ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Applications of Derivatives (Set 6): 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. A rectangle has its base on the diameter of a semicircle of radius 5 and its two upper vertices on the arc. The maximum possible area of the rectangle is:
Answer: C. 25
Explanation: With half-width 5cosθ and height 5sinθ, area = 50 sinθ cosθ = 25 sin 2θ, which has maximum 25.
Q2. Squares of side x are cut from the corners of a 12 cm × 12 cm sheet and the sides folded up to make an open box. The maximum volume of the box in cm³ is:
Answer: C. 128
Explanation: V = x(12 − 2x)², V′ = (12 − 2x)(12 − 6x) = 0 ⇒ x = 2 (x = 6 gives zero volume). V = 2·8² = 128.
Q3. A right circular cone is inscribed in a sphere of radius 6. The height of the cone of maximum volume is:
Answer: C. 8
Explanation: With height h, base radius² = h(12 − h), V = (π/3)h²(12 − h). V′ = 0 ⇒ 24h − 3h² = 0 ⇒ h = 8.
Q4. A 13 m ladder leans against a vertical wall. Its foot is pulled away at 12 m/s. When the foot is 5 m from the wall, the top of the ladder is sliding down at:
Answer: C. 5 m/s
Explanation: x² + y² = 169 ⇒ y dy/dt = −x dx/dt. Here y = 12, so dy/dt = −(5/12)(12) = −5, i.e. the top slides down at 5 m/s.
Q5. A spherical balloon's volume increases at 100π cm³/s. When its radius is 5 cm, its surface area is increasing at:
Answer: B. 40π cm²/s
Explanation: dV/dt = 4πr² dr/dt = 100π dr/dt ⇒ dr/dt = 1 cm/s at r = 5. dS/dt = 8πr dr/dt = 8π·5·1 = 40π cm²/s.
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Q6. Water is poured at 3π m³/min into an inverted conical tank whose radius at any depth h is h/2. At the instant h = 6 m, the water level is rising at:
Q7. For f(x) = x³ − 3x on [0, √3], the value of c guaranteed by Rolle's theorem is:
Q8. For f(x) = ln x on [1, e], the Mean Value Theorem gives c equal to:
Q9. The function f(x) = x e^(−x) attains its maximum value, which equals:
Q10. For f(x) = x⁴ we have f″(0) = 0. At x = 0 the graph has:
Q11. For f(x) = 2x³ − 9x² + 12x, the local maximum value minus the local minimum value is:
Q12. The sum of the absolute maximum and absolute minimum values of f(x) = x³ − 3x on [−2, 3] is:
Q13. The oblique asymptote of y = (2x² + 1)/(x − 1) is:
Q14. Newton's method is used on f(x) = x² − 7 with starting value x₀ = 3. The next approximation x₁ is:
Q15. Using differentials, the approximate value of ∛8.12 is:
Q16. A particle moves along a line with s = t³ − 6t² + 9t. The total distance it travels during 0 ≤ t ≤ 4 is:
Q17. The minimum distance from the point (4, 0) to the curve y = √x is:
Q18. A closed cylindrical can of volume 16π cm³ is to have minimum surface area. Its height should be:
Q19. The maximum value of f(x) = sin x (1 + cos x) on [0, π] is:
Q20. The tangent to y = √x at the point where x = 4 meets the x-axis at x equal to:
Q21. The normal to the parabola y = x² at the point (1, 1) cuts the y-axis at the point:
Q22. The function f(x) = x³ + kx is increasing on all of ℝ for exactly which values of k?
Q23. The equation x³ − 3x + k = 0 has three distinct real roots exactly when:
Q24. The radius of a circular oil patch grows at 2 cm/s. When its circumference is 20π cm, its area is increasing at:
Q25. The sum of the x-coordinates of all points of inflection of the curve y = x⁴ − 4x³ is:
Q26. The Mean Value Theorem is applied to f(x) = x + 1/x over the interval [1, 4]. The number c it guarantees is:
Q27. The period T = 2π√(l/g) of a pendulum is computed from a length l measured with a 4% error (g is exact). The approximate percentage error in T is:
Q28. lim (x→0) (cos x − 1 + x²/2)/x⁴ equals:
Q29. In the Maclaurin series of sec x, the coefficient of x⁴ is:
Q30. The function f(x) = (ln x)/x is increasing on the largest interval: