ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Applications of Derivatives (Set 7): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review.…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. In the Maclaurin series of e^(cos x), the coefficient of x² is:
Answer: D. −e/2
Explanation: e^(cos x) = e^(1 − x²/2 + …) = e·(1 − x²/2 + …), so the x² coefficient is −e/2.
Q2. Sand falls at 18π cm³/s, forming a cone whose height is always equal to its base diameter. When the base radius is 3 cm, the height is increasing at:
Answer: B. 2 cm/s
Explanation: h = 2r gives V = πr²h/3 = 2πr³/3. dV/dt = 2πr² dr/dt = 18π, so at r = 3: dr/dt = 1 cm/s and dh/dt = 2 dr/dt = 2 cm/s.
Q3. The side of an equilateral triangle increases at 2 cm/s. When the side is 6 cm, the area is increasing at:
Answer: D. 6√3 cm²/s
Explanation: A = (√3/4)s², so dA/dt = (√3/2)s·ds/dt = (√3/2)(6)(2) = 6√3 cm²/s.
Q4. The least distance from the point (0, 3) to the parabola x² = 4y is:
Answer: C. 2√2
Explanation: A point (x, x²/4) has squared distance x² + (x²/4 − 3)². Its derivative is 2x + x(x²/4 − 3) = x(x²/4 − 1), which is zero at x = 0 (squared distance 9) and x² = 4 (squared distance 4 + 4 = 8, the smaller). The least distance is 2√2.
Q5. The maximum value of f(x) = x e^(−x²) on [0, ∞) is:
Answer: C. 1/√(2e)
Explanation: f′ = e^(−x²)(1 − 2x²) = 0 gives x = 1/√2, and f = (1/√2)e^(−1/2) = 1/√(2e).
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Q6. For x > 0, the minimum value of f(x) = xˣ is:
Q7. The graph of f(x) = x e^(−x) has its point of inflection at x equal to:
Q8. For f(x) = ln x on [1, e²], the number c guaranteed by the Mean Value Theorem is:
Q9. The number of values of c in (0, π) for which Rolle's theorem holds for f(x) = sin 2x on [0, π] is:
Q10. Using differentials with sin 30° = 1/2, the approximate value of sin 31° is:
Q11. A pendulum's period is T = 2π√(L/g). If its length L is increased by 4% (g constant), the period increases by approximately:
Q12. Newton's method is applied to f(x) = x³ − 2 starting at x₀ = 1. The second iterate x₂ is:
Q13. The function f(x) = x³ + ax² + 3x + 5 is increasing on the whole real line for all values of a in:
Q14. The number of critical points of f(x) = x^(2/3)(x − 2), counting points where f′(x) = 0 or f′(x) does not exist, is:
Q15. The absolute minimum value of f(x) = x − 2 sin x on [0, π] is:
Q16. The points of inflection of y = ln(1 + x²) occur at x equal to:
Q17. A particle moves with s = t³ − 9t² + 24t (metres, t in seconds). The total distance travelled in the first 5 seconds is:
Q18. A tangent line drawn from the origin to the curve y = ln x has slope:
Q19. The normal to y = x² at (1, 1) meets the parabola again at the point:
Q20. The curves y² = 4x and x² = 4y intersect at (4, 4). The tangent of the acute angle between them there is:
Q21. A closed cylindrical can of volume 54π cm³ is to have the least total surface area. Its height is:
Q22. The maximum value of f(x) = sin x (1 + cos x) is:
Q23. lim (x→0) (1/x² − cot²x) equals:
Q24. In the Taylor expansion of f(x) = x⁴ about x = 2, the coefficient of (x − 2)³ is:
Q25. The oblique asymptote of y = (x² + x + 1)/(x − 2) is:
Q26. A lamp is 5.4 m above the ground. A man 1.8 m tall walks away from the point below it at 1.2 m/s. The tip of his shadow moves at:
Q27. The volume of a cube increases at 24 cm³/s. When its edge is 4 cm, its surface area is increasing at:
Q28. The coefficient of x⁶ in the Maclaurin series of sin²x is:
Q29. The function f(x) = x³ − 3x² + 3x + 7 has a stationary point at x = 1. At this point f has:
Q30. The minimum value of f(x) = x² ln x for x > 0 is: