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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Functions and Limits (Set 2)

Take the ECAT Mathematics — Functions and Limits (Set 2): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. If f(x) = 3x − 2 and g(x) = x + 4, then (g∘f)(x) equals:

    • A. 3x² + 10x − 8
    • B. 3x + 10
    • C. 3x − 6
    • D. 3x + 2 ✓

    Answer: D. 3x + 2

    Explanation: (g∘f)(x) = g(3x − 2) = (3x − 2) + 4 = 3x + 2. (3x + 10 is f∘g.)

  2. Q2. The domain of f(x) = 1/(x² − 4) is:

    • A. ℝ − {2}
    • B. ℝ − {−2, 2} ✓
    • C. ℝ − {4}
    • D. (−2, 2)

    Answer: B. ℝ − {−2, 2}

    Explanation: f is undefined where x² − 4 = 0, i.e. x = ±2; every other real number is allowed.

  3. Q3. The range of f(x) = |x| + 1, x ∈ ℝ, is:

    • A. (1, ∞)
    • B. [1, ∞) ✓
    • C. [0, ∞)
    • D. ℝ

    Answer: B. [1, ∞)

    Explanation: |x| ≥ 0 takes every non-negative value, so |x| + 1 takes every value ≥ 1.

  4. Q4. The inverse of f(x) = x³ + 2 is:

    • A. (x − 2)³
    • B. ∛x − 2
    • C. ∛(x − 2) ✓
    • D. 1/(x³ + 2)

    Answer: C. ∛(x − 2)

    Explanation: Put y = x³ + 2, so x = ∛(y − 2); hence f⁻¹(x) = ∛(x − 2).

  5. Q5. lim (x→0) sin 3x / sin 7x equals:

    • A. 1
    • B. 0
    • C. 7/3
    • D. 3/7 ✓

    Answer: D. 3/7

    Explanation: sin 3x/sin 7x = (sin 3x/3x)(7x/sin 7x)(3/7) → 1·1·3/7 = 3/7.

The other 25 questions

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  1. Q6. lim (x→0) (1 − cos 2x)/x² equals:

    • A. 4
    • B. 2
    • C. 1/2
    • D. 1

  2. Q7. lim (x→∞) (1 + 1/(3x))ˣ equals:

    • A. 3e
    • B. e^(1/3)
    • C. e³
    • D. 1

  3. Q8. lim (x→3) (x² − 5x + 6)/(x − 3) equals:

    • A. −1
    • B. 1
    • C. 0
    • D. 5

  4. Q9. lim (x→∞) (2x³ − x)/(4x² + 1):

    • A. equals 2
    • B. equals 0
    • C. equals 1/2
    • D. does not exist as a finite number (→ ∞)

  5. Q10. lim (x→0) (e^(4x) − 1)/x equals:

    • A. e⁴
    • B. 4
    • C. 1/4
    • D. 1

  6. Q11. lim (x→0) ln(1 + 2x)/x equals:

    • A. 0
    • B. ln 2
    • C. 2
    • D. 1/2

  7. Q12. lim (x→3) (x⁴ − 81)/(x − 3) equals:

    • A. 108
    • B. 81
    • C. 27
    • D. 36

  8. Q13. For f(x) = |x − 2|/(x − 2), the left-hand limit at x = 2 is:

    • A. 1
    • B. 0
    • C. −1
    • D. 2

  9. Q14. f(x) = kx + 1 for x ≤ 2 and f(x) = x² + 1 for x > 2. For what value of k is f continuous at x = 2?

    • A. 3/2
    • B. 1
    • C. 5/2
    • D. 2

  10. Q15. The hyperbolic sine function sinh x is defined as:

    • A. (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)
    • B. (eˣ + e⁻ˣ)/2
    • C. (eˣ − e⁻ˣ)/2
    • D. 2/(eˣ − e⁻ˣ)

  11. Q16. Which of the following functions is neither even nor odd?

    • A. x² + cos x
    • B. x³ − x
    • C. x sin x
    • D. x + x²

  12. Q17. If f(x) = 2x + 1, then f⁻¹(5) equals:

    • A. 2
    • B. 1/11
    • C. 11
    • D. 3

  13. Q18. lim (x→0) [√(x + 4) − 2]/x equals:

    • A. 1/4
    • B. 1/2
    • C. 4
    • D. 0

  14. Q19. lim (x→∞) (2x + 3)/√(x² + 1) equals:

    • A. 2
    • B. ∞
    • C. 1
    • D. 3

  15. Q20. lim (x→0) (1 − cos x)/(x sin x) equals:

    • A. 0
    • B. 1/2
    • C. 2
    • D. 1

  16. Q21. lim (x→0) (6ˣ − 2ˣ)/x equals:

    • A. ln 12
    • B. ln 3
    • C. 4
    • D. ln 4

  17. Q22. lim (x→∞) (1 + 3/x)^(2x) equals:

    • A. e^(3/2)
    • B. e⁵
    • C. e^(2/3)
    • D. e⁶

  18. Q23. lim (x→1) (x − 1)/ln x equals:

    • A. 1
    • B. 0
    • C. ∞
    • D. e

  19. Q24. Which of the following defines y as an implicit function of x?

    • A. y = sin x + cos x
    • B. x² + xy + y² = 7
    • C. y = x² + 3
    • D. y = √(x + 1)

  20. Q25. If f(x) = x² and g(x) = sin x, then (f∘g)(π/6) equals:

    • A. sin(π²/36)
    • B. 1/2
    • C. π²/36
    • D. 1/4

  21. Q26. If f(x) = (x + 1)/(x − 1), then for x ≠ 0, ±1, f(1/x) equals:

    • A. 1/f(x)
    • B. −1/f(x)
    • C. f(x)
    • D. −f(x)

  22. Q27. A linear function f(x) = ax + b satisfies f(1) = 5 and f(3) = 11. Then f(5) equals:

    • A. 15
    • B. 16
    • C. 18
    • D. 17

  23. Q28. lim (x→−1) (x³ + 1)/(x + 1) equals:

    • A. 3
    • B. 1
    • C. 0
    • D. −3

  24. Q29. The graph of y = |x − 2| + 1 is a V-shape with its vertex at:

    • A. (−2, 1)
    • B. (2, −1)
    • C. (2, 1)
    • D. (1, 2)

  25. Q30. lim (x→0) (tan x − sin x)/x³ equals:

    • A. 0
    • B. 1
    • C. 1/2
    • D. 1/6