ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Differentiation (Set 5)

Take the ECAT Mathematics — Differentiation (Set 5): 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. d/dx [cos(eˣ)] equals:
- A. eˣ sin eˣ
- B. −sin eˣ
- C. −eˣ sin eˣ ✓
- D. −eˣ cos eˣ
Answer: C. −eˣ sin eˣ
Explanation: Chain rule: −sin(eˣ) · eˣ.
Q2. If f(x) = x³e⁻ˣ, then f′(1) equals:
- A. 3/e
- B. 2e
- C. −2/e
- D. 2/e ✓
Answer: D. 2/e
Explanation: f′(x) = (3x² − x³)e⁻ˣ, so f′(1) = 2e⁻¹.
Q3. For x > 0 and x ≠ 1, d/dx [log_x e] equals:
- A. −1/(x ln x)
- B. −1/(x ln²x) ✓
- C. −1/(x² ln x)
- D. 1/(x ln x)
Answer: B. −1/(x ln²x)
Explanation: log_x e = 1/ln x, whose derivative is −(1/x)/(ln x)².
Q4. d/dx [tan⁻¹(eˣ)] equals:
- A. eˣ/(1 + eˣ)
- B. eˣ/(1 + e^(2x)) ✓
- C. 1/(1 + e^(2x))
- D. 1/(1 + eˣ)
Answer: B. eˣ/(1 + e^(2x))
Explanation: (1/(1 + (eˣ)²)) · eˣ = eˣ/(1 + e²ˣ).
Q5. d/dx [sec²x − tan²x] equals:
- A. 2 sec²x tan x − 2 tan x sec²x + 1
- B. 2 sec x tan x
- C. 0 ✓
- D. 1
Answer: C. 0
Explanation: sec²x − tan²x = 1 is constant, so its derivative is 0.

The other 25 questions

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Q6. d/dx [sin x/(1 + cos x)] equals:
- A. 1/(1 + cos x)
- B. cos x/(1 + cos x)²
- C. 1/(1 − cos x)
- D. (1 + sin x)/(1 + cos x)²
Q7. If y = (ln x)ˣ for x > 1, the value of dy/dx at x = e is:
- A. 1
- B. 1/e
- C. 0
- D. e
Q8. If y = ln(x² + 1), then d²y/dx² at x = 1 equals:
- A. 1/2
- B. 1
- C. 0
- D. −1/2
Q9. For the curve x²y + y³ = 10, the slope dy/dx at (1, 2) is:
- A. −2/13
- B. −1/2
- C. −4/13
- D. 4/13
Q10. If y = √((x − 1)/(x + 1)), then dy/dx at x = 3 equals:
- A. √2/16
- B. √2/4
- C. √2/8
- D. 1/16
Q11. If x = sec t and y = tan t, then dy/dx at t = π/6 equals:
- A. √3
- B. 2
- C. 1/√3
- D. 1/2
Q12. The derivative of ln x with respect to 1/x is:
- A. −1/x
- B. −x
- C. 1/x
- D. −1/x²
Q13. If f(x) = x³ + x + 1, then (f⁻¹)′(3) equals:
- A. 4
- B. 1/10
- C. 1/4
- D. 1/3
Q14. The nth derivative of y = x eˣ is:
- A. x eˣ
- B. (nx + 1)eˣ
- C. (x + 1)ⁿeˣ
- D. (x + n)eˣ
Q15. If y = (A + Bx)e⁻ˣ with constants A and B, then:
- A. y″ + y = 0
- B. y″ + 2y′ + y = 0
- C. y″ − 2y′ + y = 0
- D. y″ + 2y′ − y = 0
Q16. f(x) = x² for x ≤ 1 and f(x) = ax + b for x > 1 is differentiable at x = 1. Then a + b equals:
- A. 1
- B. −1
- C. 3
- D. 2
Q17. lim (h→0) [ln(2 + h) − ln 2]/h equals:
- A. 1
- B. 1/2
- C. ln 2
- D. 2
Q18. If f(x) = x², then lim (x→3) [x f(3) − 3 f(x)]/(x − 3) equals:
- A. 9
- B. −9
- C. 0
- D. −18
Q19. d/dx |x² − 4| at x = 1 equals:
- A. 2
- B. −4
- C. 0
- D. −2
Q20. For |x| < 1, d/dx [sin⁻¹(2x/(1 + x²))] equals:
- A. 1/(1 + x²)
- B. 2/(1 + x²)
- C. 2/√(1 − x²)
- D. 2x/(1 + x²)
Q21. d²/dx² (sin²x) equals:
- A. 2 cos 2x
- B. 2 sin 2x
- C. −2 sin 2x
- D. cos 2x
Q22. d/dx [cot⁻¹(x²)] equals:
- A. 2x/(1 + x⁴)
- B. −2x/(1 + x²)
- C. −2x/(1 + x⁴)
- D. −1/(1 + x⁴)
Q23. The number of values of x in [0, 2π] for which d/dx(sin x + cos x) = 0 is:
- A. 1
- B. 2
- C. 3
- D. 4
Q24. f(x + y) = f(x)f(y) for all x, y, with f′(0) = 3 and f(0) = 1. If f(2) = 5, then f′(2) equals:
- A. 5
- B. 15
- C. 3
- D. 10
Q25. For x > 0, d/dx [tan⁻¹x + tan⁻¹(1/x)] equals:
- A. 0
- B. 1
- C. π/2
- D. 2/(1 + x²)
Q26. If h(x) = f(g(x)), f′(2) = 3, g(1) = 2 and g′(1) = 4, then h′(1) equals:
- A. 8
- B. 3
- C. 12
- D. 6
Q27. d²/dx² (x² ln x) at x = e equals:
- A. 2e
- B. 5
- C. 2
- D. 3
Q28. For the curve xy = e^(x − y), the slope dy/dx at (1, 1) is:
- A. 1
- B. 0
- C. 1/2
- D. −1
Q29. For x² + y² = 25, the second derivative d²y/dx² at (3, 4) equals:
- A. −25/64
- B. −3/16
- C. −1/4
- D. 25/64
Q30. d/dx [x²·2ˣ] at x = 1 equals:
- A. 2
- B. 4 + ln 2
- C. 2 + 2 ln 2
- D. 4 + 2 ln 2