Practice

Question 101

If x and y are positive real numbers and x>0, there exists a positive integer n such that, by the Archimedean property:

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Correct answer: (B) nx>y

1 marks · 2024 · Set E

Question 102

Let f(x)=x³+3 for x≠0 and f(0)=1. Which is true?

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Correct answer: (E) None of the above

1 marks · 2024 · Set E

Question 103

Which statement is not correct?

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Correct answer: (A) Every continuous function is differentiable

1 marks · 2024 · Set E

Question 104

Which option is not correct?

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Correct answer: (E) None of the above

1 marks · 2024 · Set E

Question 105

The order and degree of the differential equation d²y/dx² = (dy/dx)^(1/3) + y are:

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Correct answer: (C) (2, 3)

1 marks · 2024 · Set E

Question 106

Find the equation of a curve passing through (−2,3), given that the slope of its tangent at any point (x,y) is 2x/y².

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Correct answer: (A) y³=3x²+15

1 marks · 2024 · Set E

Question 107

The solution of the differential equation dy/dx − y/x = 2x² is:

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Correct answer: (C) y=x³+cx

1 marks · 2024 · Set E

Question 108

The general solution of y=px+p², where p=dy/dx, is:

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Correct answer: (B) y=cx+c²

1 marks · 2024 · Set E

Question 109

The particular solution of dy/dx=−4xy², given y(0)=1, is:

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Correct answer: (A) (2x²+1)y=1

1 marks · 2024 · Set E

Question 110

How many arbitrary constants are there in the general solution of a third-order differential equation?

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Correct answer: (C) 3

1 marks · 2024 · Set E

Question 111

The locus of solutions of x dx−y dy=0 represents a family of:

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Correct answer: (B) Hyperbolas

1 marks · 2024 · Set E

Question 112

The solution curve of x dy+y dx=0 is symmetric with respect to:

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Correct answer: (C) y=x

1 marks · 2024 · Set E

Question 113

The area enclosed between the parabolas y²=4x and x²=4y is:

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Correct answer: (A) 16/3

1 marks · 2024 · Set E

Question 114

The nature of f(x)=sin 3x on [0, π/6] is:

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Correct answer: (B) Increasing

1 marks · 2024 · Set E

Question 115

The minimum value of f(x)=x−3 is attained at:

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Correct answer: (E) None of the above

1 marks · 2024 · Set E

Question 116

The maximum value of f(x)=3x⁴+4x³−12x²+12 is:

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Correct answer: (E) None of the above

1 marks · 2024 · Set E

Question 117

The value of ∫ log x dx is:

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Correct answer: (C) x(log x−1)+c

1 marks · 2024 · Set E

Question 118

The value of (1+i)⁴ is:

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Correct answer: (D) More than one of the above

1 marks · 2024 · Set E

Question 119

The argument of i is:

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Correct answer: (B) π/2

1 marks · 2024 · Set E

Question 120

If the modulus of a+3i is 5, the value(s) of a are:

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Correct answer: (D) More than one of the above

1 marks · 2024 · Set E

Question 121

The value of 1³+2³+3³+…+n³ is:

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Correct answer: (E) None of the above

1 marks · 2024 · Set E

Question 122

The principal value of iⁱ is:

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Correct answer: (A) e^(−π/2)

1 marks · 2024 · Set E

Question 123

If A′ is the set of all limit points of A and A is the closure of A, which option is correct?

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Correct answer: (D) More than one of the above

1 marks · 2024 · Set E

Question 124

If x∈(a,b], which option is correct?

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Correct answer: (B) a<x≤b

1 marks · 2024 · Set E

Question 125

Which one is the null set?

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Correct answer: (A) {}

1 marks · 2024 · Set E

Question 126

The set [−1,1]−{0} is equal to:

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Correct answer: (C) [−1,0)∪(0,1]

1 marks · 2024 · Set E

Question 127

Which is a unit vector in the direction of 3j+4k?

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Correct answer: (A) (1/5)(3j+4k)

1 marks · 2024 · Set E

Question 128

For what value of a are the vectors 2i−j+k, i−3j−5k, and 3i−4j+ak coplanar?

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Correct answer: (B) −4

1 marks · 2024 · Set E

Question 129

The position vectors of A and B are i+2j−k and −i+j+k respectively. Q divides AB externally in the ratio 2:1. The position vector of Q is:

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Correct answer: (A) −3i+3k

1 marks · 2024 · Set E

Question 130

If vectors a+b+c=0 and |a|=3, |b|=5, |c|=7, the angle between a and b is:

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Correct answer: (C) π/3

1 marks · 2024 · Set E

Question 131

If a=5i−j−3k and b=i+3j−5k, then a+b and a−b are:

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Correct answer: (B) Perpendicular

1 marks · 2024 · Set E

Question 132

The vertices of a triangle are A(1,1,1), B(1,2,3), and C(2,3,1). Its area in square units is:

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Correct answer: (E) None of the above

1 marks · 2024 · Set E

Question 133

The equation of the line through (1,2,3), parallel to 3i+2j−2k, is:

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Correct answer: (A) r=i+2j+3k+λ(3i+2j−2k), λ∈ℝ

1 marks · 2024 · Set E

Question 134

For what value of a are the lines (x−5)/7=(y+2)/a=z/1 and x/1=y/2=z/3 perpendicular?

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Correct answer: (A) −5

1 marks · 2024 · Set E

Question 135

The region formed by 5x+y≤100, x+y≤60, x≥0, and y≥0 is:

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Correct answer: (B) Feasible

1 marks · 2024 · Set E

Question 136

The value of maximizing Z=4x+y subject to x+y≤50, 3x+y≤90, x≥0, y≥0 is:

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Correct answer: (C) 120

1 marks · 2024 · Set E

Question 137

The optimal value of the objective function is attained at a point:

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Correct answer: (A) Inside the feasible region

1 marks · 2024 · Set E

Question 138

The simplex method was given by:

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Correct answer: (A) G. B. Dantzig

1 marks · 2024 · Set E

Question 139

The objective function of a linear programming problem is:

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Correct answer: (D) More than one of the above

1 marks · 2024 · Set E

Question 140

Which of the following sets is convex?

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Correct answer: (C) {(x,y): y≥2, y≤4}

1 marks · 2024 · Set E

Question 141

If P(A)=0.6, P(B)=0.3, and P(A∩B)=0.2, then P(A|B) is:

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Correct answer: (B) 2/3

1 marks · 2024 · Set E

Question 142

A pair of dice is thrown. The probability that their sum is 7 is:

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Correct answer: (A) 1/6

1 marks · 2024 · Set E

Question 143

The probability that a leap year has 53 Fridays is:

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Correct answer: (B) 2/7

1 marks · 2024 · Set E

Question 144

The number of permutations of the letters of ALLAHABAD is:

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Correct answer: (C) 7560

1 marks · 2024 · Set E

Question 145

If A and B are events such that P(A|B)=P(B|A), then:

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Correct answer: (E) None of the above

1 marks · 2024 · Set E

Question 146

A group consists of 8 girls and 7 boys. In how many ways can a team of 4 be selected if it has no girls?

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Correct answer: (A) 35

1 marks · 2024 · Set E

Question 147

There are 6 boys and 3 girls. In how many ways can they sit around a circular table with no restrictions?

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Correct answer: (E) None of the above

1 marks · 2024 · Set E

Question 148

The average of 10 entries is 50. If 5 is subtracted from each entry, the new mean is:

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Correct answer: (C) 45

1 marks · 2024 · Set E

Question 149

Which of the following options is correct?

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Correct answer: (D) More than one of the above

1 marks · 2024 · Set E

Question 150

The coefficient of x⁶ in the expansion of (2x−3)⁶ is:

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Correct answer: (C) 64

1 marks · 2024 · Set E

TRE3 — Language General Studies And Mathematics