ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Definite Integrals and Differential Equations (Set 7): 30 multiple-choice questions (40 minutes) with instant scoring and a…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. ∫₀^(π/2) sin⁴x cos²x dx equals:
Answer: D. π/32
Explanation: Using the Wallis/Beta form: ∫₀^(π/2) sin⁴x cos²x dx = (3·1·1)/(6·4·2)·(π/2) = π/32.
Q2. ∫₀¹ x(1 − x)⁵ dx equals:
Answer: D. 1/42
Explanation: Substitute u = 1 − x: ∫₀¹ (1 − u)u⁵ du = 1/6 − 1/7 = 1/42.
Q3. ∫₀³ |(x − 1)(x − 2)| dx equals:
Answer: B. 11/6
Explanation: The product is positive on [0, 1] and [2, 3], negative on [1, 2]. The integrals are 5/6, 1/6 (in absolute value) and 5/6, so the total is 5/6 + 1/6 + 5/6 = 11/6. (Without the modulus the answer would be 3/2.)
Q4. ∫₀^(π/2) cos⁶x dx equals:
Answer: B. 5π/32
Explanation: ∫₀^(π/2) cos⁶x dx = (5·3·1)/(6·4·2)·(π/2) = (15/48)(π/2) = 5π/32.
Q5. The area enclosed between the parabola y² = 4x and the line y = 2x − 4 is:
Answer: C. 9
Explanation: They meet at (1, −2) and (4, 4). Integrating with respect to y: ∫ from −2 to 4 of [(y + 4)/2 − y²/4] dy = [y²/4 + 2y − y³/12] = 9.
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Q6. The area enclosed between the parabola y = x² and the V-shaped graph y = 2 − |x| is:
Q7. The region between y = x and y = x² is revolved about the x-axis. The volume of the solid generated is:
Q8. The solution of (x² + y²) dx = 2xy dy with y(1) = 0 and y > 0 satisfies x² − y² = x. Then y(2) equals:
Q9. The solution of dy/dx = e^(x − y) with y(0) = ln 2 satisfies eʸ = eˣ + 1. Then y(ln 3) equals:
Q10. A liquid at 100 °C is placed in a 20 °C room (Newton's law of cooling). It cools to 60 °C in 10 minutes. Its temperature after 20 minutes is:
Q11. A radioactive substance loses 20% of its mass in the first year. The fraction of the original mass left after 3 years is:
Q12. For the equation e^(y″) + (y′)² = x, which is correct? (Degree is defined only when the equation is a polynomial in its derivatives.)
Q13. The differential equation of the family of circles passing through the origin with centres on the x-axis is:
Q14. ∫₀^(π/2) dx/(1 + tan⁵x) equals:
Q15. ∫₁³ √x/(√x + √(4 − x)) dx equals:
Q16. If F(x) = ∫ from x to x² of (t² + 1) dt, then F′(1) equals:
Q17. The mean (average) value of f(x) = 1/(1 + x²) on [0, 1] is:
Q18. lim (n→∞) Σ (k = 1 to n) n/(n² + k²) equals:
Q19. The trapezoidal rule with 4 equal strips is used to estimate ∫₀² x² dx. The estimate is:
Q20. ∫₀^(ln 2) eˣ/(1 + eˣ) dx equals:
Q21. ∫₁^e (ln x)² dx equals:
Q22. The area bounded by y = x eˣ, the x-axis and the lines x = −1 and x = 0 is:
Q23. The solution of dy/dx = (1 + y²)/(1 + x²) with y(0) = 1 is y = (1 + x)/(1 − x) for |x| < 1. Then y(1/3) equals:
Q24. ∫₀² ⌊x²⌋ dx, where ⌊·⌋ is the greatest-integer function, equals:
Q25. ∫ from −π to π of (x³ cos x + sin²x) dx equals:
Q26. ∫₀¹ dx/(eˣ + e⁻ˣ) equals:
Q27. A curve passes through (1, 2) and its slope at every point (x, y) is 2y/x. Its ordinate at x = 3 is:
Q28. ∫₀^(π/2) x sin x dx equals:
Q29. The solution of x dy/dx = y ln y with y(1) = e is y = e^x. Then y(2) equals:
Q30. A function satisfies y″ = 4y with y(0) = 0 and y′(0) = 2. Then y(ln 2) equals: