1. Value of lim (x → 0) (sin x / x) is equal to:
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Answer: A. 1
Explanation: Standard trigonometric limit identity: lim (x → 0) (sin x / x) = 1.
2. Value of lim (x → 0) (tan x / x) is equal to:
A. 1
B. 0
C. ∞
D. Not defined
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Answer: A. 1
Explanation: lim (x → 0) (tan x / x) = lim (x → 0) (sin x / x) × (1 / cos x) = 1 × 1 = 1.
3. Value of lim (x → a) [(xⁿ - aⁿ) / (x - a)] is equal to:
A. n · aⁿ⁻¹
B. n · aⁿ
C. aⁿ⁻¹
D. 0
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Answer: A. n · aⁿ⁻¹
Explanation: Standard limit theorem formula: lim (x → a) [(xⁿ - aⁿ) / (x - a)] = n aⁿ⁻¹.
4. Value of lim (x → 0) [(1 - cos x) / x] is equal to:
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Answer: A. 0
Explanation: 1 - cos x = 2 sin²(x/2), so limit reduces to lim (x → 0) [sin(x/2)/(x/2)] × sin(x/2) = 1 × 0 = 0.
5. Derivative of xⁿ with respect to x is:
A. n · xⁿ⁻¹
B. n · xⁿ⁺¹
C. xⁿ⁻¹ / n
D. xⁿ
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Answer: A. n · xⁿ⁻¹
Explanation: Power rule of differentiation gives d/dx (xⁿ) = n xⁿ⁻¹.
6. Derivative of a constant c with respect to x is:
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Answer: A. 0
Explanation: The rate of change of a constant value is always zero.
7. Derivative of sin x with respect to x is:
A. cos x
B. -cos x
C. tan x
D. -sin x
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Answer: A. cos x
Explanation: Standard trigonometric derivative: d/dx (sin x) = cos x.
8. Derivative of cos x with respect to x is:
A. -sin x
B. sin x
C. -cos x
D. sec² x
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Answer: A. -sin x
Explanation: Standard trigonometric derivative: d/dx (cos x) = -sin x.
9. Derivative of tan x with respect to x is:
A. sec² x
B. -sec² x
C. cot x
D. sec x tan x
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Answer: A. sec² x
Explanation: Standard trigonometric derivative: d/dx (tan x) = sec² x.
10. Product Rule of differentiation for d/dx [u · v] is:
A. u · (dv/dx) + v · (du/dx)
B. u · (dv/dx) - v · (du/dx)
C. (du/dx) · (dv/dx)
D. [u · (dv/dx) + v · (du/dx)] / v²
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Answer: A. u · (dv/dx) + v · (du/dx)
Explanation: Product rule statement: derivative of product equals first times derivative of second plus second times derivative of first.
11. Quotient Rule of differentiation for d/dx [u / v] is:
A. [v · (du/dx) - u · (dv/dx)] / v²
B. [u · (dv/dx) - v · (du/dx)] / v²
C. [v · (du/dx) + u · (dv/dx)] / v²
D. (du/dx) / (dv/dx)
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Answer: A. [v · (du/dx) - u · (dv/dx)] / v²
Explanation: Standard quotient rule formula for finding derivative of ratio of two functions.
12. Derivative of f(x) = x³ + 2x² - 5x + 7 is:
A. 3x² + 4x - 5
B. 3x² + 2x - 5
C. x² + 4x - 5
D. 3x² + 4x + 7
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Answer: A. 3x² + 4x - 5
Explanation: Applying power rule term by term: d/dx(x³) + d/dx(2x²) - d/dx(5x) + d/dx(7) = 3x² + 4x - 5 + 0.
13. Value of lim (x → 2) (x² - 4) / (x - 2) is:
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Answer: A. 4
Explanation: (x² - 4)/(x - 2) = (x - 2)(x + 2)/(x - 2) = x + 2. Substituting x = 2 gives 2 + 2 = 4.
14. Value of lim (x → 0) (sin 3x / x) is:
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Answer: A. 3
Explanation: lim (x → 0) 3 × [sin 3x / 3x] = 3 × 1 = 3.
15. Derivative of sec x with respect to x is:
A. sec x tan x
B. -sec x tan x
C. sec² x
D. tan² x
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Answer: A. sec x tan x
Explanation: Standard derivative formula: d/dx (sec x) = sec x tan x.
