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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Definite Integrals and Differential Equations (Set 3)

Take the ECAT Mathematics — Definite Integrals and Differential Equations (Set 3): 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. ∫₀³ (2x + 1) dx equals:

    • A. 10
    • B. 15
    • C. 12 ✓
    • D. 9

    Answer: C. 12

    Explanation: [x² + x] from 0 to 3 = 9 + 3 = 12.

  2. Q2. ∫₁⁸ ∛x dx equals:

    • A. 45/8
    • B. 21/4
    • C. 45/4 ✓
    • D. 15/4

    Answer: C. 45/4

    Explanation: ∫ x^(1/3) dx = (3/4)x^(4/3). From 1 to 8: (3/4)(16 − 1) = 45/4.

  3. Q3. ∫₀^(π/3) sin x dx equals:

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

    Answer: D. 1/2

    Explanation: [−cos x] from 0 to π/3 = −1/2 + 1 = 1/2.

  4. Q4. ∫₀^(π/2) cos 2x dx equals:

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

    Answer: D. 0

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

  5. Q5. ∫₀¹ x/(x² + 1) dx equals:

    • A. ln 2
    • B. ½
    • C. ½ ln 2 ✓
    • D. π/4

    Answer: C. ½ ln 2

    Explanation: With u = x² + 1: ½[ln(x² + 1)] from 0 to 1 = ½ ln 2.

The other 25 questions

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  1. Q6. ∫₁^e (ln x)/x dx equals:

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

  2. Q7. ∫ from −π/2 to π/2 of sin³x dx equals:

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

  3. Q8. ∫ from −2 to 2 of (x⁴ + x³) dx equals:

    • A. 16/5
    • B. 32/5
    • C. 0
    • D. 64/5

  4. Q9. ∫₀^π x sin x dx equals:

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

  5. Q10. ∫₁^e x ln x dx equals:

    • A. (e² + 1)/2
    • B. (e² − 1)/4
    • C. (e² + 1)/4
    • D. e²/2

  6. Q11. ∫₀^(ln 3) eˣ/(1 + eˣ) dx equals:

    • A. ln 3
    • B. ln 2
    • C. ln 4
    • D. ln(3/2)

  7. Q12. ∫ from −1 to 2 of |x| dx equals:

    • A. 5/2
    • B. 3
    • C. 2
    • D. 3/2

  8. Q13. The area bounded by y = x³, the x-axis and the lines x = 0 and x = 2 is:

    • A. 4
    • B. 2
    • C. 8
    • D. 16

  9. Q14. The area enclosed between the parabola y = x² and the line y = x + 2 is:

    • A. 9/2
    • B. 9
    • C. 7/2
    • D. 27/2

  10. Q15. The total area enclosed between the curve y = sin x and the x-axis from x = 0 to x = 2π is:

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

  11. Q16. The area bounded by y = 1/x, the x-axis and the lines x = 1 and x = e² is:

    • A. 2
    • B. 1
    • C. e²
    • D. e² − 1

  12. Q17. The area of the region bounded by the parabola y² = 4x and the line x = 4 is:

    • A. 16
    • B. 32/3
    • C. 128/3
    • D. 64/3

  13. Q18. The general solution of y·(dy/dx) = x is (C an arbitrary constant):

    • A. y² − x² = C
    • B. y = x + C
    • C. y² + x² = C
    • D. y = x² + C

  14. Q19. The general solution of dy/dx = eˣy² is (C an arbitrary constant):

    • A. y = eˣ + C
    • B. y = −1/(eˣ + C)
    • C. y = Ceˣ
    • D. y = 1/(eˣ + C)

  15. Q20. The general solution of dy/dx = y/x² (x ≠ 0) is (C an arbitrary constant):

    • A. y = Cx²
    • B. y = Ce^(1/x)
    • C. y = C ln x
    • D. y = Ce^(−1/x)

  16. Q21. The general solution of dy/dx = (y − 1)/(x + 2) is (C an arbitrary constant):

    • A. y = 1 + C(x + 2)
    • B. y = 1 + C ln(x + 2)
    • C. y = 1 + C/(x + 2)
    • D. y = C(x + 2) − 1

  17. Q22. The general solution of sec²x tan y dx + sec²y tan x dy = 0 is (C an arbitrary constant):

    • A. tan x / tan y = C
    • B. tan x · tan y = C
    • C. sec x · sec y = C
    • D. tan x + tan y = C

  18. Q23. The general solution of dy/dx = (x + y)/x (x > 0) is (C an arbitrary constant):

    • A. y = ln x + Cx
    • B. y = x ln x + C
    • C. y = x ln x + Cx
    • D. y = x + C ln x

  19. Q24. What are the order and degree of (d³y/dx³)² + 5(dy/dx)⁶ = x?

    • A. order 3, degree 2
    • B. order 2, degree 3
    • C. order 3, degree 6
    • D. order 1, degree 6

  20. Q25. The order and degree of d²y/dx² = √(1 + (dy/dx)²) are:

    • A. order 1, degree 2
    • B. order 2, degree 2
    • C. order 2, degree ½
    • D. order 2, degree 1

  21. Q26. Eliminating the constants A and B from y = A cos 2x + B sin 2x gives:

    • A. d²y/dx² + 4y = 0
    • B. d²y/dx² + 2y = 0
    • C. d²y/dx² − 4y = 0
    • D. dy/dx + 2y = 0

  22. Q27. A radioactive substance has a half-life of 5 years. The fraction of the original amount that remains after 15 years is:

    • A. 1/6
    • B. 1/8
    • C. 1/15
    • D. 1/3

  23. Q28. A quantity satisfies dN/dt = 0.1N. The time taken for N to become three times its initial value is:

    • A. 10 ln 3
    • B. 30
    • C. 10 ln 2
    • D. ln 3/10

  24. Q29. A curve passes through (1, 2) and has slope 3x² at every point (x, y). Its equation is:

    • A. y = x³ − 1
    • B. y = x³ + 2
    • C. y = x³ + 1
    • D. y = 3x² − 1

  25. Q30. Let f(x) = x for 0 ≤ x < 1 and f(x) = 2 for 1 ≤ x ≤ 3. Then ∫₀³ f(x) dx equals:

    • A. 11/2
    • B. 5
    • C. 9/2
    • D. 7/2