ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Applications of Derivatives (Set 2): 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 = x³ − 2x at x = 1 is:
Answer: B. −1
Explanation: dy/dx = 3x² − 2 = 1 at x = 1, so the normal's slope is −1/1 = −1.
Q2. The equation of the tangent to y = eˣ at x = 0 is:
Answer: B. y = x + 1
Explanation: At x = 0, y = 1 and slope = e⁰ = 1, so y − 1 = 1(x − 0), i.e. y = x + 1.
Q3. The equation of the tangent to y = 1/x at the point (1, 1) is:
Answer: A. x + y = 2
Explanation: dy/dx = −1/x² = −1 at x = 1, so y − 1 = −(x − 1), i.e. x + y = 2.
Q4. The equation of the normal to y = x² at the point (2, 4) is:
Answer: B. x + 4y = 18
Explanation: Tangent slope is 2x = 4, so the normal slope is −1/4: y − 4 = −(x − 2)/4, i.e. x + 4y = 18.
Q5. The critical points of f(x) = 2x³ − 9x² + 12x occur at:
Answer: B. x = 1 and x = 2
Explanation: f′(x) = 6x² − 18x + 12 = 6(x − 1)(x − 2) = 0 gives x = 1, 2.
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Q6. The local minimum value of f(x) = 2x³ − 9x² + 12x is:
Q7. The minimum value of f(x) = 2x² + 8x + 3 is:
Q8. The function f(x) = x³ − 3x² is decreasing on the interval:
Q9. Which of the following functions is increasing for every real x?
Q10. For f(x) = x⁴ at x = 0, which statement is correct?
Q11. The point of inflection of y = x³ − 3x² + 2 is:
Q12. The edge of a cube increases at 2 cm/s. How fast is its volume increasing when the edge is 5 cm?
Q13. Air is pumped into a spherical balloon at 36π cm³/s. How fast is the radius increasing when r = 3 cm?
Q14. The radius of a circle increases at 3 cm/s. The rate of increase of its circumference is:
Q15. In the Maclaurin series of cos 2x, the coefficient of x² is:
Q16. In the Maclaurin series of 1/(1 + x), |x| < 1, the coefficient of x³ is:
Q17. The first two non-zero terms of the Maclaurin series of tan x are:
Q18. In the Maclaurin series of e^(2x), the coefficient of x³ is:
Q19. In the Taylor series of eˣ about x = 1, the coefficient of (x − 1)² is:
Q20. Among all rectangles with perimeter 40 cm, the greatest possible area is:
Q21. If x + 2y = 20 with x, y > 0, the maximum value of xy is:
Q22. The least value of 5 sin x − 12 cos x is:
Q23. A particle moves along a line with s = t³ − 3t² (s in m, t in s). Its acceleration at t = 2 s is:
Q24. If s = 2t² + 3t gives the position of a body, its velocity at t = 4 is:
Q25. Using differentials, the approximate value of ∛8.06 is:
Q26. For y = x³, the differential dy at x = 2 with dx = 0.01 is:
Q27. For x > 0, the minimum value of x + 4/x is:
Q28. The local maximum value of f(x) = x²e⁻ˣ is:
Q29. How many stationary points does y = x³ + 3x have?
Q30. At which point does the curve y = x² − 4x + 5 have a horizontal tangent?