ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Differentiation (Set 7): 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 = x^(ln x), then dy/dx at x = e equals:
Answer: D. 2
Explanation: y = e^((ln x)²), so y′ = y·2 ln x/x. At x = e: y = e and y′ = e·2·1/e = 2.
Q2. For 0 < x < 1/√2, if y = sin⁻¹(2x√(1 − x²)), then dy/dx at x = 1/2 equals:
Answer: D. 4/√3
Explanation: Put x = sin θ: y = sin⁻¹(sin 2θ) = 2θ = 2 sin⁻¹x for 0 < θ < π/4. So dy/dx = 2/√(1 − x²) = 2/√(3/4) = 4/√3.
Q3. If y = tan⁻¹[(√(1 + x²) − 1)/x], then dy/dx at x = 1 equals:
Answer: C. 1/4
Explanation: Put x = tan θ: (sec θ − 1)/tan θ = tan(θ/2), so y = ½ tan⁻¹x and y′ = 1/(2(1 + x²)) = 1/4 at x = 1.
Q4. If f(x) = x^(x²) for x > 0, then f′(1) equals:
Answer: A. 1
Explanation: ln f = x² ln x, so f′/f = 2x ln x + x. At x = 1: f = 1 and f′ = 1·(0 + 1) = 1.
Q5. The number of points at which f(x) = |x² − 4x + 3| is not differentiable is:
Answer: A. 2
Explanation: x² − 4x + 3 = (x − 1)(x − 3) changes sign at x = 1 and x = 3, each a simple root, so |·| has a corner at both points: two points.
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Q6. For the curve x³ + y³ = 3xy, the value of dy/dx at the point (3/2, 3/2) is:
Q7. For x = 2(cos θ + θ sin θ), y = 2(sin θ − θ cos θ), the value of d²y/dx² at θ = π/4 is:
Q8. If y = 1/(x² − 1), the fourth derivative y⁗(0) equals:
Q9. The derivative of sin(x²) with respect to x³ at x = 1 is:
Q10. The function y = e^(2x) cos 3x satisfies y″ + py′ + qy = 0 for constants p, q. Then p + q equals:
Q11. lim (h→0) [sin(π/6 + h) − sin(π/6 − h)]/h equals:
Q12. If y = (x² + 1)^x, then dy/dx at x = 1 equals:
Q13. If f(x) = x⁵ + x³ + 2x, then (f⁻¹)′(4), the derivative of the inverse function at 4, equals:
Q14. Let y = x³ + x. Treating x as a function of y, d²x/dy² at x = 1 equals:
Q15. For y = sin⁻¹(sin x), the value of dy/dx at x = 2 (radians) is:
Q16. If y = sinh⁻¹(x²), then dy/dx at x = 1 equals:
Q17. If y = x²eˣ, the tenth derivative y⁽¹⁰⁾(0) equals:
Q18. On the ellipse 4x² + 9y² = 36, the value of d²y/dx² at the point (0, 2) is:
Q19. If f(x) = x(x − 1)(x − 2)(x − 3)(x − 4), then f′(2) equals:
Q20. For f(x) = |x|³, what is the highest order of derivative that exists at x = 0?
Q21. If y = x^(sin x), then dy/dx at x = π/2 equals:
Q22. Given f(1) = 2, f′(1) = 3, f(2) = 5, f′(2) = −1, g(1) = 2, g′(1) = 4, g(2) = 1, g′(2) = −2, the value of (f∘g)′(1) + (g∘f)′(1) is:
Q23. The derivative of ln|cosec x − cot x| at x = π/6 is:
Q24. If y = √(x + √(x + √(x + …))) (an infinite nested radical) for x > 0, then dy/dx at x = 6 equals:
Q25. If y = sin(2 sin⁻¹x), then (1 − x²)y″ − xy′ + ky = 0 where k equals:
Q26. If f(x) = x^(1/x) for x > 0, then f′(e) equals:
Q27. If y = tan⁻¹(eˣ), then dy/dx at x = 0 equals:
Q28. If y = A cos(ln x) + B sin(ln x), then x²y″ + xy′ + ky = 0 where k equals:
Q29. If f(x) = x|x|, which statement is correct at x = 0?
Q30. lim (h→0) [tan²(π/4 + h) − 1]/h equals: