ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Functions and Limits (Set 3)

Take the ECAT Mathematics — Functions and Limits (Set 3): 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) = x² − 2x and g(x) = x + 3, then (f∘g)(1) equals:
- A. 3
- B. −4
- C. 2
- D. 8 ✓
Answer: D. 8
Explanation: g(1) = 4, so f(g(1)) = f(4) = 16 − 8 = 8.
Q2. The domain of f(x) = ln(x² − 4) is:
- A. (−2, 2)
- B. (−∞, −2) ∪ (2, ∞) ✓
- C. [−2, 2]
- D. ℝ ∖ {−2, 2}
Answer: B. (−∞, −2) ∪ (2, ∞)
Explanation: The logarithm needs x² − 4 > 0, i.e. |x| > 2, so x < −2 or x > 2.
Q3. The range of f(x) = 1/(x² + 1), x ∈ ℝ, is:
- A. (0, 1)
- B. (0, 1] ✓
- C. [0, 1]
- D. (−∞, 1]
Answer: B. (0, 1]
Explanation: x² + 1 takes every value in [1, ∞), so its reciprocal takes every value in (0, 1]; the value 1 is reached at x = 0 and 0 is never reached.
Q4. If f(x) = (2x + 1)/(x − 3), then f⁻¹(4) equals:
- A. 2/13
- B. 9
- C. 7/2
- D. 13/2 ✓
Answer: D. 13/2
Explanation: Solving y = (2x + 1)/(x − 3) gives x = (3y + 1)/(y − 2). So f⁻¹(4) = 13/2. (Note f(4) = 9 is not the inverse value.)
Q5. The range of f(x) = 2 + 3 sin x is:
- A. [−1, 5] ✓
- B. [−1, 3]
- C. [2, 5]
- D. [−3, 3]
Answer: A. [−1, 5]
Explanation: sin x takes all values in [−1, 1], so 3 sin x covers [−3, 3] and adding 2 gives [−1, 5].

The other 25 questions

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Q6. lim (x→0) sin 4x / tan 2x equals:
- A. 4
- B. 2
- C. 1/2
- D. 8
Q7. lim (x→0) (1 − cos x)/x equals:
- A. 0
- B. 1/2
- C. does not exist
- D. 1
Q8. lim (x→∞) (1 + 5/x)ˣ equals:
- A. 5e
- B. e
- C. e^(1/5)
- D. e⁵
Q9. lim (x→0) (1 − 2x)^(1/x) equals:
- A. e⁻²
- B. −2e
- C. 1/2
- D. e²
Q10. lim (x→∞) (4x³ − 2x)/(2x³ + 5x²) equals:
- A. 0
- B. 4
- C. ∞
- D. 2
Q11. lim (x→2) [√(x + 7) − 3]/(x − 2) equals:
- A. 1/3
- B. 3
- C. 1/6
- D. 0
Q12. lim (x→0) (e^(3x) − e^(−3x))/x equals:
- A. 6
- B. 0
- C. 3
- D. 1
Q13. lim (x→∞) x·sin(1/x) equals:
- A. ∞
- B. 0
- C. 1
- D. does not exist
Q14. lim (x→0) sin²(3x)/x² equals:
- A. 3
- B. 9
- C. 1
- D. 6
Q15. f(x) = (x² − x − 6)/(x − 3) for x ≠ 3 and f(3) = k. For f to be continuous at x = 3, k must equal:
- A. 3
- B. 6
- C. 5
- D. −5
Q16. f(x) = ax + 2 for x < 1 and f(x) = 3x − a for x ≥ 1. The value of a that makes f continuous at x = 1 is:
- A. −1/2
- B. 1
- C. 2
- D. 1/2
Q17. Let ⌊x⌋ denote the greatest integer not exceeding x. The left-hand limit of ⌊x⌋ as x → 3 is:
- A. 2.5
- B. does not exist
- C. 2
- D. 3
Q18. The function f(x) = x·|x| is:
- A. neither even nor odd
- B. both even and odd
- C. odd
- D. even
Q19. If f is an odd function and g is an even function, then the product h(x) = f(x)·g(x) is:
- A. even
- B. odd
- C. constant
- D. neither odd nor even always
Q20. The period of f(x) = cos 3x is:
- A. 2π/3
- B. π/3
- C. 6π
- D. 2π
Q21. The value of sinh(ln 2) is:
- A. 1/4
- B. 3/4
- C. 3/2
- D. 5/4
Q22. The graph of the inverse function f⁻¹ is the reflection of the graph of f in the line:
- A. x = 0
- B. y = 0
- C. y = −x
- D. y = x
Q23. The parametric equations x = 1 + t, y = 2 − t represent:
- A. the line x + y = 3
- B. a parabola y = 2x
- C. a circle x² + y² = 3
- D. the line x − y = 3
Q24. If f(x) = 2x − 1 and (f∘g)(x) = 4x + 3, then g(x) equals:
- A. 4x + 4
- B. 2x + 1
- C. 2x + 4
- D. 2x + 2
Q25. lim (x→0) [(1 + x)^(1/3) − 1]/x equals:
- A. 1
- B. 0
- C. 1/3
- D. 3
Q26. lim (x→0) (x cos x − sin x)/x³ equals:
- A. 1/3
- B. −1/3
- C. −1/2
- D. 0
Q27. lim (x→∞) [√(x² + 4x) − x] equals:
- A. ∞
- B. 4
- C. 2
- D. 1
Q28. lim (x→0) sin(x²)/x equals:
- A. 0
- B. 2
- C. ∞
- D. 1
Q29. The domain of f(x) = √(x² − 5x + 6) is:
- A. [2, 3]
- B. (−∞, 2) ∪ (3, ∞)
- C. (−∞, 2] ∪ [3, ∞)
- D. ℝ
Q30. If f(x) = (x − 1)/(x + 1), then f(f(x)) equals:
- A. −1/x
- B. x
- C. 1/x
- D. −x