ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Differentiation (Set 2)

Take the ECAT Mathematics — Differentiation (Set 2): 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.

1. Q1. If f(x) = x⁵ − 3x² + 7, then f′(1) equals:

   • A. 11
   • B. 5
   • C. −1 ✓
   • D. 2

   Answer: C. −1

   Explanation: f′(x) = 5x⁴ − 6x, so f′(1) = 5 − 6 = −1.

2. Q2. d/dx (tan 2x) equals:

   • A. 2 sec 2x tan 2x
   • B. 2 sec² 2x ✓
   • C. sec² 2x
   • D. ½ sec² 2x

   Answer: B. 2 sec² 2x

   Explanation: Chain rule: d/dx tan u = sec²u · u′ with u = 2x, giving 2 sec² 2x.

3. Q3. d/dx [e^(x²)] equals:

   • A. 2x·e^(x²) ✓
   • B. 2e^(2x)
   • C. e^(x²)
   • D. x²·e^(x² − 1)

   Answer: A. 2x·e^(x²)

   Explanation: Chain rule: e^(x²) multiplied by the derivative of x², which is 2x.

4. Q4. For x > 0, d/dx [ln(3x)] equals:

   • A. ln 3/x
   • B. 3/x
   • C. 1/x ✓
   • D. 1/(3x)

   Answer: C. 1/x

   Explanation: ln(3x) = ln 3 + ln x, and ln 3 is a constant, so the derivative is 1/x.

5. Q5. d/dx (x² sin x) equals:

   • A. 2x sin x − x² cos x
   • B. 2x sin x + x² cos x ✓
   • C. 2x cos x
   • D. x² cos x

   Answer: B. 2x sin x + x² cos x

   Explanation: Product rule: (x²)′ sin x + x² (sin x)′ = 2x sin x + x² cos x.

The other 25 questions

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6. Q6. d/dx [(x + 1)/(x − 1)] equals:

   • A. 2/(x − 1)²
   • B. −2/(x − 1)²
   • C. −2/(x + 1)²
   • D. 1

7. Q7. d/dx (cos⁻¹x), for −1 < x < 1, equals:

   • A. −1/√(1 − x²)
   • B. −1/(1 + x²)
   • C. 1/√(1 − x²)
   • D. −sin⁻¹x

8. Q8. For x > 1, d/dx (sec⁻¹x) equals:

   • A. −1/(x√(x² − 1))
   • B. 1/√(x² − 1)
   • C. x/√(x² − 1)
   • D. 1/(x√(x² − 1))

9. Q9. d/dx (2^(3x)) equals:

   • A. 3x·2^(3x − 1)
   • B. 3·2^(3x)·ln 2
   • C. 3·2^(3x)
   • D. 2^(3x)·ln 2

10. Q10. For x > 1, d/dx [ln(ln x)] equals:

    • A. 1/ln x
    • B. ln x/x
    • C. 1/x²
    • D. 1/(x ln x)

11. Q11. If y = √(x² + 9), the value of dy/dx at x = 4 is:

    • A. 4/5
    • B. 1/10
    • C. 8/5
    • D. 5/4

12. Q12. If x³ + y³ = 3xy, then dy/dx equals:

    • A. (y + x²)/(y² + x)
    • B. −x²/y²
    • C. (x² − y)/(y² − x)
    • D. (y − x²)/(y² − x)

13. Q13. For the parabola y² = 4ax, dy/dx equals:

    • A. 4a/y
    • B. 2ay
    • C. 2a/y
    • D. a/y

14. Q14. If x = a cos t and y = a sin t, then dy/dx equals:

    • A. −cot t
    • B. tan t
    • C. −tan t
    • D. cot t

15. Q15. If x = 2t + 1 and y = t², the value of dy/dx at t = 3 is:

    • A. 6
    • B. 3/2
    • C. 9
    • D. 3

16. Q16. If y = sin 2x, which relation is true?

    • A. d²y/dx² = −4y
    • B. d²y/dx² + y = 0
    • C. d²y/dx² = 4y
    • D. d²y/dx² = −2y

17. Q17. d/dx (cosec x) equals:

    • A. cosec x cot x
    • B. −cosec²x
    • C. −cosec x cot x
    • D. sec x tan x

18. Q18. d/dx [sin²(3x)] equals:

    • A. 2 sin 3x
    • B. 3 sin²(6x)
    • C. 6 sin 3x
    • D. 3 sin 6x

19. Q19. If y = x^(sin x) (x > 0), then dy/dx equals:

    • A. x^(sin x)[cos x·ln x + (sin x)/x]
    • B. x^(sin x)[sin x·ln x + (cos x)/x]
    • C. sin x · x^(sin x − 1)
    • D. x^(sin x)·cos x·ln x

20. Q20. d/dx [sinh(2x)] equals:

    • A. −2 cosh 2x
    • B. 2 cosh 2x
    • C. 2 sinh 2x
    • D. cosh 2x

21. Q21. d/dx (tanh x) equals:

    • A. sech x tanh x
    • B. sech²x
    • C. −sech²x
    • D. cosh²x

22. Q22. The limit lim (h→0) [sin(x + h) − sin x]/h equals:

    • A. cos x
    • B. 0
    • C. −cos x
    • D. sin x

23. Q23. For x ≠ 0, d/dx [tan⁻¹(1/x)] equals:

    • A. x²/(1 + x²)
    • B. −1/(1 + x²)
    • C. −1/x²
    • D. 1/(1 + x²)

24. Q24. d/dx (eˣ/x) equals:

    • A. eˣ(1 − x)/x²
    • B. eˣ(x − 1)/x²
    • C. eˣ(x + 1)/x²
    • D. eˣ/x²

25. Q25. d/dx (x ln x − x) equals:

    • A. 1/x − 1
    • B. ln x
    • C. x ln x
    • D. ln x − 1

26. Q26. The nth derivative of y = 1/x is:

    • A. (−1)ⁿ n!/xⁿ⁺¹
    • B. n!/xⁿ⁺¹
    • C. (−1)ⁿ⁺¹ n!/xⁿ⁺¹
    • D. (−1)ⁿ n!/xⁿ

27. Q27. If f(x) = (2x − 1)⁴, then f′(1) equals:

    • A. 4
    • B. 8
    • C. 2
    • D. 16

28. Q28. d/dx [sin⁻¹(2x)] equals:

    • A. 2/√(1 − 2x²)
    • B. 2/(1 + 4x²)
    • C. 2/√(1 − 4x²)
    • D. 1/√(1 − 4x²)

29. Q29. If y = ln(sec x + tan x), then dy/dx equals:

    • A. tan x
    • B. sec²x
    • C. sec x
    • D. sec x tan x

30. Q30. The derivative of x³ with respect to x² is:

    • A. 3x²/2
    • B. 2x/3
    • C. 3x/2
    • D. 3x²