ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Functions and Limits (Set 1)

Take the ECAT Mathematics — Functions and Limits (Set 1): 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) = 2x + 3 and g(x) = x², what is (f∘g)(2)?
- A. 49
- B. 19
- C. 7
- D. 11 ✓
Answer: D. 11
Explanation: (f∘g)(2) = f(g(2)) = f(4) = 2(4) + 3 = 11. (49 is g(f(2)), the reverse composition.)
Q2. What is the domain of f(x) = √(x − 3)?
- A. (−∞, 3]
- B. All real numbers
- C. [3, ∞) ✓
- D. (3, ∞)
Answer: C. [3, ∞)
Explanation: The expression under the square root must be non-negative: x − 3 ≥ 0, so x ≥ 3, i.e. [3, ∞).
Q3. The inverse of f(x) = (2x − 1)/3 is:
- A. f⁻¹(x) = (2x + 1)/3
- B. f⁻¹(x) = 3/(2x − 1)
- C. f⁻¹(x) = (3x − 1)/2
- D. f⁻¹(x) = (3x + 1)/2 ✓
Answer: D. f⁻¹(x) = (3x + 1)/2
Explanation: Put y = (2x − 1)/3, so 3y = 2x − 1 and x = (3y + 1)/2. Swapping letters gives f⁻¹(x) = (3x + 1)/2.
Q4. Which of the following functions is even?
- A. f(x) = x³ + x
- B. f(x) = x² + cos x ✓
- C. f(x) = sin x + x²
- D. f(x) = eˣ
Answer: B. f(x) = x² + cos x
Explanation: x² and cos x are both unchanged when x is replaced by −x, so f(−x) = f(x). The others fail this test.
Q5. lim (x→0) sin 5x / x equals:
- A. 1/5
- B. 1
- C. 5 ✓
- D. 0
Answer: C. 5
Explanation: sin 5x / x = 5 · (sin 5x)/(5x) → 5 × 1 = 5.

The other 25 questions

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Q6. lim (x→0) (1 − cos x)/x² equals:
- A. 0
- B. 1/2
- C. 2
- D. 1
Q7. lim (x→∞) (1 + 2/x)ˣ equals:
- A. √e
- B. e
- C. 2e
- D. e²
Q8. lim (x→2) (x² − 4)/(x − 2) equals:
- A. Does not exist
- B. 2
- C. 4
- D. 0
Q9. lim (x→∞) (3x² + 2x)/(5x² − 1) equals:
- A. 0
- B. ∞
- C. 3/5
- D. 2/5
Q10. lim (x→0) (2ˣ − 1)/x equals:
- A. 2
- B. ln 2
- C. log₂ e
- D. 1
Q11. lim (x→0) (1 + 3x)^(1/x) equals:
- A. e^(1/3)
- B. e³
- C. 1
- D. 3e
Q12. lim (x→0) tan 2x / sin 3x equals:
- A. 2/3
- B. 1
- C. 3/2
- D. 6
Q13. For f(x) = |x|/x, the right-hand limit at x = 0 is:
- A. Undefined because f(0) is undefined
- B. 1
- C. 0
- D. −1
Q14. f(x) = (x² − 9)/(x − 3) for x ≠ 3 and f(3) = k. For what value of k is f continuous at x = 3?
- A. 0
- B. 9
- C. 6
- D. 3
Q15. The range of f(x) = 3 − x², x ∈ ℝ, is:
- A. (−∞, 3)
- B. [3, ∞)
- C. (−∞, 3]
- D. All real numbers
Q16. For the hyperbolic functions, cosh²x − sinh²x equals:
- A. cosh 2x
- B. 1
- C. −1
- D. 0
Q17. Which of the following functions is odd?
- A. f(x) = x³ − sin x
- B. f(x) = x² + sin x
- C. f(x) = |x| + x
- D. f(x) = x³ + cos x
Q18. If f(x) = x² − 1 and g(x) = √x, the domain of g∘f is:
- A. −1 ≤ x ≤ 1
- B. x ≥ 1 only
- C. x ≥ 0
- D. x ≤ −1 or x ≥ 1
Q19. lim (x→∞) [√(x² + x) − x] equals:
- A. 1
- B. 1/2
- C. ∞
- D. 0
Q20. If x is measured in degrees, lim (x→0) sin x / x equals:
- A. 1
- B. 0
- C. 180/π
- D. π/180
Q21. lim (x→1) (x³ − 1)/(x − 1) equals:
- A. 3
- B. 2
- C. 1
- D. 0
Q22. lim (x→2) (x⁵ − 32)/(x − 2) equals:
- A. 32
- B. 80
- C. 16
- D. 40
Q23. If f(x) = 1/(1 − x), then f(f(x)) for x ≠ 0, 1 equals:
- A. x
- B. 1 − x
- C. x/(x − 1)
- D. (x − 1)/x
Q24. If f(x) = x² + x, then f(x + 1) − f(x) equals:
- A. 2x + 1
- B. 2x + 2
- C. x + 2
- D. 1
Q25. lim (x→0) [√(1 + x) − 1]/x equals:
- A. 0
- B. 1/2
- C. 1
- D. 2
Q26. lim (x→∞) (1 − 1/x)^(2x) equals:
- A. e⁻²
- B. e²
- C. e⁻¹
- D. 1
Q27. The parametric equations x = at², y = 2at (a > 0) represent which curve?
- A. The parabola y² = 4ax
- B. The line y = 2x
- C. The circle x² + y² = a²
- D. The parabola x² = 4ay
Q28. If f(x) = eˣ, the domain of f⁻¹ is:
- A. (1, ∞)
- B. All real numbers
- C. (0, ∞)
- D. [0, ∞)
Q29. lim (x→0) x·sin(1/x) equals:
- A. 1
- B. ∞
- C. 0
- D. Does not exist
Q30. lim (x→π/2) cos x / (x − π/2) equals:
- A. 0
- B. π/2
- C. 1
- D. −1