ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Definite Integrals and Differential Equations (Set 1)

Take the ECAT Mathematics — Definite Integrals and Differential Equations (Set 1): 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. ∫₀¹ x² dx equals:
- A. 1
- B. 1/3 ✓
- C. 2/3
- D. 1/2
Answer: B. 1/3
Explanation: [x³/3] from 0 to 1 = 1/3.
Q2. ∫₀^π sin x dx equals:
- A. 2 ✓
- B. 1
- C. 0
- D. π
Answer: A. 2
Explanation: [−cos x] from 0 to π = −(−1) − (−1) = 2.
Q3. ∫₁^e (1/x) dx equals:
- A. 1 ✓
- B. 0
- C. e − 1
- D. 1/e
Answer: A. 1
Explanation: [ln x] from 1 to e = ln e − ln 1 = 1.
Q4. ∫₀¹ eˣ dx equals:
- A. e − 1 ✓
- B. e + 1
- C. 1
- D. e
Answer: A. e − 1
Explanation: [eˣ] from 0 to 1 = e − 1.
Q5. ∫₋₁¹ x³ dx equals:
- A. 1/4
- B. 2
- C. 0 ✓
- D. 1/2
Answer: C. 0
Explanation: x³ is an odd function, so its integral over a symmetric interval [−a, a] is zero.

The other 25 questions

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Q6. ∫₀² (3x² + 1) dx equals:
- A. 8
- B. 14
- C. 10
- D. 12
Q7. The area bounded by y = x², the x-axis and the lines x = 0 and x = 3 is:
- A. 27 square units
- B. 3 square units
- C. 6 square units
- D. 9 square units
Q8. The area of the region enclosed between y = x and y = x² is:
- A. 5/6 square unit
- B. 1/2 square unit
- C. 1/3 square unit
- D. 1/6 square unit
Q9. The area enclosed by the parabola y = 4 − x² and the x-axis is:
- A. 8 square units
- B. 32/3 square units
- C. 16/3 square units
- D. 16 square units
Q10. ∫₀^(π/4) sec²x dx equals:
- A. 1
- B. π/4
- C. √2
- D. 2
Q11. ∫₀¹ dx/(1 + x²) equals:
- A. π/4
- B. 1
- C. ln 2
- D. π/2
Q12. ∫₀^(1/2) dx/√(1 − x²) equals:
- A. π/3
- B. π/6
- C. π/4
- D. 1/2
Q13. If ∫₁³ f(x) dx = 7, then ∫₃¹ f(x) dx equals:
- A. −7
- B. 7
- C. 14
- D. 0
Q14. ∫₀^(π/2) sin²x dx equals:
- A. 1
- B. π/4
- C. 1/2
- D. π/2
Q15. ∫₀¹ x·eˣ dx equals:
- A. e
- B. 1/2
- C. 1
- D. e − 1
Q16. ∫₁² dx/x² equals:
- A. −1/2
- B. 1/2
- C. 7/3
- D. 3/4
Q17. ∫₋₂² |x| dx equals:
- A. 2
- B. 0
- C. 4
- D. 8
Q18. The solution of dy/dx = 2x that satisfies y(0) = 3 is:
- A. y = x² + 3
- B. y = x² − 3
- C. y = 2x + 3
- D. y = 3x²
Q19. The general solution of dy/dx = y is:
- A. y = eˣ + c
- B. y = c·eˣ
- C. y = cx
- D. y = x²/2 + c
Q20. The general solution of dy/dx = y/x is:
- A. y = cx
- B. y = ceˣ
- C. y = c/x
- D. y = x + c
Q21. The order and degree of the differential equation (d²y/dx²)³ + (dy/dx)⁴ + y = 0 are:
- A. Order 3, degree 2
- B. Order 2, degree 3
- C. Order 4, degree 2
- D. Order 2, degree 4
Q22. The solution curves of dy/dx = −x/y form a family of:
- A. Parabolas
- B. Circles centred at the origin
- C. Straight lines through the origin
- D. Rectangular hyperbolas
Q23. The general solution of dy/dx = e^(x − y) is:
- A. eʸ = e⁻ˣ + c
- B. e⁻ʸ = eˣ + c
- C. y = eˣ + c
- D. eʸ = eˣ + c
Q24. If dy/dx = 2y and y(0) = 5, then y(1) equals:
- A. 5e²
- B. 10e
- C. e¹⁰
- D. 5e
Q25. ∫₀^π cos x dx equals:
- A. 0
- B. 2
- C. −2
- D. 1
Q26. ∫₀^(ln 2) eˣ dx equals:
- A. 1
- B. 2
- C. ln 2
- D. e − 1
Q27. ∫₀³ √(x + 1) dx equals:
- A. 16/3
- B. 2
- C. 7/3
- D. 14/3
Q28. ∫₀^(π/2) sin x cos x dx equals:
- A. 0
- B. 1/2
- C. 1
- D. π/4
Q29. If ∫₀ᵏ 2x dx = 9 and k > 0, then k equals:
- A. 4.5
- B. 3
- C. √3
- D. 9
Q30. A population grows according to dP/dt = kP and doubles in 10 years. The value of k (per year) is:
- A. (ln 2)/10
- B. ln 10 / 2
- C. 2/10
- D. 10 ln 2