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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Definite Integrals and Differential Equations (Set 2)

Take the ECAT Mathematics — Definite Integrals and Differential Equations (Set 2): 30 multiple-choice questions (40 minutes) with instant scoring and a…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. ∫₀² x³ dx equals:

    • A. 4 ✓
    • B. 8
    • C. 16/3
    • D. 2

    Answer: A. 4

    Explanation: [x⁴/4] from 0 to 2 = 16/4 = 4.

  2. Q2. ∫₀^(π/2) cos x dx equals:

    • A. π/2
    • B. −1
    • C. 1 ✓
    • D. 0

    Answer: C. 1

    Explanation: [sin x] from 0 to π/2 = 1 − 0 = 1.

  3. Q3. ∫₁⁴ dx/√x equals:

    • A. 2 ✓
    • B. 3
    • C. 1
    • D. 4

    Answer: A. 2

    Explanation: [2√x] from 1 to 4 = 4 − 2 = 2.

  4. Q4. ∫₀¹ (2x + 1) dx equals:

    • A. 3
    • B. 1
    • C. 3/2
    • D. 2 ✓

    Answer: D. 2

    Explanation: [x² + x] from 0 to 1 = 2.

  5. Q5. ∫₋₁¹ (x² + x³) dx equals:

    • A. 1/3
    • B. 0
    • C. 2/3 ✓
    • D. 4/3

    Answer: C. 2/3

    Explanation: x³ is odd so its part is 0; x² is even, giving 2∫₀¹ x² dx = 2/3.

The other 25 questions

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  1. Q6. ∫₀^(π/2) cos²x dx equals:

    • A. π/2
    • B. 1/2
    • C. π
    • D. π/4

  2. Q7. ∫₀¹ x(1 − x) dx equals:

    • A. 1/3
    • B. 1/6
    • C. 1/2
    • D. 1/12

  3. Q8. The area bounded by y = √x, the x-axis and the line x = 4 is:

    • A. 4
    • B. 16/3
    • C. 8/3
    • D. 32/3

  4. Q9. The area of the region enclosed by y = x² and y = 2x is:

    • A. 4/3
    • B. 2/3
    • C. 4
    • D. 8/3

  5. Q10. The area under y = eˣ from x = 0 to x = ln 3 is:

    • A. ln 3
    • B. 2
    • C. e³ − 1
    • D. 3

  6. Q11. ∫₀^(π/4) tan x dx equals:

    • A. ln √2 + 1
    • B. ln 2
    • C. ½ ln 2
    • D. 1

  7. Q12. ∫₀¹ dx/(1 + x) equals:

    • A. ln 2
    • B. 1
    • C. ln 3
    • D. 1/2

  8. Q13. ∫₀^√3 dx/(1 + x²) equals:

    • A. π/6
    • B. π/3
    • C. π/4
    • D. √3

  9. Q14. Which statement about definite integrals is always true?

    • A. ∫ₐᵃ f(x) dx = 0
    • B. ∫ₐᵇ f(x) dx = ∫ᵇₐ f(x) dx
    • C. ∫₋ₐᵃ f(x) dx = 0 for every f
    • D. ∫ₐᵇ f(x)·g(x) dx = ∫ₐᵇ f dx · ∫ₐᵇ g dx

  10. Q15. If ∫₀² f(x) dx = 3 and ∫₂⁵ f(x) dx = 4, then ∫₀⁵ f(x) dx equals:

    • A. 1
    • B. 7
    • C. 12
    • D. −1

  11. Q16. ∫₀^π |cos x| dx equals:

    • A. 1
    • B. π
    • C. 0
    • D. 2

  12. Q17. ∫₁^e ln x dx equals:

    • A. 0
    • B. e
    • C. e − 1
    • D. 1

  13. Q18. ∫₀^(π/2) x cos x dx equals:

    • A. π/2 − 1
    • B. 1
    • C. π/2 + 1
    • D. π/2

  14. Q19. ∫₀¹ x·e^(x²) dx equals:

    • A. 2(e − 1)
    • B. e − 1
    • C. (e − 1)/2
    • D. e/2

  15. Q20. ∫₀^(π/2) sin³x cos x dx equals:

    • A. 1/4
    • B. 3/4
    • C. 1/3
    • D. 1/2

  16. Q21. ∫₀⁴ |x − 2| dx equals:

    • A. 2
    • B. 0
    • C. 4
    • D. 8

  17. Q22. The solution of dy/dx = 3x² with y(1) = 2 is:

    • A. y = x³ + 1
    • B. y = x³ − 1
    • C. y = x³ + 2
    • D. y = 3x³ − 1

  18. Q23. The general solution of dy/dx = −2y is (C an arbitrary constant):

    • A. y = Ce^(−2x)
    • B. y = −2x + C
    • C. y = Ce^(2x)
    • D. y = −x² + C

  19. Q24. The general solution of dy/dx = xy is (C an arbitrary constant):

    • A. y = x²/2 + C
    • B. y = Ce^(x²/2)
    • C. y = Ce^x + x²/2
    • D. y = Ce^(x²)

  20. Q25. The general solution of dy/dx = 1 + y² is (C an arbitrary constant):

    • A. y = x + y³/3 + C
    • B. y = tan(x + C)
    • C. y = sec(x + C)
    • D. y = tan⁻¹x + C

  21. Q26. The order and degree of d³y/dx³ + 2(d²y/dx²)² − y = 0 are:

    • A. order 3, degree 2
    • B. order 2, degree 2
    • C. order 2, degree 3
    • D. order 3, degree 1

  22. Q27. Eliminating the constant c from y = cx² gives the differential equation:

    • A. x·dy/dx = 2y
    • B. x·dy/dx = y
    • C. y·dy/dx = 2x
    • D. dy/dx = 2y

  23. Q28. A radioactive substance decays according to dN/dt = −kN and its half-life is 5 years. What fraction of it remains after 15 years?

    • A. 1/8
    • B. 1/6
    • C. 1/3
    • D. 1/16

  24. Q29. ∫₀^(π/2) sin x/(sin x + cos x) dx equals:

    • A. ln 2
    • B. 1
    • C. π/2
    • D. π/4

  25. Q30. If ∫₁ᵏ (1/x) dx = 2 with k > 1, then k equals:

    • A. √e
    • B. e
    • C. 2
    • D. e²