NDA/NA(I) Exam 2015 Mathematics
Q. 1 Let X be the set of all persons living in a city, Persons x, in y in X are said to be related as x < y if y is at least 5 years older than x. Which one of the following is correct?
A. The relation is an equivalence relation on X
B. The relation is transitive but neither reflexive nor symmetric
C. The relation is reflexive but neither transitive nor symmetric
D. The relation is symmetric but neither transitive not reflexive
Q. 2 Which one of the following matrices is an elementary matrix?
A. (a)
B. (b)
C. (c)
D. (d)
Q. 3 Consider the following statements in respect of the given equation : (x^2 + 2)^2 + 8 x^2 = 6 x ( x^2 + 2) . Given : 1.) All the roots of the equation are complex. 2.) The sum of all the roots of the equation is 6. 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. 4 In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds -9 and -1 for roots. The correct equation is
A. x^2 – 10 x + 9 = 0
B. x^2 + 10 x + 9 = 0
C. x^2 – 10 x + 16 = 0
D. x^2 – 8 x -9 = 0
Q. 5 What is A + 3 A^(-1) equal to?
A. 3I
B. 5I
C. 7I
D. None of the above
Q. 6 In a class of 60 students, 45 students like music, 50 students like dancing, 5 students like neither. Then the number of students in the class who like both music and dancing is
A. 35
B. 40
C. 50
D. 55
Q. 7 If log [base 10]( 2 ), log [base 10]( 2^x – 1 ) and log [base 10]( 2^x + 3 ) are three consecutive terms of an AP, then the value of x is
A. 1
B. log [base 5]( 2 )
C. log [base 2]( 5 )
D. log [base 10] (5)
Q. 8 The Matrix A is
A. symmetric
B. skew – symmetric
C. Hermitian
D. skew – Hermitian
Q. 9 Let Z be the set of integers and aRb, where a, b ∈ Z if and only if ( a – b ) is divisible by 5.
Consider the following statements :
1. The relation R partitions Z into five equivalent classes.
2. Any two equivalent classes are either equal or disjoint
A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 not 2
Q. 10 If z = -2( 1 + 2i ) / 3 + i where i = √-1, then the argument θ( -Π < θ ≤ Π ) of z is
A. 3Π/4
B. Π/4
C. 5Π/6
D. -3Π/4
Q. 11 If m and n are the roots of the equation ( x + p )( x + q ) – k = 0, then the roots of the equation ( x – m )( x – n ) + k = 0 are
A. p and q
B. 1/p and 1/q
C. -p and -q
D. p + q and p – q
Q. 12 What is the sum of the series 0.5 + 0.55 + 0.555 + … to n terms?
A. 5/9[ n – ( 2/9)( 1 – 1/(10^n) ) ]
B. 1/9[ 5 – ( 2/9)( 1 – 1/(10^n) ) ]
C. 1/9[ n – ( 5/9)( 1 – 1/(10^n) ) ]
D. 5/9[ n – ( 1/9)( 1 – 1/(10^n) ) ]
Q. 13 If 1, ω, ω^2 are the cube roots of unity, then the value of ( 1 + ω )( 1 + ω^2 )( 1 + ω^4 )( 1 + ω^8 ) is
A. -1
B. 0
C. 1
D. 2
Q. 14 Let A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }. Then the number of subsets of A containing exactly two elements is
A. 20
B. 40
C. 45
D. 90
Q. 15 What is the square root of i, where i = √-1
A. ( 1 + i )/2
B. ( 1 – i )/2
C. ( 1 + i )/√2
D. None of the above
Q. 16 The decimal number (127.25)(decimal) , when converted to binary number, takes the form
A. ( 1 1 1 1 1 1 1 . 1 1)(binary)
B. ( 1 1 1 1 1 1 0 . 0 1)(binary)
C. ( 1 1 1 0 1 1 1 . 1 1)(binary)
D. ( 1 1 1 1 1 1 1 .0 1)(binary)
Q. 17 Consider the following in respect of two non-singular matrices A and B of same order :
1. det( A + B ) = det A + det B
2. ( A + B )^(-1) = A^(-1) + B^(-1) . Which of the above is/are correct?
A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2
Q. 18 If X, A and B satisfy the equation AX = B, then the matrix A is equal to:
A. (a)
B. (b)
C. (c)
D. (d)
Q. 19 The equation given in figure equals to
A. ⁿ⁺²C₁
B. ⁿ⁺²Cₙ
C. ⁿ⁺³Cₙ
D. ⁿ⁺²Cₙ₋₁
Q. 20 How many words can be formed using all the letters of the word ‘NATION’ so that all the three vowels should never come together?
A. 354
B. 348
C. 288
D. None of the above
Q. 21 ( x^3 – 1 ) can be factorized as : ( where ω is one of the cube roots of unity. )
A. ( x- 1 )( x – ω )( x + ω^2 )
B. ( x- 1 )( x – ω )( x – ω^2 )
C. ( x- 1 )( x + ω )( x + ω^2 )
D. ( x- 1 )( x + ω )( x – ω^2 )
Q. 22 What is { [sin Π/6 + i( 1 – cos Π/6 )] / [sin Π/6 – i( 1 – cos Π/6 )] }^3 where i = √-1, equal to ?
A. 1
B. -1
C. i
D. -i
Q. 23 For given A, B and C matrices. If AB = C, then what is A² equal to?
A. (a)
B. (b)
C. (c)
D. (d)
Q. 24 The value of A is
A. x + y
B. x – y
C. xy
D. 1 + x + y
Q. 25 If A = { x : x is a multiple of 3 } and B = { x : x is a multiple of 4 } and C = { x : x is multiple of 12 }, then which one of the following is a null set?
A. ( A \ B ) ∪ C
B. ( A \ B ) \ C
C. ( A ∩ B ) ∩ C
D. ( A ∩ B ) \ C
Q. 26 If ( 1 1 1 0 1 0 1 1)(binary) is converted to decimal system, then the resulting number is
A. 235
B. 175
C. 160
D. 126
Q. 27 What is the real part of (sin x + i cos x )^3 where i = √-1?
A. -cos 3x
B. -sin 3x
C. sin 3x
D. cos 3x
Q. 28 then E(α) E(β) is equal to
A. E(αβ)
B. E(α – β)
C. E(α + β)
D. – E(α + β)
Q. 29 Let A = { x, y, z} and B = { p, q, r, s}. What is the number of distinct relations from B to A?
A. 4096
B. 4094
C. 128
D. 126
Q. 30 If 2p + 3q = 18 and 4p² + 4pq – 3q² – 36 = 0. Then what is (2p + q) equal to?
A. 6
B. 7
C. 10
D. 20
Q. 31 Let θ be a positive angle. If the number of degrees in θ is divided by the number of radians in θ, then an irrational number 180 / Π results. If the number of degrees in θ is multiplied by the number of radians in θ, then an irrational number 125π / 9 results. The angle θ must be equal to
A. 30°
B. 45°
C. 50°
D. 60°
Q. 32 In a triangle ABC, a = (1 + √3 ) cm, b = 2 cm and angle C = 60°. Then the other two angles are
A. 45° and 75°
B. 30° and 90°
C. 105° and 15°
D. 100° and 20°
Q. 33 Let α be the root of the equation 25cos² θ + 5cos θ – 12 = C, where Π/2 < α < Π. What is tan α equal to?
A. -3 / 4
B. 3 / 4
C. – 4 / 3
D. – 4 / 5
Q. 34 Let α be the root of the equation 25cos² θ + 5cos θ – 12 = C, where Π/2 < α < Π. What is sin 2α equal to?
A. 24 / 25
B. -24 / 25
C. -5 / 12
D. -21 / 25
Q. 35 The angle of elevation of the top of a tower from a point of 20 m away from its base is 45°. What is the height of the tower?
A. 10 m
B. 20 m
C. 30 m
D. 40 m
Q. 36 The equation tan^(-1)( 1 + x ) + tan^(-1)( 1 – x ) = Π/2 is satisfied by
A. x = 1
B. x = -1
C. x = 0
D. x = 1/2
Q. 37 The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances 49 m and 36 m are 43° and 47° respectively. What is the height of the tower?
A. 40 m
B. 43 m
C. 45 m
D. 47 m
Q. 38 ( 1 – sin A + cos A )² is equal to
A. 2( 1 – cos A )( 1 + sin A )
B. 2( 1 – sin A )( 1 + cos A )
C. 2( 1 – cos A )( 1 – sin A )
D. None of the above
Q. 39 What is cos θ/( 1 + tanθ ) + sinθ/( 1 – cotθ ) equal to ?
A. sinθ – cosθ
B. sinθ + cosθ
C. 2sinθ
D. 2cosθ
Q. 40 Consider x = 4 tan^(-1) ( 1 / 5 ), y = tan^(-1) ( 1 / 70 ) and z = tan^(-1) ( 1/99 ). What is x equal to?
A. tan^(-1) ( 60/119 )
B. tan^(-1) ( 120/119 )
C. tan^(-1) ( 90/169 )
D. tan^(-1) ( 170/169 )
Q. 41 Consider x = 4 tan^(-1) ( 1 / 5 ), y = tan^(-1) ( 1 / 70 ) and z = tan^(-1) ( 1/99 ). What is x – y equal to?
A. tan^(-1) ( 828/845 )
B. tan^(-1) ( 8287/8450 )
C. tan^(-1) ( 8281/8450 )
D. tan^(-1) ( 8287/8471 )
Q. 42 Consider x = 4 tan^(-1) ( 1 / 5 ), y = tan^(-1) ( 1 / 70 ) and z = tan^(-1) ( 1/99 ). What is x – y – z equal to ?
A. Π/2
B. Π/3
C. Π/6
D. Π/4
Q. 43 Consider the triangle ABC with vertices A(-2, 3), B(2, 1) and C(1, 2). What is the circumcentre of the triangle ABC?
A. ( -2, -2)
B. ( 2, 2)
C. ( -2, 2)
D. ( 2, -2)
Q. 44 Consider the triangle ABC with vertices A(-2, 3), B(2, 1) and C(1, 2). What is the centroid of the triangle ABC?
A. ( 1/3, 1 )
B. ( 1/3, 2 )
C. ( 1, 2/3 )
D. ( 1/2, 3 )
Q. 45 Consider the triangle ABC with vertices A(-2, 3), B(2, 1) and C(1, 2). What is foot of the altitude from the vertex A of the triangle ABC?
A. ( 1, 4)
B. ( -1, 3)
C. ( -2, 4)
D. ( – 1, 4)
Q. 46 The point on the parabola y^2 = 4ax nearest to the focus has its abscissa
A. x = 0
B. x = a
C. x = a/2
D. x = 2a
Q. 47 A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is
A. 3/4
B. 4/3
C. 1/3
D. 3
Q. 48 The hyperbola (x / a)^2 – (y / b)^2 = 1 passes through the point ( 3√5, 1 ) and the length of its latus rectum is 4/3 units. The length of the conjugate axis is
A. 2 units
B. 3 units
C. 4 units
D. 5 units
Q. 49 The perpendicular distance between the straight lines 6x + 8y + 15 = 0 and 3x + 4y + 9 = 0 is
A. 3/2 unit
B. 3/10 unit
C. 3/4 unit
D. 2/7 unit
Q. 50 The area of a triangle, whose vertices are (3, 4), (5, 2) and the point of intersection of the lines x = a and y = 5, is 3 square units. What is the value of a?
A. 2
B. 3
C. 4
D. 5
Q. 51 The length of perpendicular from the origin to a line is 5 units and the line makes an angle 120° with the positive direction of x-axis. The equation of the line is
A. x + √3y = 5
B. √3 + y = 10
C. √3x – y = 10
D. None of the above
Q. 52 The equation of the line joining the origin to the point of intersection of the lines x/a + y/b = 1 and x/b + y/a = 1 is
A. x – y = 0
B. x + y = 0
C. x = 0
D. y= 0
Q. 53 The projections of a directed line segment on the coordinate axes are 12, 4, 3 respectively. What is the length of the line segment?
A. 19 units
B. 17 units
C. 15 units
D. 13 units
Q. 54 The projections of a directed line segment on the coordinate axes are 12, 4, 3 respectively. What are the direction cosines of the line segment?
A. < 12/13, 4/13, 3/13 >
B. < 12/13, – 4/13, 3/13 >
C. < 12/13, -4/13, -3/13 >
D. < -12/13, -4/13, 3/13 >
Q. 55 From the point P(3, -1, 11), a perpendicular is drawn on the line L given by the equation x/2 = (y – 2)/3 = (z – 3)/4. Let Q be the foot of the perpendicular. What are the direction ratios of the line segment PQ?
A. < 1, 6, 4 >
B. < -1, 6, -4 >
C. < -1, -6, 4 >
D. < 1, -6, 4 >
Q. 56 From the point P(3, -1, 11), a perpendicular is drawn on the line L given by the equation x/2 = (y – 2)/3 = (z – 3)/4. Let Q be the foot of the perpendicular.What is the length of the line segment PQ?
A. √47 units
B. 7 units
C. √53 units
D. 8 units
Q. 57 A triangular plane ABC with the centroid (1, 2, 3) cuts the coordinate axes at A,B,C respectively. What are the intercepts made by the plane ABC on the axes?
A. 3,, 6, 9
B. 1, 2, 3
C. 1, 4, 9
D. 2, 4, 6
Q. 58 A triangular plane ABC with the centroid (1, 2, 3) cuts the coordinate axes at A,B,C respectively. What is the equation of the plane ABC?
A. x + 2y + 3z = 1
B. 3x + 2y + z = 3
C. 2x + 3y + 6z = 18
D. 6x + 3y + 2z = 18
Q. 59 A point P(1, 2, 3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes. What is the length of one of the diagonals of the cuboid?
A. √10 units
B. √14 units
C. 4 units
D. 5 units
Q. 60 A point P(1, 2, 3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes. What is the equation of the plane passing through P(1, 2, 3) and parallel to xy-plane?
A. x + y = 3
B. x – y = -1
C. z = 3
D. x + 2y + 3z = 14
Q. 61 If G(x) = √(25 – x²) , then what is lim x→1 (G(x) – G(1)) / (x – 1) equal to?
A. – 1/ 2√6
B. 1 / 5
C. – 1 / √6
D. 1 / √6
Q. 62 Consider the following statements: 1. y = ( e^x + e^(-x))/ 2 is an increasing function on [0, ∞). 2. y = ( e^x – e^(-x))/ 2 is an increasing function on (-∞, ∞). 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. 63 For each non-zero real number x, let f(x) = x / |x|. The range of f is
A. a null set
B. a set consisting of only one element
C. a set consisting of two elements
D. a set consisting of infinitely many elements
Q. 64 Consider the following statements: 1. f(x) = [ x ], where [.] is the greatest integer function, is discontinuous at x = n, there n ∈ Z. 2. f(x) = cot x is discontinuous at x = n.Π, where n ∈ Z. 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. 65 What is the derivative of y with respect to tan^(-1)x?
A. 0
B. 1/2
C. 1
D. x
Q. 66 If f(x) = loge( (1 + x)/(1 – x) ), g(x) = (3x + x^3)/(1 + 3x^2) and g∘f(t) = g( f(t) ), then what is g∘f( (e – 1)/(e + 1)) equal to
A. 2
B. 1
C. 0
D. 1/2
Q. 67 Given a function f(x) = -1 if x ≤ 0, ax + b if 0 < x < 1; 1 if x ≥ 1 where a, b are constants. The function is continuous everywhere. What is the value of a?
A. -1
B. 0
C. 1
D. 2
Q. 68 Given a function f(x) = -1 if x ≤ 0, ax + b if 0 < x < 1; 1 if x ≥ 1 where a, b are constants. The function is continuous everywhere. What is the value of b?
A. -1
B. 1
C. 0
D. 2
Q. 69 Consider the following functions : 1. f(x) = x^3, x ∈ ℝ. 2. f(x) = sin x, 0 < x < 2Π. 3. f(x) = e^x , x ∈ ℝ. Which of the above functions have inverse defined on their ranges
A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3
Q. 70 The integral ∫( 1 /( acosx + bsinx) )dx is of the form (1 / r) [ tan( x+a/ 2 ) ].. what is r equal to?
A. a^2 + b^2
B. √( a^2 + b^2 )
C. a + b
D. √( a^2 – b^2 )
Q. 71 The integral ∫( 1 /( acosx + bsinx) )dx is of the form (1 / r) [ tan( x+a/ 2 ) ]. What is a equal to?
A. tan^(-1) ( a / b )
B. tan^(-1) ( b / a )
C. tan^(-1) ( a + b / a – b )
D. tan^(-1) ( a – b / a + b )
Q. 72 Consider the function f(x) = ( x^2 – 1)/ ( x^2 + 1 ), where x ∈ ℝ. At what value of x does f(x) attain minimum value?
A. -1
B. 0
C. 1
D. 2
Q. 73 Consider the function f(x) = ( x^2 – 1)/ ( x^2 + 1 ), where x ∈ ℝ. What is the minimum value of f(x)?
A. 0
B. 1/2
C. -1
D. 2
Q. 74 Consider the function, f(x) = { acosx / Π – 2x if x ≠ Π/2; 3 if x = Π/2 } which is continuous at x = Π/2, where ‘a’ is a constant. What is the value of a ?
A. 6
B. 3
C. 2
D. 1
Q. 75 Consider the function, f(x) = { acosx / Π – 2x if x ≠ Π/2; 3 if x = Π/2 } which is continuous at x = Π/2, where ‘a’ is a constant. What is lim x→0 f(x) equal to?
A. 0
B. 3
C. 3/Π
D. 6/Π
Q. 76 Consider the line x = √(3y) and the circle x^2 + y^2 = 4. What is the area of the region in the first quadrant enclosed by the x-axis, the line x = √3 and the circle?
A. Π/3 – √3/2
B. Π/2 – √3/2
C. Π/3 – 1/2
D. None of the above
Q. 77 Consider the line x = √(3y) and the circle x^2 + y^2 = 4. What is the area of the region in the first quadrant enclosed by the x-axis, the line x = √3y and the circle?
A. Π/3
B. Π/6
C. Π/3 = √3/2
D. None of the above
Q. 78 Consider the curves y = sin x and y = cos x. What is the area of the region bounded by the above two curves and the lines x = 0 and x = Π/4 ?
A. √2 – 1
B. √2 + 1
C. √2
D. 2
Q. 79 Consider the curves y = sin x and y = cos x. What is the area of the region bounded by the above two curves and the lines x = Π/4 and x = Π/2 ?
A. √2 – 1
B. √2 + 1
C. 2√2
D. 2
Q. 80 Consider the function f( x ) = 0.75 x^4 – x^3 – 9x^2 + 7. What is the maximum value of the function?
A. 1
B. 3
C. 7
D. 9
Q. 81 Consider the function f( x ) = 0.75 x^4 – x^3 – 9x^2 + 7. Consider the following statements :
(1) The function attains local minima at x = -2 and x = 3.
(2) The function increases in the interval (-2, 0). 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. 82 Consider the parametric equation x = a(1 – t²) / (1 – t²), y = 2at / (1 – t²). What does the equation represent?
A. It represents a circle of diameter a
B. It represents a circle of radius a
C. it represents a parabola
D. None of the above
Q. 83 Consider the parametric equation x = a(1 – t²) / (1 – t²), y = 2at / (1 – t²). What is dy/dx equal to?
A. y/x
B. – y/x
C. x/y
D. – x/y
Q. 84 Consider the parametric equation x = a(1 – t²) / (1 – t²), y = 2at / (1 – t²). What is d²y / dx² equal to?
A. (a / y)^2
B. (a / x)^2
C. – (a / y)^2
D. – a^2 / y^3
Q. 85 Consider the following statements : 1. The general solution of dy/dx = f(x) + x is of the form y = g(x) + c, where c is an arbitrary constant. 2. The degree of (fy/dx)^2 = f(x) is 2. Which fo the above statements is/are correct?
A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2
Q. 86 What is ∫( 1 / √( x^2 + a^2 ) ) equal to? ( where c is the constant of integration. )
A. ln| ( x + √(x^2 + a^2) )/ a | + c
B. ln| ( x – √(x^2 + a^2) )/ a | + c
C. ln| ( x^2 + √(x^2 + a^2) )/ a | + c
D. None of the above
Q. 87 Consider the integral Im, where m is a positive integer. What is I1 equal to?
A. 0
B. 1/2
C. 1
D. 2
Q. 88 Consider the integral I(m), where m is a positive integer. What is I2 + I3 equal to?
A. 4
B. 2
C. 1
D. 0
Q. 89 Consider the integral I(m), where m is a positive integer. What is Im equal to?
A. 0
B. 1
C. m
D. 2m
Q. 90 Consider the integral I(m), where m is a positive integer. Consider the following : 1.) I( m ) – I( m – 1) is equal to 0. 2.) I(2m) > Im. Which of the above is/are correct?
A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2
Q. 91 Given that d/dx( ( 1 + x^2 + s^4 )/( 1 + x + x^2 ) ) = Ax + B. What sis the value of A?
A. -1
B. 1
C. 2
D. 4
Q. 92 Given that d/dx( ( 1 + x² + s⁴ )/( 1 + x + x² ) ) = Ax + B. What is the value of B?
A. -1
B. 1
C. 2
D. 4
Q. 93 Given that lim x→∞( (2 + x^2)/(1 + x) – Ax – B ) = 3. What is value of A?
A. -1
B. 1
C. 2
D. 3
Q. 94 Given that lim x→∞( (2 + x^2)/(1 + x) – Ax – B ) = 3. What is the value of B?
A. -1
B. 3
C. -4
D. -3
Q. 95 What is the solution of the differential equation ydx – xdy / y^2 = 0? ( where c is an arbitrary constant )
A. xy = c
B. y = cx
C. x + y = c
D. x – y = c
Q. 96 What is the solution of the differential equation sin(dy/dx) – a = 0 ? ( where c is an arbitrary constant )
A. y = x sin^(-1) a + c
B. x = y sin^(-1) a + c
C. y = x + x sin^(-1) a + c
D. y = sin^(-1) a + c
Q. 97 What is the solution of the differential equation dx/dy + x/y – y^2 = 0? ( where c is an arbitrary constant )
A. xy = x^4 + c
B. xy = y^4 + c
C. 4 xy = y^4 + c
D. 3xy = y^3 + c
Q. 98 What is ∫( (x e^x)/(x + 1)^2 )dx equal to? ( where c is the constant of integration. )
A. (x + 1)^2 e^x + c
B. (x + 1) e^x + c
C. e^x/(x + 1) + c
D. e^x/(x + 1)^2 + c
Q. 99 The adjacent sides AB and Ac of a triangle ABC are represented by the vectors -2 i^ + 3j^ + 2k^ and -4i^ + 5j^ +2k^ respectively. The area of the triangle ABC is
A. 6 square units
B. 5 square units
C. 4 square units
D. 3 square units
Q. 100 A force F ⃗ = 3i^ + 4 j^ – 3k^ is applied at the point P, whose position vector is r ⃗ = 2i^ – 2j^ – 3k^. What is the magnitude of the moment of the force about the origin?
A. 23 units
B. 19 units
C. 18 units
D. 21 units
Q. 101 Given that the vectors α ⃗ and β ⃗ are non-collinear. The values of x and y for which u ⃗ – v ⃗ = w ⃗ holds true if u ⃗ = 2xα ⃗ + yβ ⃗, v ⃗ = 2yα ⃗ + 3xβ ⃗ and w ⃗ = 2α ⃗ – 5β ⃗, are
A. x = 2, y = 1
B. x = 1, y = 2
C. x = -2, y = 1
D. x = -2, y = -1
Q. 102 If |a ⃗| = 7, |b ⃗| = 11 and | a ⃗ + b ⃗ | = 10√3, then | a ⃗ + b ⃗ | is equal to
A. 40
B. 10
C. 4√10
D. 2√10
Q. 103 Let α, β, γ be distinct real numbers. The points with position vectors αi^ + βj^ + γk^, βi^ + γj^ + αk^ and γi^ + αj^ + βk^
A. are collinear
B. form an equilateral triangle
C. form a scalene triangle
D. form a right-angled triangle
Q. 104 If a ⃗ + b ⃗ + c ⃗ = 0 ⃗, then which of the following is/are correct? 1. a ⃗,b ⃗,c ⃗ are coplanar 2. a ⃗ × b ⃗ = b ⃗ × c ⃗ = c ⃗ × a ⃗. 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. 105 If |a ⃗ + b ⃗ | = |a ⃗ – b ⃗ |, then which one of the following is correct?
A. a ⃗ = λb ⃗ for some scalar λ
B. a ⃗ is parallel to b ⃗
C. a ⃗ is perpendicular to b ⃗
D. a ⃗ = b ⃗ = 0 ⃗
Q. 106 The mean and the variance of 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be respectively
A. 8 and 20
B. 8 and 4
C. 8 and 8
D. 80 and 40
Q. 107 Which one of the following measure of central tendency is used in construction of index numbers?
A. Harmonic mean
B. Geometric mean
C. Median
D. Mode
Q. 108 The correlation coefficient between two variables X and Y is found to be 0.6. All the observations on X and Y are transformed using the transformations U = 2 – 3X and V = 4Y + 1. The correlation coefficient between the transformed variables U and V will be
A. -0.5
B. +0.5
C. -0.6
D. +0.6
Q. 109 Which of the following statements is/are correct in respect of regression coefficients?
1.) It measures the degree of linear relationship between two variables.
2.) It gives the value by which one variable changes for a unit change in the other variable. 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. 110 A set of annual numerical data, comparable over the years, is given for the last 12 years. Consider the following statements: 1.The data is best represented by a broken line graph, each corner ( turning point ) representing the data of one year. 2. Such a graph depicts the chronological change and also enables one to make a short-term forecast/ 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. 111 Two men hit a target with probabilities 1/2 and 1/3 respectively. What is the probability that exactly one of them hits the target?
A. 1/2
B. 1/3
C. 1/6
D. 2/3
Q. 112 Two similar boxes B(i) where ( i= 1, 2) contain ( i + 1) red and ( 5 – i -1) black balls. One box is chosen at random and two balls are drawn randomly. What is the probability that both the balls are of different colours?
A. 1/2
B. 3/10
C. 2/5
D. 3/5
Q. 113 In an examination. The probability of a candidate solving a question is 1/2. Out of given 5 question in the examination, that is the probability that the candidate was able to solve at least 2 questions?
A. 1/64
B. 3/16
C. 1/2
D. 13/16
Q. 114 If A ⊆B, then which one of the following is not correct?
A. P( A ∩ B’ ) = 0
B. P( A | B ) = P(A)/P(B)
C. P( B | A ) = P(B)/P(A)
D. P( A | (A ∪ B) ) = P(A)/P(B)
Q. 115 The mean and the variance in a binomial distribution are found to be 2 and 1 respectively. The probability P( X = 0) is
A. 1/2
B. 1/4
C. 1/8
D. 1/16
Q. 116 The mean of five numbers is 30. If one number is excluded, their mean becomes 28. The excluded number is
A. 28
B. 30
C. 35
D. 38
Q. 117 If A and B are two events such that P(A∪B) = 3/4, P(A∩B) = 1/4 and P(A’) = 2/3, then what is P(B) equal to?
A. 1/3
B. 2/3
C. 1/8
D. 2/9
Q. 118 The ‘less than’ ogive curve and the ‘more than’ ogive curve intersect at
A. median
B. mode
C. arithmetic mean
D. None of the above
Q. 119 In throwing of two dice, the number of exhaustive events that ‘5’ will never appear on any one of the dice is
A. 5
B. 18
C. 25
D. 36
Q. 120 Two cards are drawn successively without replacement from a well-shuffled pack of 52 cards. The probability of drawing two aces is
A. 1/26
B. 1/221
C. 4/223
D. 1/13
| Answer Sheet |
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| Question | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Answer | B | B | C | A | C | B | D | C | D | A |
| Question | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Answer | D | C | C | C | C | D | B | A | D | C |
| Question | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| Answer | B | D | A | C | D | A | C | C | A | C |
| Question | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| Answer | C | C | A | B | B | C | B | B | B | B |
| Question | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| Answer | C | D | B | B | D | A | B | C | B | D |
| Question | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| Answer | D | A | D | A | B | C | A | D | B | A |
| Question | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| Answer | A | C | C | C | B | B | D | A | A | B |
| Question | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| Answer | B | B | C | A | A | D | C | A | A | C |
| Question | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| Answer | C | B | D | D | C | D | A | C | A | A |
| Question | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
| Answer | A | C | B | B | B | A | C | C | D | A |
| Question | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 |
| Answer | A | D | A | C | C | C | D | B | C | C |
| Question | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 |
| Answer | A | B | D | C | D | D | B | A | C | B |