ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Functions and Limits (Set 4)

Take the ECAT Mathematics — Functions and Limits (Set 4): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers…

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. If f(x) = x/(x − 2), x ≠ 2, then f⁻¹(x) is
- A. (x − 2)/x
- B. 2x/(x − 1) ✓
- C. x/(x − 1)
- D. 2x/(x + 1)
Answer: B. 2x/(x − 1)
Explanation: Put y = x/(x−2): xy − 2y = x, so x(y − 1) = 2y and x = 2y/(y − 1). Hence f⁻¹(x) = 2x/(x − 1).
Q2. lim(x→0) sin 5x / sin 3x equals
- A. 3/5
- B. 15
- C. 5/3 ✓
- D. 1
Answer: C. 5/3
Explanation: sin 5x ≈ 5x and sin 3x ≈ 3x near 0, so the limit is 5/3.
Q3. lim(x→∞) (4x² − 3x)/(6x² + x) equals
- A. 4
- B. 2/3 ✓
- C. 3/2
- D. 0
Answer: B. 2/3
Explanation: Divide numerator and denominator by x²: the limit is the ratio of leading coefficients, 4/6 = 2/3.
Q4. lim(x→0) (3ˣ − 2ˣ)/x equals
- A. ln 3/ln 2
- B. ln(3/2) ✓
- C. 1
- D. ln 6
Answer: B. ln(3/2)
Explanation: (3ˣ − 1)/x → ln 3 and (2ˣ − 1)/x → ln 2, so the limit is ln 3 − ln 2 = ln(3/2).
Q5. lim(x→0) (1 − cos 4x)/x² equals
- A. 2
- B. 4
- C. 8 ✓
- D. 16
Answer: C. 8
Explanation: 1 − cos 4x = 2sin²2x ≈ 2(2x)² = 8x², so the limit is 8.

The other 25 questions

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Q6. lim(x→∞) (1 + 2/x)^(3x) equals
- A. e⁶
- B. e³
- C. e²
- D. e⁵
Q7. lim(x→1) (x³ − 1)/(x² − 1) equals
- A. 1
- B. 3/2
- C. 3
- D. 2/3
Q8. f(x) = ax + 1 for x < 1, f(1) = 3, f(x) = x² + b for x > 1. If f is continuous at x = 1, then a + b equals
- A. 5
- B. 2
- C. 3
- D. 4
Q9. The domain of f(x) = √(x² − 7x + 10) is
- A. (−∞, 2) ∪ (5, ∞)
- B. [2, 5]
- C. (2, 5)
- D. (−∞, 2] ∪ [5, ∞)
Q10. The range of f(x) = 3 − 2 sin x is
- A. [1, 5]
- B. [−2, 2]
- C. [−1, 5]
- D. [1, 3]
Q11. The function f(x) = ln(x + √(x² + 1)) is
- A. neither even nor odd
- B. periodic
- C. odd
- D. even
Q12. If f(x) = (4x + 3)/(2x − 5), then f⁻¹(3) equals
- A. −9
- B. 9
- C. 15
- D. 1/3
Q13. lim(x→0) (sin 2x − 2 sin x)/x³ equals
- A. −2
- B. 1
- C. 0
- D. −1
Q14. lim(x→0) (∛(8 + x) − 2)/x equals
- A. 1/12
- B. 1/3
- C. 1/4
- D. 1/6
Q15. lim(x→∞) (√(x² + 6x) − x) equals
- A. 0
- B. 6
- C. ∞
- D. 3
Q16. For f(x) = |x − 2|/(x − 2), lim(x→2) f(x) is
- A. 0
- B. does not exist
- C. 1
- D. −1
Q17. For f(x) = 2x − 1 and g(x) = x², the solution set of (f∘g)(x) = (g∘f)(x) is
- A. {1, 2}
- B. {−1, 1}
- C. {1}
- D. {0, 1}
Q18. lim(x→0) (x − sin x)/x³ equals
- A. 0
- B. 1/6
- C. 1/3
- D. 1/2
Q19. The function f(x) = x|x| is
- A. odd
- B. constant
- C. even
- D. neither even nor odd
Q20. Which of the following functions is one-to-one on ℝ?
- A. f(x) = sin x
- B. f(x) = x³ + x
- C. f(x) = x²
- D. f(x) = |x|
Q21. The period of f(x) = sin(2x/3) + cos(x/2) is
- A. 4π
- B. 3π
- C. 6π
- D. 12π
Q22. If sinh x = 3/4, then eˣ equals
- A. 7/4
- B. 3/2
- C. 2
- D. 1/2
Q23. lim(x→0) sin(x²)/(1 − cos x) equals
- A. 2
- B. 1
- C. 1/2
- D. 0
Q24. lim(x→∞) x sin(π/x) equals
- A. 0
- B. π
- C. 1
- D. 2π
Q25. lim(x→∞) ((x + 1)/(x − 1))ˣ equals
- A. e
- B. 1
- C. e⁻²
- D. e²
Q26. Which of the following limits does NOT exist?
- A. lim(x→0) x² sin(1/x)
- B. lim(x→0) x sin(1/x)
- C. lim(x→0) x² cos(1/x)
- D. lim(x→0) sin(1/x)
Q27. Let f(x) = x² − 4x + 3 for x ≥ 2. Then f⁻¹(8) equals
- A. 3
- B. −1
- C. 1
- D. 5
Q28. f(x) = (1 − cos 2x)/x² for x ≠ 0 is continuous at 0 if f(0) equals
- A. 1/2
- B. 4
- C. 2
- D. 1
Q29. The number of real values of x at which f(x) = (x² − 1)/((x² − 4)(x + 1)) is undefined is
- A. 2
- B. 3
- C. 4
- D. 1
Q30. lim(x→0) (e^(x²) − cos x)/x² equals
- A. 2
- B. 3/2
- C. 1/2
- D. 1