ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Definite Integrals and Differential Equations (Set 4)

Take the ECAT Mathematics — Definite Integrals and Differential Equations (Set 4): 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.

Q1. ∫₁² (4x³ − 3) dx equals
- A. 9
- B. 13
- C. 12 ✓
- D. 15
Answer: C. 12
Explanation: Antiderivative x⁴ − 3x: (16 − 6) − (1 − 3) = 12.
Q2. ∫₀^(π/6) cos 3x dx equals
- A. 1
- B. 3
- C. 1/3 ✓
- D. 1/2
Answer: C. 1/3
Explanation: [sin 3x/3] from 0 to π/6 = sin(π/2)/3 = 1/3.
Q3. The differential equation (y″)³ + (y′)⁴ + y = 0 has
- A. order 3 and degree 4
- B. order 2 and degree 3 ✓
- C. order 3 and degree 2
- D. order 2 and degree 4
Answer: B. order 2 and degree 3
Explanation: The highest derivative is y″, so the order is 2. Its power is 3, so the degree is 3.
Q4. ∫₁^e dx/x equals
- A. 0
- B. e − 1
- C. 1 ✓
- D. e
Answer: C. 1
Explanation: [ln x] from 1 to e = 1 − 0 = 1.
Q5. ∫ from −π to π of x² sin x dx equals
- A. π³
- B. 2π³/3
- C. 0 ✓
- D. 2π
Answer: C. 0
Explanation: x² sin x is an odd function (even × odd) and the interval is symmetric about 0, so the integral is 0.

The other 25 questions

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Q6. ∫₀¹ x eˣ dx equals
- A. e − 1
- B. e
- C. 1
- D. 2e − 1
Q7. ∫₀^(π/2) cos³x dx equals
- A. 3/4
- B. 1/3
- C. 2/3
- D. 1
Q8. ∫₀¹ x/(1 + x²) dx equals
- A. ln 2
- B. (1/2) ln 2
- C. 1/2
- D. π/4
Q9. ∫₀¹ x e⁻ˣ dx equals
- A. 1/e
- B. 1 − 1/e
- C. 2/e − 1
- D. 1 − 2/e
Q10. The area enclosed between y = x² and y = x is
- A. 1/2
- B. 1/12
- C. 1/6
- D. 1/3
Q11. The area enclosed by y = x² and the line y = 4 is
- A. 8
- B. 32/3
- C. 16/3
- D. 64/3
Q12. The solution of x dy/dx = y with y(1) = 2 gives y(3) equal to
- A. 2/3
- B. 5
- C. 18
- D. 6
Q13. If dy/dx = xy and y(0) = 1, then y(2) equals
- A. e²
- B. e
- C. 2e
- D. e⁴
Q14. A population of 1000 doubles every 5 years under exponential growth. After 15 years it is
- A. 7500
- B. 4500
- C. 8000
- D. 6000
Q15. The differential equation of the family y = Ae^(2x) + Be^(−2x) is
- A. y′ = 2y
- B. y″ = 4y
- C. y″ + 2y′ = 0
- D. y″ = −4y
Q16. ∫₀⁴ √x/(√x + √(4 − x)) dx equals
- A. 1
- B. 2
- C. 2√2
- D. 4
Q17. ∫ from −π/2 to π/2 of (x³ cos x + 2) dx equals
- A. 0
- B. 2π
- C. π
- D. 4π
Q18. ∫₀² |x − 1| dx equals
- A. 0
- B. 2
- C. 1
- D. 1/2
Q19. ∫₀^(π/4) tan³x sec²x dx equals
- A. 1/3
- B. 3/4
- C. 1/2
- D. 1/4
Q20. ∫₀³ x√(x + 1) dx equals
- A. 62/5
- B. 116/15
- C. 14/3
- D. 58/15
Q21. The area enclosed by the parabola y² = 4x and the line y = x is
- A. 8/3
- B. 32/3
- C. 16/3
- D. 4
Q22. The total area enclosed between the curves y = sin x and y = cos x for 0 ≤ x ≤ π/2 is
- A. √2 − 1
- B. 2(√2 − 1)
- C. 2
- D. 0
Q23. The solution of dy/dx = (1 + y²)/(1 + x²) with y(0) = 1 gives y(1/3) equal to
- A. 4/3
- B. 1
- C. 2
- D. 3/2
Q24. The solution of dy/dx = e^(x − y) with y(0) = ln 3 gives y(ln 2) equal to
- A. ln 3
- B. 2 ln 2
- C. ln 5
- D. ln 4 + ln 3
Q25. A radioactive substance has a half-life of 4 hours. The fraction remaining after 10 hours is
- A. 1/8
- B. 1/5
- C. 1/(4√2)
- D. 1/4
Q26. If ∫₀³ f(x) dx = 7 and ∫₂³ f(x) dx = 2, then ∫₀² (2f(x) + 1) dx equals
- A. 16
- B. 11
- C. 12
- D. 10
Q27. ∫₀^π x sin x/(1 + cos²x) dx equals
- A. π/2
- B. π/4
- C. π²/4
- D. π²/2
Q28. ∫₀¹ x(1 − x)¹⁰ dx equals
- A. 1/121
- B. 1/66
- C. 1/11
- D. 1/132
Q29. The differential equation [1 + (y′)²]^(3/2) = y″ has
- A. order 2 and degree 2
- B. order 2 and degree 3/2
- C. order 1 and degree 2
- D. order 2 and degree 3
Q30. The general solution of dy/dx = −2xy is
- A. y = C − x²
- B. y = Ce^(−x²)
- C. y = Ce^(−2x)
- D. y = Ce^(x²)