The solution of the inequality (4x + 3)/(2x − 5) < 6 is
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Correct answer: (A) (−∞, 5/2) ∪ (33/8, ∞)
1 marks · 2023 · Set A
Question 52
A line lying in the yz-plane is inclined at an angle α with the z-axis. Its direction cosines are
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Correct answer: (A) 0, sin α, cos α
1 marks · 2023 · Set A
Question 53
If (2n)!/[3!(2n − 3)!] and n!/[2!(n − 2)!] are in the ratio 44:3, then n is
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Correct answer: (B) 6
1 marks · 2023 · Set A
Question 54
Writing Lagrange’s mean value theorem as f(b) − f(a) = (b − a)f′(c), a < c < b, find c if f(x) = x(x − 1), a = 0 and b = 1/2.
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Correct answer: (A) 1/4
1 marks · 2023 · Set A
Question 55
If ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁, then the values of n and r are, respectively
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Correct answer: (A) 3 and 2
1 marks · 2023 · Set A
Question 56
The radius of an air bubble is increasing at the rate of 1/2 cm/second. At what rate is the volume of the bubble increasing when its radius is 1 cm?
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Correct answer: (B) 2π cm³/second
1 marks · 2023 · Set A
Question 57
From 6 boys and 4 girls, 5 students are to be selected for admission to a course. In how many ways can this be done if there must be exactly 2 girls?
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Correct answer: (E) None of the above
1 marks · 2023 · Set A
Question 58
Find the points on the curve x²/9 + y²/16 = 1 at which the tangents are parallel to the x-axis.
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Correct answer: (A) (0, ±4)
1 marks · 2023 · Set A
Question 59
If C₀, C₁, C₂, …, Cₙ are the coefficients in the expansion of (1 + x)ⁿ, then C₁ + 2C₂ + 3C₃ + ⋯ + nCₙ is equal to
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Correct answer: (A) n × 2ⁿ⁻¹
1 marks · 2023 · Set A
Question 60
The radius of the right circular cylinder of maximum curved surface inscribed in a right circular cone of radius r is
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Correct answer: (C) r/2
1 marks · 2023 · Set A
Question 61
For what value of n is (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) the geometric mean of a and b?
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Correct answer: (E) None of the above
1 marks · 2023 · Set A
Question 62
The gradient of the normal to the curve y = 2x² + 3 sin x at x = 0 is
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Correct answer: (B) −1/3
1 marks · 2023 · Set A
Question 63
The sum of three numbers in AP is −3 and their product is 8. The numbers are
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Correct answer: (D) More than one of the above
1 marks · 2023 · Set A
Question 64
An appropriate substitution for the integral ∫[(1 + x^(1/2))/(1 + x^(1/3))] dx is
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Correct answer: (C) x = t⁶
1 marks · 2023 · Set A
Question 65
The radius of a circular disc in which an arc of 12π cm subtends an angle of 36° at the centre is
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Correct answer: (C) 60 cm
1 marks · 2023 · Set A
Question 66
Integrate 1/[x(xⁿ + 1)] with respect to x.
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Correct answer: (B) (1/n) log |xⁿ/(xⁿ + 1)| + c
1 marks · 2023 · Set A
Question 67
The value of the determinant [[1, x, y], [0, cos x, sin y], [0, sin x, cos y]] is
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Correct answer: (C) cos(x + y)
1 marks · 2023 · Set A
Question 68
The general solution of the differential equation dy/dx = e^(x−y) + x²e^(−y) is
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Correct answer: (A) eʸ = eˣ + x³/3 + c
1 marks · 2023 · Set A
Question 69
The value of tan⁻¹5 − tan⁻¹3 + tan⁻¹(7/9) is
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Correct answer: (A) π/4
1 marks · 2023 · Set A
Question 70
Find the differential equation representing the family of curves y = a sin(x + b), where a and b are arbitrary constants.
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Correct answer: (C) d²y/dx² + y = 0
1 marks · 2023 · Set A
Question 71
If sin(π cos q) = cos(π sin q), then q is
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Correct answer: (C) (1/2) sin⁻¹(3/4)
1 marks · 2023 · Set A
Question 72
The differential equation dy/dx + Py = Qyⁿ, where P and Q are functions of x alone, can be reduced to linear form by dividing the equation by yⁿ and putting
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Correct answer: (A) 1/yⁿ⁻¹ = v
1 marks · 2023 · Set A
Question 73
If the points (at₁², 2at₁) and (at₂², 2at₂) are the vertices of a focal chord of the parabola y² = 4ax, then
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Correct answer: (E) None of the above
1 marks · 2023 · Set A
Question 74
The general solution of the differential equation sec²y (dy/dx) + 2x tan y = x³ is
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Correct answer: (B) tan y = ce^(−x²) + (1/2)(x² − 1)
1 marks · 2023 · Set A
Question 75
If e₁ and e₂ are the eccentricities of the hyperbolas x²/a² − y²/b² = 1 and x²/a² − y²/b² = −1, respectively, then 1/e₁² + 1/e₂² is
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Correct answer: (A) 1
1 marks · 2023 · Set A
Question 76
The value of i × (j × k) + j × (k × i) + k × (i × j) is
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Correct answer: (C) 3
1 marks · 2023 · Set A
Question 77
The function f(x) = |cos x| is
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Correct answer: (B) Continuous at every x, but not differentiable at x = (2n + 1)π/2, n ∈ Z
1 marks · 2023 · Set A
Question 78
The condition under which the roots of the cubic equation x³ − px² + qx − r = 0 are in a geometrical progression is
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Correct answer: (C) p³r − q³ = 0
1 marks · 2023 · Set A
Question 79
If the function f(x) = √(1 − √(1 − x²)), then f(x) is
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Correct answer: (B) Continuous on [−1, 1] and differentiable on (−1, 0) ∪ (0, 1)
1 marks · 2023 · Set A
Question 80
What is the probability that a leap year selected at random will have 53 Sundays?
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Correct answer: (C) 2/7
1 marks · 2023 · Set A
Question 81
If y = x + x³/3 + x⁵/5 + ⋯, then dy/dx is equal to
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Correct answer: (C) 1/(1 − x²)
1 marks · 2023 · Set A
Question 82
There are two children in a family. If it is known that at least one of the children is a boy, what is the probability that both children are boys?
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Correct answer: (B) 1/3
1 marks · 2023 · Set A
Question 83
The points A(1, 2, 3), B(−1, −1, −1) and C(3, 5, 7) are
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Correct answer: (A) Collinear
1 marks · 2023 · Set A
Question 84
Let A and B be independent events. If P(A) = 0.3 and P(B) = 0.4, what is P(A ∪ B)?
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Correct answer: (B) 0.58
1 marks · 2023 · Set A
Question 85
The differential coefficient of tan⁻¹(2x/(1 − x²)) with respect to sin⁻¹(2x/(1 + x²)) is
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Correct answer: (B) 1
1 marks · 2023 · Set A
Question 86
Which term of the progression 5, √5, 1, … is 1/625?
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Correct answer: (B) 11th
1 marks · 2023 · Set A
Question 87
If y = e^(a cos⁻¹x), −1 < x < 1, then the value of (1 − x²)d²y/dx² − x(dy/dx) is
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Correct answer: (B) a²y
1 marks · 2023 · Set A
Question 88
The sum of the roots of the equation x² − 4|x − 2| − 4x + 8 = 0 is
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Correct answer: (E) None of the above x² − 4x + 8 को (x − 2)² + 4 लिखिए और t = x − 2 रखिए। तब समीकरण बनता है t² − 4|t| + 4 = 0, यानी (|t| − 2)² = 0, जिससे |t| = 2 और t = ±2। इसलिए x = 4 अथवा x = 0, और मूलों का योग 4 + 0 = 4 है—जो 8, 9 या 6 में से कोई नहीं। आयोग की कुंजी में विकल्प A (8) है, पर हल करने पर योग 4 ही आता है।
1 marks · 2023 · Set A
Question 89
The angle between the lines (5 − x)/3 = (y + 3)/(−4), z = 7 and (x − 1)/1 = (y − 2)/2 = (z − 6)/2 is
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Correct answer: (D) More than one of the above
1 marks · 2023 · Set A
Question 90
If the mapping f: R → R is defined by f(x) = x² − 6, then what is the value of f⁻¹(−2)?
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Correct answer: (A) ±2
1 marks · 2023 · Set A
Question 91
∫xⁿ dx = xⁿ⁺¹/(n + 1) + c is not true if n is equal to
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Correct answer: (E) None of the above
1 marks · 2023 · Set A
Question 92
The value of tan⁻¹√((1 + sin x)/(1 − sin x)) is
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Correct answer: (C) π/4 + x/2
1 marks · 2023 · Set A
Question 93
∫₀^(π/2) [tan x/(√tan x + √cot x)] dx is
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Correct answer: (E) None of the above
1 marks · 2023 · Set A
Question 94
The function sinᵖθ cosᑫθ has maximum value when
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Correct answer: (A) θ = tan⁻¹√(p/q)
1 marks · 2023 · Set A
Question 95
The differential equation of y = eˣ(A cos x + B sin x) is
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Correct answer: (B) d²y/dx² − 2dy/dx + 2y = 0
1 marks · 2023 · Set A
Question 96
The equation of the normal to the ellipse x²/a² + y²/b² = 1 at the point (a cos θ, b sin θ) is