GS: Hindi / English · Language questions stay in their original language.
120 questions · 120 marks · 2 hours
Choose one option per question.
Tap an option to check it, or open “See answer”.
Correct answer: (A) (−∞, 5/2) ∪ (33/8, ∞)
Solving (4x + 3)/(2x − 5) < 6 gives (−8x + 33)/(2x − 5) < 0, hence x < 5/2 or x > 33/8.
Correct answer: (A) 0, sin α, cos α
A line in the yz-plane has zero x-direction cosine, and with the angle α to the z-axis the cosines are 0, sin α, cos α.
Correct answer: (B) 6
Setting [2n(2n−1)(2n−2)/6] ÷ [n(n−1)/2] = 44/3 and solving gives n = 6.
Correct answer: (A) 1/4
With f(x) = x² − x, f′(x) = 2x − 1; f(1/2) − f(0) = (1/2)f′(c) gives c = 1/4.
Correct answer: (A) 3 and 2
ⁿCᵣ = ⁿCᵣ₋₁ gives n = 2r − 1 and ⁿPᵣ = ⁿPᵣ₊₁ gives n − r = 1; solving both, n = 3 and r = 2.
Correct answer: (B) 2π cm³/second
From V = (4/3)πr³, dV/dt = 4πr²(dr/dt) = 4π × 1 × 1/2 = 2π cm³ per second.
Correct answer: (E) None of the above
Exactly two girls can be chosen in C(4,2) × C(6,3) = 6 × 20 = 120 ways, which is not among the options—so the answer is ‘None of the above’.
Correct answer: (A) (0, ±4)
The tangent is parallel to the x-axis where dy/dx = 0, that is at x = 0—so the points are (0, ±4).
Correct answer: (A) n × 2ⁿ⁻¹
C₁ + 2C₂ + ⋯ + nCₙ = n·2ⁿ⁻¹, got by differentiating (1 + x)ⁿ and putting x = 1.
Correct answer: (C) r/2
The curved surface of the inscribed cylinder is greatest when its radius is half the cone's, that is r/2.
Correct answer: (E) None of the above
(aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) equals √(ab) when n = −1/2, which is not among the options—so the answer is ‘None of the above’.
Correct answer: (B) −1/3
y′ = 4x + 3 cos x, so at x = 0 the tangent's gradient is 3 and the normal's is −1/3.
Correct answer: (D) More than one of the above
The numbers with sum −3 and product 8 are −4, −1, 2, and the same set is given in reverse order too, so more than one option is right.
Correct answer: (C) x = t⁶
The LCM of 2 and 3 is 6, so x = t⁶ clears both roots at once.
Correct answer: (C) 60 cm
36° is π/5 radian; arc = radius × angle gives 12π = r × π/5, so r = 60 cm.
Correct answer: (B) (1/n) log |xⁿ/(xⁿ + 1)| + c
The integral works out to (1/n) log |xⁿ/(xⁿ + 1)| + c.
Correct answer: (C) cos(x + y)
Expanding along the first column, the determinant is cos x cos y − sin x sin y = cos(x + y).
Correct answer: (A) eʸ = eˣ + x³/3 + c
Writing dy/dx = e^(x−y) + x²e^(−y) as eʸ dy = (eˣ + x²) dx and integrating gives eʸ = eˣ + x³/3 + c.
Correct answer: (A) π/4
tan⁻¹5 − tan⁻¹3 = tan⁻¹(1/8); adding tan⁻¹(7/9) gives tan⁻¹1 = π/4.
Correct answer: (C) d²y/dx² + y = 0
Differentiating y = a sin(x + b) twice gives y″ = −y, that is d²y/dx² + y = 0.
Correct answer: (C) (1/2) sin⁻¹(3/4)
sin(π cos q) = cos(π sin q) gives π cos q + π sin q = π/2, so sin q + cos q = 1/2, and solving further q = (1/2) sin⁻¹(3/4).
Correct answer: (A) 1/yⁿ⁻¹ = v
In Bernoulli's equation, dividing by yⁿ and putting 1/yⁿ⁻¹ = v reduces it to linear form.
Correct answer: (E) None of the above
For a focal chord of y² = 4ax, t₁t₂ = −1, which is not among the options—so the answer is ‘None of the above’.
Correct answer: (B) tan y = ce^(−x²) + (1/2)(x² − 1)
Putting tan y = v makes the equation linear, and the solution is tan y = ce^(−x²) + (1/2)(x² − 1).
Correct answer: (A) 1
For two conjugate hyperbolas 1/e₁² + 1/e₂² = 1.
Correct answer: (C) 3
What is meant is the scalar triple product—i·(j × k), j·(k × i) and k·(i × j), each of which is 1, so the sum is 3.
Correct answer: (B) Continuous at every x, but not differentiable at x = (2n + 1)π/2, n ∈ Z
|cos x| is continuous everywhere, but it cannot be differentiated where cos x vanishes.
Correct answer: (C) p³r − q³ = 0
If the roots are in GP the middle root is r^(1/3), and substituting gives the condition p³r − q³ = 0.
Correct answer: (B) Continuous on [−1, 1] and differentiable on (−1, 0) ∪ (0, 1)
The function is continuous on [−1, 1] but not differentiable at x = 0 and at the end points, which is what the second option says.
Correct answer: (C) 2/7
A leap year has 52 weeks and two odd days; of the seven possible pairs two contain a Sunday, so the probability is 2/7.
Correct answer: (C) 1/(1 − x²)
Differentiating the series gives 1 + x² + x⁴ + ⋯, whose sum is 1/(1 − x²).
Correct answer: (B) 1/3
Given at least one boy, the cases are BB, BG, GB, so the probability that both are boys is 1/3.
Correct answer: (A) Collinear
The direction ratios of AB and AC are proportional, so the three points are collinear.
Correct answer: (B) 0.58
For independent events P(A ∪ B) = 0.3 + 0.4 − 0.12 = 0.58.
Correct answer: (B) 1
Both expressions equal 2 tan⁻¹x, so the differential coefficient of one with respect to the other is 1.
Correct answer: (B) 11th
The common ratio is 1/√5; solving 5 × (1/√5)^(n−1) = 1/625 gives n = 11.
Correct answer: (B) a²y
Differentiating y = e^(a cos⁻¹x) twice gives (1 − x²)y″ − xy′ = a²y.
Correct answer: (E) None of the above
Putting x − 2 = u the equation becomes u² − 4|u| + 4 = 0, so |u| = 2 and x = 0 or 4; their sum is 4, which is not among the options—so the answer is ‘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।
Correct answer: (D) More than one of the above
The direction ratios (−3, −4, 0) and (1, 2, 2) give cos θ = 11/15; besides, cos⁻¹(1/3) and sec⁻¹3 are the same thing, so the answer is ‘more than one’.
Correct answer: (A) ±2
f(x) = x² − 6 = −2 gives x² = 4, so f⁻¹(−2) = ±2.
Correct answer: (E) None of the above
The formula breaks down at n = −1, where the denominator vanishes and the integral is log|x|; that value is not among the options, so the answer is ‘None of the above’.
Correct answer: (C) π/4 + x/2
By the half-angle formulae √((1 + sin x)/(1 − sin x)) = tan(π/4 + x/2), so the value is π/4 + x/2.
Correct answer: (E) None of the above
This standard integral comes to π/4, which is not among the options—so the answer is ‘None of the above’.
Correct answer: (A) θ = tan⁻¹√(p/q)
Setting the derivative to zero gives tan θ = √(p/q), so the maximum is at θ = tan⁻¹√(p/q).
Correct answer: (B) d²y/dx² − 2dy/dx + 2y = 0
For y = eˣ(A cos x + B sin x) the differential equation is y″ − 2y′ + 2y = 0.
Correct answer: (C) ax sec θ − by cosec θ = a² − b²
The normal to the ellipse at (a cos θ, b sin θ) is ax sec θ − by cosec θ = a² − b².
Correct answer: (E) None of the above
The solution of x dy/dx + y = y² comes out as y(1 − cx) = 1, which is not among the options—so the answer is ‘None of the above’.
Correct answer: (B) 60°
a × (b × c) = (a·c)b − (a·b)c; the given condition makes a·c = 1/2, so the angle between the unit vectors is 60°.
Correct answer: (B) (15/√10)i + (5/√10)j
a = 3i + j has magnitude √10, so a vector of 5 units in that direction is (15/√10)i + (5/√10)j.
Correct answer: (B) x = 1/6
tan⁻¹2x + tan⁻¹3x = π/4 gives 5x/(1 − 6x²) = 1, so 6x² + 5x − 1 = 0 and the admissible root is x = 1/6.