ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Applications of Derivatives (Set 4): 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. The slope of the normal to y = x² at the point (1, 1) is
Answer: C. −1/2
Explanation: Tangent slope = 2x = 2 at x = 1, so the normal slope is −1/(2) = −1/2.
Q2. The function f(x) = x³ − 12x is increasing on
Answer: B. (−∞, −2) and (2, ∞)
Explanation: f′(x) = 3x² − 12 = 3(x − 2)(x + 2) > 0 when x < −2 or x > 2.
Q3. The critical points of f(x) = x³ − 3x² are
Answer: B. x = 0 and x = 2
Explanation: f′(x) = 3x² − 6x = 3x(x − 2) = 0 gives x = 0 and x = 2.
Q4. The minimum value of f(x) = x² − 6x + 5 is
Answer: B. −4
Explanation: f(x) = (x − 3)² − 4, so the minimum is −4 at x = 3.
Q5. In the Maclaurin series of ln(1 + x), the coefficient of x⁴ is
Answer: B. −1/4
Explanation: ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + …, so the coefficient of x⁴ is −1/4.
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Q6. A 10 m ladder leans on a wall. Its foot, 6 m from the wall, is pulled away at 2 m/s. How fast is the top sliding down the wall at that instant?
Q7. The edge of a cube increases at 0.5 cm/s. When the edge is 4 cm, the volume is increasing at
Q8. The local maximum value of f(x) = 2x³ − 9x² + 12x + 1 is
Q9. The function f(x) = x⁴ − 4x³ has
Q10. A farmer fences a rectangular field against a long straight wall, using 80 m of fencing on the three other sides. The maximum area is
Q11. For f(x) = x² − 4x on [1, 5], the value of c guaranteed by the Mean Value Theorem is
Q12. For f(x) = x³ − 3x on [0, √3], Rolle's theorem gives c equal to
Q13. In the Taylor series of ln x about x = 1, the coefficient of (x − 1)³ is
Q14. In the Maclaurin series of eˣ sin x, the coefficient of x³ is
Q15. Which of these functions has a point of inflection at x = 0?
Q16. Using differentials, ∛8.06 is approximately
Q17. For f(x) = 2x³ − 3x² − 12x on [0, 3], the absolute maximum minus the absolute minimum equals
Q18. A particle moves with s = t³ − 6t² + 9t. The total distance it travels in the first 4 seconds is
Q19. The curve y = x² + ax + b touches the line y = 2x at the point (1, 2). Then a + b equals
Q20. The normal to y = √x at x = 4 has equation
Q21. A closed cylindrical can must hold 54π cm³. The radius that minimises its total surface area is
Q22. The minimum distance from the point (2, 0) to the parabola y² = x is
Q23. The function f(x) = x/(x² + 1) is increasing on the interval
Q24. At x = 0, the function f(x) = x^(2/3) has
Q25. The graph of y = x e^(−x) has a point of inflection at x equal to
Q26. In the Taylor expansion of sin x about x = π/4, the coefficient of (x − π/4)² is
Q27. In the Maclaurin series of cos²x, the coefficient of x⁴ is
Q28. Water is poured at 9π cm³/s into an inverted cone whose radius equals half its height (r = h/2). How fast is the water level rising when the depth is 6 cm?
Q29. If f′(x) = (x − 1)²(x − 2)(x − 3), the number of local extrema of f is
Q30. The minimum value of f(x) = x + 36/x for x > 0 is