ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Definite Integrals and Differential Equations (Set 5)

Take the ECAT Mathematics — Definite Integrals and Differential Equations (Set 5): 30 multiple-choice questions (40 minutes) with instant scoring and a…

What this test covers

• 30 ECAT Mathematics

Sample questions with answers

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

1. Q1. ∫₀^(π/2) sin x cos x/(1 + sin²x) dx equals:

   • A. ½ ln 2 ✓
   • B. 1/2
   • C. ln 2
   • D. π/4

   Answer: A. ½ ln 2

   Explanation: Put u = 1 + sin²x (from 1 to 2): ½∫ du/u = ½ ln 2.

2. Q2. ∫₀⁴ dx/(1 + √x) equals:

   • A. 4 − 2 ln 3 ✓
   • B. 2 ln 3
   • C. 2 − 2 ln 3
   • D. 4 − ln 3

   Answer: A. 4 − 2 ln 3

   Explanation: Put x = u²: ∫₀² 2u/(1 + u) du = 2[u − ln(1 + u)]₀² = 2(2 − ln 3).

3. Q3. ∫₀¹ x√(1 − x²) dx equals:

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

   Answer: C. 1/3

   Explanation: Put u = 1 − x²: ½∫₀¹ √u du = ½·(2/3) = 1/3.

4. Q4. ∫₀^(π/2) cos x/(1 + sin²x) dx equals:

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

   Answer: B. π/4

   Explanation: Put u = sin x: ∫₀¹ du/(1 + u²) = tan⁻¹1 = π/4.

5. Q5. ∫₀^(π/2) dx/(1 + tan³x) equals:

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

   Answer: B. π/4

   Explanation: By the property ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a − x)dx, the integrals of 1/(1 + tan³x) and 1/(1 + cot³x) = tan³x/(1 + tan³x) are equal and sum to π/2, so each is π/4.

The other 25 questions

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6. Q6. The definite integral of ln[(2 − x)/(2 + x)] over the interval [−1, 1] is:

   • A. 0
   • B. 2 ln 3
   • C. −2 ln 3
   • D. ln 3

7. Q7. ∫₀^(π/2) sin 2x e^(sin²x) dx equals:

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

8. Q8. ∫₁^e (ln x)²/x dx equals:

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

9. Q9. ∫₀^(ln 3) eˣ/√(1 + eˣ) dx equals:

   • A. 2√2
   • B. 4√2 − 4
   • C. 2 − √2
   • D. 4 − 2√2

10. Q10. ∫₀¹ x²e^(x³) dx equals:

    • A. e/3
    • B. (e³ − 1)/3
    • C. (e − 1)/3
    • D. e − 1

11. Q11. ∫₀² |x² − 1| dx equals:

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

12. Q12. ∫₀^π √(1 + cos 2x) dx equals:

    • A. √2
    • B. 2√2
    • C. 4
    • D. 2

13. Q13. The area enclosed between the curves y = x³ and y = x is:

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

14. Q14. The area under y = ln x from x = 1 to x = e is:

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

15. Q15. The area bounded by y = 4x − x² and y = x is:

    • A. 9/2
    • B. 9
    • C. 27/2
    • D. 3

16. Q16. The area between y = sin 2x and y = sin x from x = 0 to x = π/3 (where sin 2x ≥ sin x) is:

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

17. Q17. The area enclosed by the ellipse x²/9 + y²/4 = 1 is:

    • A. 6π
    • B. 3π
    • C. 12π
    • D. 36π

18. Q18. The general solution of dy/dx = (x² + y²)/(2xy) is (C an arbitrary constant):

    • A. x² − y² = C
    • B. x² + y² = Cx
    • C. x² − y² = Cx²
    • D. x² − y² = Cx

19. Q19. Solving dy/dx = y tan x (with C an arbitrary constant) gives:

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

20. Q20. The solution of dy/dx = (1 + x)/(xy) (x, y > 0) is:

    • A. y² = 2(x + ln x) + C
    • B. y = x + ln x + C
    • C. y² = x²/2 + x + C
    • D. y² = 2x + ln x + C

21. Q21. If x dy + y dx = 0 and y(1) = 2, then y(4) equals:

    • A. 1/8
    • B. 1/2
    • C. 8
    • D. 2

22. Q22. The solution of dy/dx = 2√y with y(0) = 1 gives y(2) equal to:

    • A. 4
    • B. 5
    • C. 9
    • D. 25

23. Q23. A cup of tea cools according to dT/dt = −k(T − 20) (°C, min). It cools from 100 to 60 in 10 minutes, so after a further 10 minutes (20 minutes in all) its temperature is:

    • A. 20 °C
    • B. 45 °C
    • C. 30 °C
    • D. 40 °C

24. Q24. A culture triples in 3 hours under exponential growth. The time needed for it to grow to 9 times its initial size is:

    • A. 4.5 hours
    • B. 27 hours
    • C. 9 hours
    • D. 6 hours

25. Q25. The order and degree of the differential equation y″ + sin(y′) = 0 are (degree is defined only when the equation is a polynomial in its derivatives):

    • A. order 2, degree not defined
    • B. order 2, degree 2
    • C. order 2, degree 1
    • D. order 1, degree 2

26. Q26. Eliminating the constants a and b from y = ax² + b gives:

    • A. y″ = y′
    • B. y″ = 2a
    • C. x y″ = y′
    • D. y′ = 2ax

27. Q27. Which differential equation has y = x ln x as a solution (x > 0)?

    • A. x y′ = y − x
    • B. x y′ = y + x
    • C. y′ = y + x
    • D. x y′ = y

28. Q28. ∫₀^(π/2) sin⁵x dx equals:

    • A. 4/5
    • B. 8/15
    • C. 16/15
    • D. 3/8

29. Q29. lim (n→∞) [1/(n + 1) + 1/(n + 2) + … + 1/(2n)] equals:

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

30. Q30. ∫₀³ x/√(x + 1) dx equals:

    • A. 4/3
    • B. 16/3
    • C. 8/3
    • D. 14/3