ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Differentiation (Set 6): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. If y = xˣ (x > 0), the value of dy/dx at x = 2 is:
Answer: B. 4(1 + ln 2)
Explanation: ln y = x ln x ⇒ y′/y = ln x + 1, so y′ = xˣ(1 + ln x). At x = 2: 4(1 + ln 2).
Q2. If y = (sin x)^(cos x), then dy/dx at x = π/2 is:
Answer: D. 0
Explanation: y′ = (sin x)^(cos x)[cos x·cot x − sin x·ln(sin x)]. At x = π/2: sin x = 1, cos x = 0, ln 1 = 0, so y′ = 1·[0 − 0] = 0.
Q3. For |x| < 1, the derivative of tan⁻¹[2x/(1 − x²)] with respect to x is:
Answer: C. 2/(1 + x²)
Explanation: Put x = tan θ: the expression is tan⁻¹(tan 2θ) = 2θ = 2 tan⁻¹x (valid for |x| < 1). Hence the derivative is 2/(1 + x²).
Q4. For |x| < 1/2, the derivative of sin⁻¹(3x − 4x³) with respect to x is:
Answer: B. 3/√(1 − x²)
Explanation: Put x = sin θ: 3 sin θ − 4 sin³θ = sin 3θ, and for |x| < 1/2 we have 3θ in [−π/2, π/2], so the expression equals 3θ = 3 sin⁻¹x. Derivative: 3/√(1 − x²).
Q5. If x = eᵗ cos t and y = eᵗ sin t, then dy/dx at t = π/2 equals:
Answer: D. −1
Explanation: dx/dt = eᵗ(cos t − sin t) and dy/dt = eᵗ(sin t + cos t). At t = π/2 these are −e^(π/2) and e^(π/2), so dy/dx = −1.
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Q6. For the curve x³ + y³ = 3xy, the slope of the tangent at the point (3/2, 3/2) is:
Q7. If y = x ln x, then x²y″ − xy′ + y equals:
Q8. If y = 1/(1 − 2x), the value of the third derivative y‴ at x = 0 is:
Q9. The derivative of sinh(ln x) with respect to x (x > 0) is:
Q10. If f(x) = ln(cosh x), then f′(ln 2) equals:
Q11. For f(x) = |x² − 4|, the value of f′(1) + f′(3) is:
Q12. Let g(x) = f(x²) where f is differentiable with f′(4) = 3. Then g′(2) equals:
Q13. If f(x) = x³ + x + 1, then the derivative of the inverse function (f⁻¹)′(3) equals:
Q14. Taking sec⁻¹x with x > 1, the derivative of sec⁻¹x at x = 2 is:
Q15. If y = (x + 1)²(x + 2)³/(x + 3)⁴, then dy/dx at x = 0 equals:
Q16. If y = e^(2x) cos x satisfies y″ − 4y′ + ky = 0, then k equals:
Q17. If y = x^(ln x) (x > 0), then dy/dx at x = e equals:
Q18. If y = x^(1/x) (x > 0), then dy/dx at x = e equals:
Q19. For the curve x² + xy + y² = 7, dy/dx at the point (2, 1) equals:
Q20. For f(x) = |sin x|, the value of f′(−π/4) is:
Q21. The value of d⁵/dx⁵ (x⁵ + x⁴ + 7x²) is:
Q22. The derivative of ln(sin⁻¹x) at x = 1/2 is:
Q23. If y = x²eˣ, then y″ at x = 0 equals:
Q24. If y = sin⁻¹x/√(1 − x²), then (1 − x²)y′ − xy equals:
Q25. Given f(1) = 2, f′(1) = 3, g(1) = 4, g′(1) = −1, the derivative of f/g at x = 1 is:
Q26. For x in (π/2, 3π/2), the derivative of sin⁻¹(sin x) with respect to x is:
Q27. For 0 < x < π, the derivative of tan⁻¹√[(1 − cos x)/(1 + cos x)] with respect to x is:
Q28. For f(x) = (x − 1)^(1/3), the derivative f′(1):
Q29. For the cycloid x = a(θ − sin θ), y = a(1 − cos θ), the slope dy/dx at θ = π/2 equals:
Q30. For f(x) = x|x|, which statement is correct at x = 0?