ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Definite Integrals and Differential Equations (Set 6): 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. ∫ from −1 to 1 of (x⁵ + x² + x) dx equals:
Answer: A. 2/3
Explanation: The odd terms x⁵ and x integrate to zero over [−1, 1]. The even term gives 2∫₀¹x²dx = 2/3.
Q2. ∫₀³ ⌊x⌋ dx, where ⌊x⌋ is the greatest integer function, equals:
Answer: C. 3
Explanation: ⌊x⌋ = 0 on [0, 1), 1 on [1, 2), 2 on [2, 3). The integral = 0 + 1 + 2 = 3.
Q3. ∫₀¹ x e^(x²) dx equals:
Answer: B. (e − 1)/2
Explanation: Let u = x²: ½∫₀¹eᵘdu = (e − 1)/2.
Q4. ∫₀¹ x² eˣ dx equals:
Answer: B. e − 2
Explanation: By parts: ∫x²eˣ = eˣ(x² − 2x + 2). Evaluate: e(1 − 2 + 2) − 2 = e − 2.
Q5. The area under y = 1/(1 + x²) from x = 0 to x = √3 is:
Answer: B. π/3
Explanation: Area = [tan⁻¹x]₀^√3 = π/3 − 0 = π/3.
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Q6. The area bounded by y = ln x, the x-axis and the line x = e is:
Q7. The region under y = √x from x = 0 to x = 4 is rotated about the x-axis. The volume generated is:
Q8. The region bounded by y = x² and the line y = 4 is rotated about the y-axis (take x ≥ 0). The volume generated is:
Q9. The region between y = x and y = x² (0 ≤ x ≤ 1) is rotated about the x-axis. The volume generated is:
Q10. ∫₀^(π/4) tan²x dx equals:
Q11. ∫₀¹ dx/(1 + √x) equals:
Q12. If ∫₀² f(x)dx = 5 and ∫₀² [f(x) + kx] dx = 11, then k equals:
Q13. The average value of f(x) = x² on the interval [0, 3] is:
Q14. The trapezoidal rule with 4 equal sub-intervals is used to approximate ∫₀² x² dx. The result is:
Q15. The solution of dy/dx = e^(x − y) satisfying y(0) = ln 2 is such that y(1) equals:
Q16. The differential equation (y″)³ = (y′ + 1)² has (after the usual clearing of radicals/fractions) order and degree respectively:
Q17. Eliminating the constants c₁ and c₂ from y = c₁eˣ + c₂e⁻ˣ gives the differential equation:
Q18. The solution of xy dy/dx = x² + y² with y(1) = 1 gives y(e) equal to:
Q19. A radioactive sample decays at a rate proportional to the amount present. If 20% decays in 4 days, the fraction remaining after 8 days is:
Q20. ∫₀^(π/2) ln(sin x) dx equals:
Q21. ∫₀² √(4x − x²) dx equals:
Q22. If F(x) = ∫₀^(x²) √(1 + t²) dt, then F′(1) equals:
Q23. ∫₀^(π/4) ln(1 + tan x) dx equals:
Q24. ∫₀³ |x² − x − 2| dx equals:
Q25. The area enclosed between the curves y = x³ and y = 4x is:
Q26. The total area between y = sin x and y = cos x from x = 0 to x = π/2 is:
Q27. ∫₀^π x/(1 + sin x) dx equals:
Q28. The solution of x dy/dx = y + x tan(y/x) satisfying y(1) = π/2 has y(1/2) equal to:
Q29. The solution of dy/dx = y cot x with y(π/2) = 3 gives y(π/6) equal to:
Q30. Water drains from a tank so that its depth h (in cm) satisfies dh/dt = −k√h. If h = 100 initially and h = 64 after 10 minutes, the tank becomes empty after a total time of: