is the greatest value of m?

A. -9/8

B. 9/8

C. -8/9

D. 8/9

Q. 109 What is the number of four-digit decimal numbers (<1) in which no digit is repeated?

A. 3024

B. 4536

C. 5040

D. None of the above

Q. 110 What is a vector of unit length orthogonal to both the vectors î + ĵ + k̂ and 2 î + 3 ĵ -k̂?

A. (-4 î + 3 ĵ – k̂)/√26

B. (-4 î + 3 ĵ + k̂)/√26

C. (-3 î + 2 ĵ – k̂)/√14

D. (-3 î + 2 ĵ +k̂)/√14

Q. 111 If a̅, b̅ and c̅ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?

A. a̅ + b̅ + c̅ = 0̅

B. a̅ + b̅ + c̅ = unit vector

C. a̅ + b̅ = c̅

D. a̅ = b̅ + c̅

Q. 112 What is the area of the parallelogram having diagonals 3î + ĵ – 2k̂ and î – 3ĵ + 4k̂?

A. 5√5 square units

B. 4√5 square units

C. 5√3 square units

D. 15√2 square units

Q. 113 Consider the following in respect of the matrix A.

1. A² = -A

2. A³ = 4A .

Which of the above is/are correct?

A. 1 only

B. 2 only

C. Both 1 and 2

D. Neither 1 nor 2

Q. 114 Which of the following determine have value zero? Select the correct answer using the code given below.

A. 1 and 2 only

B. 2 and 3 only

C. 1 and 3 only

D. 1, 2 and 3

Q. 115 What is the acute angle between the lines represented by the equations y – √3 x – 5 = 0 and √3 y – x + 6 = 0?

A. 30°

B. 45°

C. 60°

D. 75°

Q. 116 The system of linear equations kx + y + z = 1, x + ky + z = 1 and x + y + kz = 1 has a unique solution under which one of the following conditions?

A. k ≠ 1 and k ≠ -2

B. k ≠ 1 and k ≠ 2

C. k ≠ -1 and k ≠ -2

D. k ≠ -1 and k ≠ -2

Q. 117 What is the number of different messages that can be represented by three 0’s and two 1’s?

A. 10

B. 9

C. 8

D. 7

Q. 118 If log(base a) (ab) = x, then what is log(base b)(ab) equal to?

A. 1/x

B. x/(x+1)

C. x/(1-x)

D. x/(x-1)

Q. 119 If y = log₁₀x + logₓ10 + logₓx + log₁₀10 then what is (dy/dx)ₓ₌₁₀ equal to?

A. 10

B. 2

C. 1

D. 0

Q. 120 Suppose ω₁ and ω₂ are two distinct cube roots of unity different from 1. Then what is (ω₁ – ω₂)² equal to?

A. 3

B. 1

C. -1

D. -3