ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Applications of Derivatives (Set 1): 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 tangent to y = x² − 3x at x = 2 is:
Answer: D. 1
Explanation: dy/dx = 2x − 3 = 4 − 3 = 1 at x = 2.
Q2. The equation of the tangent to y = x² at the point (1, 1) is:
Answer: C. y = 2x − 1
Explanation: Slope = 2x = 2 at x = 1; y − 1 = 2(x − 1) gives y = 2x − 1.
Q3. The equation of the normal to y = x³ at the point (1, 1) is:
Answer: C. x + 3y = 4
Explanation: Tangent slope = 3x² = 3, so the normal slope is −1/3: y − 1 = −(x − 1)/3, i.e. x + 3y = 4.
Q4. The critical points of f(x) = x³ − 3x are at:
Answer: C. x = −1 and x = 1
Explanation: f′(x) = 3x² − 3 = 0 gives x² = 1, so x = ±1.
Q5. The maximum value of f(x) = −x² + 4x + 1 is:
Answer: C. 5
Explanation: f′(x) = −2x + 4 = 0 at x = 2, and f″ = −2 < 0. f(2) = −4 + 8 + 1 = 5.
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Q6. For x > 0, the minimum value of x + 1/x is:
Q7. The local maximum value of f(x) = x³ − 3x is:
Q8. f(x) = x² − 4x is increasing on the interval:
Q9. Which of the following functions is decreasing for every real x?
Q10. If f′(c) = 0 and f″(c) < 0, then at x = c the function f has:
Q11. The point of inflection of y = x³ − 6x² occurs at:
Q12. The radius of a circle increases at 2 cm/s. How fast is the area increasing when r = 5 cm?
Q13. The radius of a sphere increases at 1 cm/s. The rate of increase of its volume when r = 3 cm is:
Q14. The side of a square grows at 3 cm/s. At what rate is its area changing when the side is 10 cm?
Q15. In the Maclaurin series of eˣ, the coefficient of x³ is:
Q16. In the Maclaurin series of sin x, the coefficient of x⁵ is:
Q17. The first three non-zero terms of the Maclaurin series of cos x are:
Q18. In the expansion of ln(1 + x) in powers of x, the coefficient of x⁴ is:
Q19. In the Taylor series of f(x) about x = a, the coefficient of (x − a)² is:
Q20. Two positive numbers have sum 20. The maximum possible value of their product is:
Q21. The maximum value of sin x + cos x is:
Q22. At which x-values does the curve y = x³ − 12x have horizontal tangents?
Q23. At which point on y = x² is the tangent parallel to the line y = 4x + 1?
Q24. The tangent to y = x²/2 at x = 1 makes what angle with the positive x-axis?
Q25. A particle moves so that s = t³ − 6t² + 9t. At what times is it momentarily at rest?
Q26. The maximum value of f(x) = x·e⁻ˣ is:
Q27. Using differentials, an approximate value of √4.01 is:
Q28. If y = x², the differential dy at x = 3 with dx = 0.1 is:
Q29. In the expansion of 1/(1 − 2x) for |x| < 1/2, the coefficient of x⁵ is:
Q30. The curve y = x² − 6x + 10 has a stationary point that is: