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QUESTION: 1

If n ∈ N, then 121^{n} - 25^{n} 1900^{n} − (−4)^{n} is divisible by which of the following?

Solution:

Putting n = 1, 121^{n} − 25^{n} + 1900^{n} − (−4)^{n} = 121 − 25 + 1900 + 4 = 2000

Which is divisible by 2000.

QUESTION: 2

If n = (2017)!, then, what is

Solution:

QUESTION: 3

In the expansion of (1 + x)^{43}, if the coefficients of (2r + 1)^{th} and (r + 2)^{th} terms are equal, then what is the value of r(r ≠ 1)?

Solution:

QUESTION: 4

What is the principal argument of (−1 − i), where

Solution:

QUESTION: 5

Let α and β be real numbers and z be a complex number. If z^{2} + αz + β = 0 has two distinct non-real roots with real roots Re(z) = 1, then it is necessary that

Solution:

Let z = x + iy

(x + iy)^{2}+ α(x + iy) + β = 0

⇒ x^{2} − y^{2} + 2ixy + αx + iαy + β = 0

Equating real and imaginary parts separately, we get

x^{2} − y^{2} + αx + β = 0 , (2x + α)y = 0

Now, 2x + α = 0 (∵ y = 0)

⇒ α = −2 (∵ x = Re z = 1)

Now, 1 − y^{2} − 2 + β = 0

⇒ β = 1 + y^{2} > 1 (∵ y ∈ R, y ≠ 0)

⇒ β ∈ (1, ∞)

QUESTION: 6

Let A and B be subsets of X and C = (A ∩ B′) ∪ (A′ ∩ B), where A’ and B’ are complements of A and B respectively in X. What is C equal to?

Solution:

C = (A ∩ B′) ∪ (A′ ∩ B)

= (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B)

QUESTION: 7

How many numbers between 100 and 1000 can be formed with the digits 5, 6, 7, 8, 9 if the r epetition of digits is not allowed?

Solution:

No. of ways = 5 × 4 × 3 = 60

QUESTION: 8

The number of non-zero integral solutions of the equation |1 − 2i|^{x }− 5^{x} i

Solution:

|1 − 2i|^{x} = 5^{x }

⇒ 5^{x/2} = 5^{x}

⇒ x = 0

There is no non-zero integral solution.

QUESTION: 9

If the ratio of AM to GM of two positive numbers a and b is 5:3, then a:b is equal to

Solution:

QUESTION: 10

If coefficients of a^{m} and a^{n} in the expansion of (1 + a)^{m+n} are α and β, then which one of the following is correct?

Solution:

QUESTION: 11

If x + log_{15}(1 + 3^{x}) = x log_{15}5 + log_{15}12, where x is an integer, then what is x equal to?

Solution:

QUESTION: 12

How many four-digit numbers divisible by 10 can be formed using 1, 5, 0, 6, 7 without repetition of digits?

Solution:

The last digit is fixed as ‘0’.

∴ No. of ways = 4 × 3 × 2 = 24

QUESTION: 13

In a class, 54 students are good in Hindi only, 63 students are good in Mathematics only, and 41 students are good in English only. There are 18 students who are good in both Hindi and Mathematics. 10 students are good in all three subjects.

Q. What is the number of students who are good in either Hindi or Mathematics but not in English?

Solution:

Required No. = (54 + 63) + (18 − 10) = 125

QUESTION: 14

In a class, 54 students are good in Hindi only, 63 students are good in Mathematics only, and 41 students are good in English only. There are 18 students who are good in both Hindi and Mathematics. 10 students are good in all three subjects.

Q. What is the number of students who are good in Hindi and Mathematics but not in English?

Solution:

Required No. = 18 − 10 = 8

QUESTION: 15

If α and β are different complex numbers with |α| = 1, then what is

Solution:

QUESTION: 16

The equation |1 − x| + x^{2} = 5 has

Solution:

Equation has a rational root and an irrational root.

QUESTION: 17

The binary number expression of the decimal number 31 is

Solution:

31 = 16 + 8 + 4 + 2 + 1

∴ Binary expression of decimal number 31 = 11111

QUESTION: 18

What is i^{1000 }+ i^{1001} + i^{1002} + i^{1003} equal to

Solution:

i^{1000} + i^{1001} + i^{1002} + i^{1003} = 1 + i + i^{2} + i3

⇒ 1 + i − 1 − i = 0

QUESTION: 19

What is equal to (n ≠ 1)?

Solution:

QUESTION: 20

The modulus-amplitude form of

Solution:

QUESTION: 21

What is the number of non-zero terms in the expansion of (after simplification)?

Solution:

Let y = 2√3 x

Now, (1 + y)^{11} + (1 − y)^{11} has no. of terms

QUESTION: 22

What is the greatest integer among the following by which the number 5^{5} + 7^{5} is divisible?

Solution:

5^{5} + 7^{5} is divisible by 5 + 7 = 12

QUESTION: 23

If x = 1 − y + y^{2} − y^{3} + … up to infinite terms, where |y| < 1, then which one of the following is correct?

Solution:

Using formula for sum of infinite terms of GP

QUESTION: 24

What is the inverse of the matrix A =

Solution:

In this case, A^{−1} = adj A = (co − factor A)^{T}

QUESTION: 25

If A is a 2 × 3 matrix and AB is a 2 × 5 matrix, then B must be a

Solution:

(A)_{(2×3) }× (B)_{(3×5)} = (AB)_{(2×5)}

∴ B must be 3 × 5 matrix

QUESTION: 26

and A^{2} − kA − I2 = O, where I2 is the 2 × 2 identity matrix, then what is the value of k?

Solution:

QUESTION: 27

What is the number of triangles that can be formed by choosing the vertices from a set of 12 points in a plane, seven of which lie on the same straight line?

Solution:

No. of triangle = ^{12}C_{3} − ^{7}C_{3} = 220 − 35 = 185

QUESTION: 28

What is C(n, r) + 2C (n, r − 1) + C(n, r) equal to?

Solution:

QUESTION: 29

Let [x] denote the greatest integer function. What is the number of solutions of the equation x^{2} + 4x + [x] = 0 in the interval [0,2]?

Solution:

No. of solution = one

QUESTION: 30

A survey of 850 students in a University yields that 680 students like music and 215 like dance. What is the least number of students who like both music and dance?

Solution:

Required No. = 680 + 215 − 850 = 45

QUESTION: 31

What is the sum of all two-digit numbers which when divided by 3 leave 2 as the remainder?

Solution:

Sum = 11 + 14+. . . . . +98

QUESTION: 32

If 0 < a < 1, the value of log_{10}a is negative. This is justified by

Solution:

Negative power of 10 will always be between 0 & 1.

QUESTION: 33

The third term of a GP is 3. What is the product of the first 5 terms?

Solution:

QUESTION: 34

are in AP; x, 3, z are in GP; then which one of the following will be in HP?

Solution:

x + z = 3 , xz = 9

QUESTION: 35

What is the value of the sum

Solution:

Sum = i^{2} + 2i^{3} + 2i^{4}+. . . . . +2i^{10} + 2i^{11 }+ i^{12} = 2i^{11} = 2i^{3} = −2i

QUESTION: 36

If sin where , then what is (x + y) equal

Solution:

sin(x + y) = sin x . cos y + cos x sin y

QUESTION: 37

Solution:

QUESTION: 38

What is sin 105° + cos 105° equal to?

Solution:

sin(90° + 15°) + cos 105° = cos 15° + cos 105°

QUESTION: 39

In a triangle ABC if a = 2, b = 3 and sin A = 2/3, then what is angle B equal to?

Solution:

QUESTION: 40

What is the principal value of

Solution:

QUESTION: 41

If x, x − y and x + y are the angles of a triangle (not an equilateral triangle) such that tan(x − y), tan x, and tan(x + y) are in GP, what is x equal to?

Solution:

Sum of angles of a triangle = π

⇒ x − y + x + x + y = π

⇒ x = π/3

QUESTION: 42

ABC is a triangle inscribed in a circle with centre O. Let α = ∠BAC, where 45° < α < 90°. Let β = ∠BOC. Which one of the following is correct?

Solution:

QUESTION: 43

If a flag-staff 6 m height placed on the top of a tower throws a shadow of 2√3 m along the ground, then what is the angle that the sun makes with the ground?

Solution:

QUESTION: 44

What is equal to?

Solution:

QUESTION: 45

A spherical balloon of radius r subtends an angle α at the eye of an observer, while the angle of elevation of its centre is β. What is the height of the centre of the balloon (neglecting the height of the observer)?

Solution:

Let H be the height and R be the distance of centre of balloon from the observer.

QUESTION: 46

then what is equal to?

Solution:

QUESTION: 47

If sin α + sin β = 0 = cos α + cos β, where 0 < β < α < 2π, then which one of the following is correct?

Solution:

(sin α + sin β)^{2}+(cos α + cos β)^{2} = 0

⇒ 2 + 2 cos(α − β) = 0

⇒ cos(α − β) = −1 = cos π

⇒ α = π + β

QUESTION: 48

Suppose cos A is given. If only one value of cos (A/2) is possible, then A must be

Solution:

A must be odd multiple of 180°

QUESTION: 49

If cos α + cos β + cos γ = 0, where then what is the value of sin α + sin β + sin γ?

Solution:

QUESTION: 50

The maximum value of where x ∈ is attained at

Solution:

QUESTION: 51

What is the distance between the points which divide the line segment joining (4, 3) and (5, 7) internally and externally in the ratio 2:3?

Solution:

QUESTION: 52

What is the angle between the straight lines (m^{2} − mn) y = (mn + n^{2}) x + n^{3} and (mn + m^{2})y = (mn − n^{2})x + m^{3}, where m > n?

Solution:

QUESTION: 53

What is the equation of the straight line cutting off an intercept 2 from the negative direction of y-axis and inclined at 30° with the positive direction of x-axis?

Solution:

Equation of line is

QUESTION: 54

What is the equation of the line passing through the point of intersection of the lines x + 2y − 3 = 0 and 2x − y + 5 = 0 and parallel to the line y − x + 10 = 0?

Solution:

Equation of line is

x + 2y − 3 + λ (2x − y + 5) = 0

⟹ (1 + 2λ)x + (2 − λ)y + 5λ − 3 = 0

∴ Equation is −5x + 5y − 18 = 0

⇒ 5x − 5y + 18 = 0

QUESTION: 55

Consider the following statements:

1. The length p of the perpendicular from the origin to the line ax + by = c satisfies the relation

2. The length p of the perpendicular from the origin to the line satisfies the relation

3. The length of the perpendicular from the origin to the line y = mx + c satisfies the relation

Q. Which of the above is/are correct?

Solution:

Statement 1

It is true.

Statement 2

It is true.

Statement 3

It is false.

QUESTION: 56

What is the equation of the ellipse whose vertices are (± 5,0) and foci are at (± 4,0)?

Solution:

c = 4, a = 5

b^{2} = 25 − 16 = 9

∴ Equation of ellipse is

QUESTION: 57

What is the equation of the straight line passing through the point (2, 3) and making an intercept on the positive yaxis equal to twice its intercept on the positive x-axis?

Solution:

⟹ 2x + y = 2a ...(1)

Putting (2, 3), we get 2a = 7

∴ Equation of line is 2x + y = 7

QUESTION: 58

Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?

Solution:

Equation of BC is

⇒ x = 2λ , y = 8λ − 11, z = −3λ + 4

Now, 2(x − 1) + 8(y − 8) − 3(z − 4) = 0

⇒ 2x + 8y − 3z = 54

⇒ 4λ + 64λ − 88 + 9λ − 12 = 54

⇒ λ = 2

∴ foot = (4, 5, −2)

QUESTION: 59

What is the equation of the plane passing through the points (-2, 6, -6), (-3, 10, -9) and (-5, 0, -6)?

Solution:

Equation of plane is

⇒ (z + 6)(18) + 3[−6(x + 2) + 3(y − 6)] = 0

⇒ 18z + 108 + 3(−6x − 12 + 3y − 18) = 0

⇒ 2x − y − 2z = 2

QUESTION: 60

A sphere of constant radius r through the origin intersects the coordinate axes in A, B, and C. What is the locus of the centroid of the triangle ABC?

Solution:

Let A (a, 0, 0), B (0, b, 0), C (0, 0, c) Equation sphere is x^{2} + y^{2} + z^{2} − ax − by − cz = 0

Let, (α, β, γ) be the centroid of triangle.

QUESTION: 61

The coordinates of the vertices P, Q, and R of a triangle PQR are (1, -1, 1), (3, -2, 2) and (0, 2, 6) respectively. If ∠RQP = θ, then what is ∠PRQ equal to?

Solution:

⇒ ∠QPR = 90°

We have, ∠RQP = θ

⇒ ∠PRQ = 90° − θ

QUESTION: 62

The perpendiculars that fall from any point of the straight line 2x + 11y = 5 upon the two straight lines 24x + 7y = 20 and 4x − 3y = 2 are

Solution:

be a point on 2x + 11y = 5.

Now, perpendicular from

24x + 7y = 20 is 8/5

Perpendicular from

4x − 3y = 2 is 8/5

QUESTION: 63

The equation of the line, when the portion of it intercepted between the axes is divided by the point (2,3) in the ratio of 3:2, is

Solution:

Equation of line is x + y = 5.

Equation of line is

QUESTION: 64

What is the distance between the straight lines 3x + 4y = 9 and 6x + 8y = 15?

Solution:

QUESTION: 65

What is the equation to the sphere whose centre is at (-2, 3, 4) and radius is 6 units?

Solution:

(x + 2)^{2}+(y − 3) 2+(z − 4)^{2} = 6^{2}

⟹ x^{2} + y^{2} + z^{2} + 4x − 6y − 8z

= 6^{2} − 2^{2} − 3^{2} − 4^{2}

⟹ x^{2} + y^{2} + z^{2} + 4x − 6y − 8z = 7

QUESTION: 66

If are vectors such that and then what is the acute angle between

Solution:

QUESTION: 67

be the position vectors of the points P and Q respectively with respect to origin O. The points R and S divide PQ internally and externally respectively in the ratio 2:3. If and are perpendicular, then which one of the following is correct?

Solution:

QUESTION: 68

What is the moment about the point of a force represented by acting through the point

Solution:

QUESTION: 69

and then what is the value of λ?

Solution:

QUESTION: 70

If the vectors are parallel to each other, then what is equal to ?

Solution:

Cross product of parallel vectors

QUESTION: 71

Which one of the following is correct in respect of the function f: R → R^{+} defined as f(x) = |x + 1|?

Solution:

f(x) = |x + 1|

going through options

(a) f(x^{2}) = |x^{2} + 1|

{f(x)}^{2} = (x + 1)^{2}

Which implies f(x^{2}) ≠ {f(x)}^{2}

(b) f(|x|) = ||x| + 1|

|f(x)| = ||x + 1|| = |x + 1|

Which implies f(|x|) ≠ |f(x)|

(c) f(x + y) = |x + y + 1|

f(x) + f(y) = |x + 1| + |y + 1|

Which implies

f(x + y) ≠ f(x) + f(y)

Option d is correct.

QUESTION: 72

Suppose f: R → R^{+} is defined by What is the range of the function?

Solution:

Clearly y ≥ 0 , Again x^{2} < 1 + x^{2}

So, Range is [0, 1)

QUESTION: 73

If f(x) = |x| + |x + 1| , then which one of the following is correct?

Solution:

Clearly f(x) is continuous at x = 0 and 1.

QUESTION: 74

Consider the function f(x) = What is f′(0) equal to?

Solution:

QUESTION: 75

What is the area of the region bounded by the parabola y^{2} = 6(x − 1) and y^{2} = 3x ?

Solution:

Solving y^{2} = 6(x − 1) and y^{2} = 3x

We get 6x − 6 = 3x

⇒ x = 2

QUESTION: 76

Three sides of a trapezium are each equal to 6cm. Let α∈(0,π/2) be the angle between a pair of adjacent sides. If the area of the trapezium is the maximum possible, then what is α equal to?

Solution:

QUESTION: 77

If the area of the trapezium is maximum, what is the length of the fourth side?

Solution:

Fourth side = 6 + 6 = 12

QUESTION: 78

What is the maximum area of the trapezium?

Solution:

Maximum area = 9 × 3√3 = 27√3

QUESTION: 79

What is equal to?

Solution:

QUESTION: 80

continuous at x = 3, then which one of the following is correct?

Solution:

QUESTION: 81

What is equal to?

Solution:

QUESTION: 82

What is equal to (where [.] is the greatest integer function)?

Solution:

QUESTION: 83

What is the maximum value of 16sinθ − 12sin^{2}θ?

Solution:

Let, sin θ = x , clearly x ∈ [−1, 1]

QUESTION: 84

If f: R → S defined by f(x) = 4 sin x − 3 cos x + 1 is onto, then what is S equal to?

Solution:

QUESTION: 85

For f to be a function, what is the domain of f, if f(x)

Solution:

f(x) is defined if

|x| − x > 0 ⟹ |x| > x

For x > 0 , x > x , (not possible)

For x < 0, −x > x ⟹ 2x < 0 ⟹ x < 0 (possible)

So, domain of f = (−∞, 0)

QUESTION: 86

What is the solution of the differential equation x dy − y dx = 0?

Solution:

QUESTION: 87

What is the derivative of the function f(x) = e^{tan x} + ℓn(sec x) − e^{ℓn x} at x = π/4 ?

Solution:

QUESTION: 88

Which one of the following differential equations has a periodic solution?

Solution:

It is obvious.

QUESTION: 89

What is the period of the function f(x) = sin x ?

Solution:

Period of f(x) = sin x is 2π

QUESTION: 90

Solution:

QUESTION: 91

The order and degree of the differential equation y^{2} = 4a(x − a), where ‘a’ is an arbitrary constant, are respectively

Solution:

y^{2} = 4a(x − a) … . . (1)

Order = 1.

Differentiating, both sides, we get

Putting in (1), we get

QUESTION: 92

What is the value of

Solution:

f(x) = sin x − tan x

⇒ f(−x) = − sin x + tan x = −f(x)

⇒ f(x) is odd function.

QUESTION: 93

then what are the values of a and b respectively?

Solution:

QUESTION: 94

What is equal to?

Solution:

QUESTION: 95

Solution:

QUESTION: 96

Solution:

QUESTION: 97

If f(x) is an even function, where f(x) ≠ 0, then which one of the following is correct?

Solution:

f(x) is even function ⟹ f ′ (x) is odd function.

QUESTION: 98

sin 2x , then what is dy/dx at x = π equal to?

Solution:

QUESTION: 99

What is the solution of (1 + 2x) dy (1 − 2y) dx = 0 ?

Solution:

(1 + 2x)dy = (1 − 2y)dx

QUESTION: 100

What are the order and degree, respectively, of the differential equation

Solution:

Order = 3, Degree = 2

QUESTION: 101

In a Binomial Distribution, the mean is three times its variance. What is the probability of exactly 3 successes out of 5 trials?

Solution:

np= 3 npq, where n= no. of trials

QUESTION: 102

Consider the following statements

Q. Which of the above statements are correct?

Solution:

QUESTION: 103

If the correlation coefficient between x and y is 0.6, covariance is 27 and variance of y is 25, then what is the variance of x?

Solution:

QUESTION: 104

The probabilities that a student will solve Question A and Question B are 0.4 and 0.5 respectively. What is the probability that he solves at least one of the two questions?

Solution:

P = (A ∪ B) = 1 − P(A′ ∩ B′)

= 1 − [(1 − 0.4) × (1 − 0.5)]

= 1 − 0.3 = 0.7

QUESTION: 105

be the mean of x_{1} , x_{2} , x_{3} , …., x_{n} . If x_{i} = a + cy_{i} , for some constants a and c, then what will be the mean of y_{1}, y_{2}, y_{3}, …., y_{n}?

Solution:

QUESTION: 106

Consider the following statements:

1. If the correlation coefficient r_{xy} = 0, then the two lines of regression are parallel to each other.

2. If the correlation coefficient r_{xy} = +1 , then the two lines of regression are perpendicular to each other.

Q. Which of the above statements is/are correct?

Solution:

If r = 0, lines of regression are perpendicular and when r = 1, lines of regression are so, both statements are wrong

QUESTION: 107

If 4x − 5y + 33 = 0 and 20x − 9y = 107 are two lines of regression, then what are the values of x̅ and y̅ respectively?

Solution:

QUESTION: 108

Consider the following statements:

1. Mean is independent of change in scale and changes in origin.

2. Variance is independent of change in scale but not in origin.

Q. Which of the above statements is / are correct?

Solution:

Mean changes with changes in origin.

Variance is independent to the choice of origin.

QUESTION: 109

Consider the following statements:

1. The sum of deviations from mean is always zero.

2. The sum of absolute deviations is minimum when taken around median.

Q. Which of the above statements is/ are correct?

Solution:

By the properties of deviation

QUESTION: 110

What is the median of the numbers 4.6, 0, 9.3, -4.8, 7.6, 2.3, 12.7, 3.5, 8.2, 6.1, 3.9, 5.2?

Solution:

On arranging these 12 numbers in ascending order, the sixth and seventh terms are 4.6 and 5.2.

QUESTION: 111

In a test in Mathematics, 20% of the students obtained “first class”. If the data are represented by a Pie-Chart, what is the central angle corresponding to “first class”?

Solution:

20% of 360° = 72°

QUESTION: 112

The mean and standard deviation of a set of values are 5 and 2 respectively. If 5 is added to each value, then what is the coefficient of variation for the new set of values?

Solution:

New mean = 5 + 5 = 10

New σ = Old σ = 2

QUESTION: 113

A train covers the first 5 km of its journey at a speed of 30 km/hr and the next 15km at speed of 45 km/hr. What is the average speed of the train?

Solution:

QUESTION: 114

Two fair dice are rolled. What is the probability of getting a sum of 7?

Solution:

E = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

n(E) = 6 , n(S) = 36 ⟹ P(E) = 6/36 = 1/6

QUESTION: 115

If A and B are two events such that 2P(A) = 3P(B), where 0 < P(A) < P(B) < 1 , then which one of the following is correct?

Solution:

QUESTION: 116

A box has ten chits numbered 0, 1, 2, 3, …., 9. First, one chit is drawn at random and kept aside. From the remaining, a second chit is drawn at random. What is the probability that the second chit drawn is “9”?

Solution:

n(E) = 1, n(S) = 10 × 9 = 90 P(E) = 1/90

QUESTION: 117

One bag contains 3 white and 2 black balls, another bag contains 5 white and 3 black balls. If a bag is chosen at random and a ball is drawn from it, what is the chance that it is white?

Solution:

QUESTION: 118

Consider the following in respect of two events A and B:

1. P (A occurs but not B) = P(A) − P(B) if B ⊂ A

2. P (A alone or B alone occurs) = P(A) + P(B) − P(A ∩ B)

3. P(A ∪ B) = P(A) + P(B) if A and B are mutually exclusive

Q. Which of the above is/ are correct?

Solution:

It B ⊂ A, then P(A − B) = P(A) − P(A ∩ B) = P(A) + P(B) − 2P(A ∩ B)

Statement 1 is correct P (A alone or B alone)

= P(A) − P(A ∩ B) + P(B) − P(A ∩ B)

= P(A) + P(B) − 2P(A ∩ B)

Statement 2 is false.

It A and B are mutually exclusive, then P(A ∩ B) = 0

⟹ − P(A ∪ B) = P(A) + P(B)

Statement 3 is correct.

QUESTION: 119

A committee of three has to be chosen from a group of 4 men and 5 women. If the selection is made at random, what is the probability that exactly two members are men?

Solution:

n(E) = C(4, 2) × C(5, 1) = 6 × 5 = 30 n(S) = C(9, 3) = 84

QUESTION: 120

The standard deviation σ of the first N natural numbers can be obtained using which one of the following formulae?

Solution:

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