16. Derivative of cot x with respect to x is:
A. -cosec² x
B. cosec² x
C. -cosec x cot x
D. -tan x
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Answer: A. -cosec² x
Explanation: Standard trigonometric derivative: d/dx (cot x) = -cosec² x.
17. Derivative of cosec x with respect to x is:
A. -cosec x cot x
B. cosec x cot x
C. -cosec² x
D. -cot² x
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Answer: A. -cosec x cot x
Explanation: Standard trigonometric derivative: d/dx (cosec x) = -cosec x cot x.
18. Value of lim (x → 0) [(1 - cos 2x) / x²] is:
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Answer: A. 2
Explanation: 1 - cos 2x = 2 sin² x. Limit = lim (x → 0) 2 (sin x / x)² = 2(1)² = 2.
19. The derivative of f(x) at x = a from first principle is defined as:
A. lim (h → 0) [f(a+h) - f(a)] / h
B. lim (h → 0) [f(a+h) + f(a)] / h
C. lim (h → 0) [f(a-h) - f(a)] / h
D. [f(a+h) - f(a)] / h
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Answer: A. lim (h → 0) [f(a+h) - f(a)] / h
Explanation: Definition of derivative at a point using first principles.
20. Value of lim (x → 3) (x⁵ - 243) / (x - 3) is:
A. 405
B. 243
C. 135
D. 81
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Answer: A. 405
Explanation: Using formula n aⁿ⁻¹ where n=5, a=3: 5 × 3⁴ = 5 × 81 = 405.
21. Derivative of f(x) = x · sin x with respect to x is:
A. x cos x + sin x
B. x cos x - sin x
C. cos x
D. x sin x + cos x
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Answer: A. x cos x + sin x
Explanation: By product rule: x · d/dx(sin x) + sin x · d/dx(x) = x cos x + sin x(1).
22. Derivative of f(x) = 1 / x with respect to x is:
A. -1 / x²
B. 1 / x²
C. -1 / x
D. log x
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Answer: A. -1 / x²
Explanation: d/dx (x⁻¹) = -1 · x⁻² = -1 / x².
23. Derivative of √x with respect to x is:
A. 1 / (2√x)
B. 1 / √x
C. 2√x
D. 1 / (2x)
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Answer: A. 1 / (2√x)
Explanation: d/dx (x^(1/2)) = (1/2) x^(-1/2) = 1 / (2√x).
24. Value of lim (x → 0) [sin ax / sin bx] is:
A. a / b
B. b / a
C. 1
D. 0
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Answer: A. a / b
Explanation: lim (x → 0) [(sin ax / ax) · a] / [(sin bx / bx) · b] = (1 · a) / (1 · b) = a / b.
25. Value of lim (x → 1) (x³ - 1) / (x - 1) is:
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Answer: A. 3
Explanation: Using n aⁿ⁻¹ with n=3, a=1: 3(1)² = 3. Or factorizing (x-1)(x²+x+1)/(x-1) = 1+1+1 = 3.
26. Derivative of f(x) = (sin x) / x at x = π/2 is:
A. -2 / π²
B. 2 / π
C. 0
D. 1
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Answer: A. -2 / π²
Explanation: f'(x) = (x cos x - sin x)/x². At x = π/2: [(π/2)(0) - 1] / (π/2)² = -1 / (π²/4) = -4/π².
27. If f(x) = 2x - 5, then derivative f'(3) is equal to:
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Answer: A. 2
Explanation: f'(x) = 2 for all x. Therefore, f'(3) = 2.
28. Value of lim (x → 0) [(tan 2x) / x] is:
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Answer: A. 2
Explanation: lim (x → 0) 2 × [tan 2x / 2x] = 2 × 1 = 2.
29. Derivative of (x + 1) / (x - 1) with respect to x is:
A. -2 / (x - 1)²
B. 2 / (x - 1)²
C. -2 / (x + 1)²
D. 1 / (x - 1)²
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Answer: A. -2 / (x - 1)²
Explanation: By quotient rule: [(x-1)(1) - (x+1)(1)] / (x-1)² = (x - 1 - x - 1) / (x-1)² = -2 / (x-1)².
30. Value of lim (x → 0) (eˣ - 1) / x is equal to:
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Answer: A. 1
Explanation: Standard exponential limit identity: lim (x → 0) (eˣ - 1) / x = 1.
