ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Differentiation (Set 1)

Take the ECAT Mathematics — Differentiation (Set 1): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30…

What this test covers

30 ECAT Mathematics

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

Q1. If f(x) = x³ − 4x, then f′(2) equals:
- A. 8 ✓
- B. 4
- C. 12
- D. 0
Answer: A. 8
Explanation: f′(x) = 3x² − 4, so f′(2) = 12 − 4 = 8.
Q2. d/dx (sin 3x) equals:
- A. 3 sin 3x
- B. −3 cos 3x
- C. 3 cos 3x ✓
- D. cos 3x
Answer: C. 3 cos 3x
Explanation: By the chain rule, d/dx sin(3x) = cos(3x) · 3 = 3 cos 3x.
Q3. d/dx (e^(2x)) equals:
- A. 2x·e^(2x−1)
- B. e^(2x)/2
- C. e^(2x)
- D. 2e^(2x) ✓
Answer: D. 2e^(2x)
Explanation: d/dx e^(u) = eᵘ·u′ with u = 2x, giving 2e^(2x).
Q4. d/dx [ln(x² + 1)] equals:
- A. x/(x² + 1)
- B. 2x/(x² + 1) ✓
- C. 1/(x² + 1)
- D. 2x·ln(x² + 1)
Answer: B. 2x/(x² + 1)
Explanation: d/dx ln u = u′/u with u = x² + 1, so the derivative is 2x/(x² + 1).
Q5. d/dx (x·eˣ) equals:
- A. eˣ(x + 1) ✓
- B. x·eˣ
- C. eˣ
- D. eˣ(x − 1)
Answer: A. eˣ(x + 1)
Explanation: Product rule: 1·eˣ + x·eˣ = eˣ(x + 1).

The other 25 questions

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Q6. d/dx (sin x / x) equals:
- A. (x cos x − sin x)/x²
- B. (x cos x + sin x)/x²
- C. cos x
- D. (sin x − x cos x)/x²
Q7. d/dx (sin⁻¹x) equals:
- A. −1/√(1 − x²)
- B. 1/(1 + x²)
- C. 1/√(x² − 1)
- D. 1/√(1 − x²)
Q8. d/dx [tan⁻¹(2x)] equals:
- A. 2/(1 + 2x²)
- B. 2/(1 + 4x²)
- C. 2/√(1 − 4x²)
- D. 1/(1 + 4x²)
Q9. d/dx (3ˣ) equals:
- A. 3ˣ
- B. 3ˣ ln 3
- C. 3ˣ / ln 3
- D. x·3ˣ⁻¹
Q10. d/dx (log₁₀ x) equals:
- A. ln 10 / x
- B. 1/(x ln 10)
- C. 10/x
- D. 1/x
Q11. If y = √x, then dy/dx at x = 4 equals:
- A. 1/4
- B. 1/2
- C. 1/8
- D. 2
Q12. For the circle x² + y² = 25, dy/dx equals:
- A. 2x + 2y
- B. −y/x
- C. x/y
- D. −x/y
Q13. If x² + xy + y² = 7, the value of dy/dx at the point (1, 2) is:
- A. 4/5
- B. −4/5
- C. −5/4
- D. −1
Q14. If x = t² and y = t³, then dy/dx equals:
- A. 3/(2t)
- B. 3t²
- C. 2t/3
- D. 3t/2
Q15. If y = x⁴, then d²y/dx² at x = 1 equals:
- A. 4
- B. 24
- C. 12
- D. 16
Q16. If y = e^(3x), which relation is true?
- A. y″ = −9y
- B. y″ = 9y
- C. y′ = y
- D. y″ = 3y
Q17. d/dx (sec x) equals:
- A. −sec x tan x
- B. sec²x
- C. sec x tan x
- D. tan²x
Q18. d/dx (cos²x) equals:
- A. 2 cos x
- B. −sin 2x
- C. sin 2x
- D. −2 sin x
Q19. If y = xˣ (x > 0), then dy/dx equals:
- A. xˣ(1 + ln x)
- B. xˣ ln x
- C. xˣ(1 − ln x)
- D. x·xˣ⁻¹
Q20. d/dx (cosh x) equals:
- A. sinh x
- B. −sinh x
- C. cosh x
- D. sech²x
Q21. The value of lim (h→0) [(2 + h)³ − 8]/h is:
- A. 12
- B. 6
- C. 8
- D. 0
Q22. d/dx [ln(sin x)] equals:
- A. tan x
- B. 1/sin x
- C. −cot x
- D. cot x
Q23. If y = u³ and u = 2x + 1, then dy/dx equals:
- A. 6(2x + 1)
- B. 3(2x + 1)²
- C. 2(2x + 1)³
- D. 6(2x + 1)²
Q24. d/dx (sin⁻¹x + cos⁻¹x) equals:
- A. 0
- B. π/2
- C. 1
- D. 2/√(1 − x²)
Q25. d/dx (sin x cos x) equals:
- A. 2 sin x cos x
- B. cos²x + sin²x
- C. −sin 2x
- D. cos 2x
Q26. If f(x) = x² ln x, then f′(e) equals:
- A. e² + 1
- B. e
- C. 3e
- D. 2e
Q27. The nth derivative of e^(2x) is:
- A. e^(2x)
- B. 2ⁿ⁻¹ e^(2x)
- C. 2ⁿ e^(2x)
- D. n e^(2x)
Q28. d/dx (e^(sin x)) equals:
- A. e^(sin x)
- B. e^(cos x)
- C. cos x · e^(sin x)
- D. sin x · e^(sin x − 1)
Q29. If xy = 4, then dy/dx equals:
- A. −y/x
- B. −x/y
- C. y/x
- D. 4/x
Q30. If y = sin x, then the fourth derivative d⁴y/dx⁴ equals:
- A. cos x
- B. −sin x
- C. −cos x
- D. sin x