ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Applications of Derivatives (Set 3): 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 equation of the tangent to y = √x at the point (4, 2) is:
Answer: D. x − 4y + 4 = 0
Explanation: dy/dx = 1/(2√x) = 1/4 at x = 4. Tangent: y − 2 = (x − 4)/4, i.e. x − 4y + 4 = 0.
Q2. The equation of the normal to y = ln x at the point (1, 0) is:
Answer: D. x + y − 1 = 0
Explanation: dy/dx = 1/x = 1 at x = 1, so the normal has slope −1: y − 0 = −(x − 1), i.e. x + y − 1 = 0.
Q3. The tangent to y = x³ has slope 12 at the points:
Answer: B. (2, 8) and (−2, −8)
Explanation: 3x² = 12 gives x = ±2, and y = x³ gives (2, 8) and (−2, −8).
Q4. For f(x) = x² − 4x + 3 on [1, 3], the value of c guaranteed by Rolle's theorem is:
Answer: D. 2
Explanation: f(1) = f(3) = 0. f′(c) = 2c − 4 = 0 gives c = 2, which lies in (1, 3).
Q5. For f(x) = x³ on [0, 2], the value c in (0, 2) given by the mean value theorem is:
Answer: C. 2/√3
Explanation: Average slope = (8 − 0)/2 = 4. f′(c) = 3c² = 4 gives c = 2/√3 ≈ 1.15, which lies in (0, 2).
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Q6. The function f(x) = x³ − 3x² − 9x + 5 is increasing for:
Q7. The local maximum value of f(x) = x³ − 3x² − 9x + 5 is:
Q8. The greatest value of f(x) = x³ − 3x on the interval [0, 2] is:
Q9. The least value of f(x) = x² − 4x + 5 on the interval [3, 5] is:
Q10. For f(x) = x⁴ − 4x³, the point x = 3 is:
Q11. The points of inflection of y = x⁴ − 6x² occur at:
Q12. The curve y = e^(−x²) has points of inflection at:
Q13. A rectangle has area 100 cm². Its least possible perimeter is:
Q14. A square of side x cm is cut from each corner of a 12 cm × 12 cm sheet and the sides are folded to form an open box. The maximum volume of the box is:
Q15. A farmer has 200 m of fencing to enclose a rectangular field along a straight river, with no fence needed along the river. The maximum area is:
Q16. A 13 m ladder leans against a wall. Its foot, 5 m from the wall, slides away at 2 m/s. The top of the ladder is sliding down the wall at:
Q17. The radius of a circular oil patch grows at 0.5 cm/s. The rate of increase of its circumference is:
Q18. The edge of a cube is 6 cm and is increasing at 0.5 cm/s. The rate of increase of its total surface area is:
Q19. In the Maclaurin series of √(1 + x), the coefficient of x² is:
Q20. The first two non-zero terms of the Maclaurin series of sin 2x are:
Q21. In the Maclaurin series of cosh x, the coefficient of x⁴ is:
Q22. In the Taylor series of ln x about x = 1, the coefficient of (x − 1)² is:
Q23. Using differentials, the approximate value of √25.2 is:
Q24. The radius of a circle is measured as 10 cm with a possible error of 0.1 cm. The approximate error in the area is:
Q25. A particle moves with s = t³ − 9t² + 24t. Its acceleration at the instant it first comes to rest is:
Q26. A particle moves with s = t³ − 6t². Its velocity at the instant its acceleration is zero is:
Q27. For f(x) = x^(2/3), the point x = 0 is:
Q28. For x > 0, the maximum value of f(x) = x/(1 + x²) is:
Q29. For x > 0, the minimum value of x² + 16/x² is:
Q30. The function f(x) = x³ + kx + 1 has no stationary points exactly when: