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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Functions and Limits (Set 5)

Take the ECAT Mathematics — Functions and Limits (Set 5): 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) = (x + 1)/(x − 1) for x ≠ 1, then f(f(f(2))) equals:

    • A. 2
    • B. 3 ✓
    • C. −3
    • D. 1/3

    Answer: B. 3

    Explanation: f(2) = 3, f(3) = 4/2 = 2, f(2) = 3. In general f(f(x)) = x, so f(f(f(x))) = f(x).

  2. Q2. lim (x→0) (cos x − cos 3x)/x² equals:

    • A. 4 ✓
    • B. 8
    • C. 2
    • D. −4

    Answer: A. 4

    Explanation: cos x − cos 3x = 2 sin 2x sin x ≈ 2(2x)(x) = 4x², so the limit is 4. (Or (9 − 1)/2.)

  3. Q3. lim (x→0) (eˣ − 1 − x)/x² equals:

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

    Answer: A. 1/2

    Explanation: eˣ − 1 − x = x²/2 + x³/6 + …, so the ratio tends to 1/2.

  4. Q4. lim (x→0) (tan x − x)/x³ equals:

    • A. 0
    • B. 1/3 ✓
    • C. 1/2
    • D. 1/6

    Answer: B. 1/3

    Explanation: tan x = x + x³/3 + …, so (tan x − x)/x³ → 1/3.

  5. Q5. lim (x→∞) x(√(x² + 1) − x) equals:

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

    Answer: C. 1/2

    Explanation: Multiply by the conjugate: x/(√(x²+1) + x) → x/(2x) = 1/2.

The other 25 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

  1. Q6. lim (x→π/4) (sin x − cos x)/(x − π/4) equals:

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

  2. Q7. lim (x→0) (1 + sin x)^(1/x) equals:

    • A. 1
    • B. e
    • C. 1/e
    • D. e²

  3. Q8. lim (x→∞) (x/(x + 3))ˣ equals:

    • A. 0
    • B. 1
    • C. e³
    • D. e⁻³

  4. Q9. lim (x→0) (cos x)^(1/x²) equals:

    • A. 1
    • B. e^(−1/2)
    • C. e^(1/2)
    • D. e⁻¹

  5. Q10. lim (x→2) [1/(x − 2) − 4/(x² − 4)] equals:

    • A. 0
    • B. does not exist
    • C. 1/2
    • D. 1/4

  6. Q11. The domain of f(x) = 1/√(x² − x − 2) is:

    • A. (−∞, −1) ∪ (2, ∞)
    • B. (−∞, −2) ∪ (1, ∞)
    • C. (−∞, −1] ∪ [2, ∞)
    • D. (−1, 2)

  7. Q12. The range of f(x) = x²/(1 + x²), x ∈ ℝ, is:

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

  8. Q13. If f(x) = e^(2x − 1), then f⁻¹(e³) equals:

    • A. (1 + e³)/2
    • B. 2
    • C. 1
    • D. 3/2

  9. Q14. f(x) = x² + 4x is restricted to x ≤ −2. Then f⁻¹(12) equals:

    • A. −4
    • B. −6
    • C. −2
    • D. 2

  10. Q15. f(x) = ⌊x⌋ is the greatest-integer function. On the interval [0, 3), f is discontinuous at exactly how many points?

    • A. 4
    • B. 1
    • C. 2
    • D. 3

  11. Q16. f(x) = 3 for x ≤ 1, f(x) = ax + b for 1 < x < 3 and f(x) = 11 for x ≥ 3. If f is continuous everywhere, ab equals:

    • A. 12
    • B. 4
    • C. −4
    • D. −12

  12. Q17. lim (x→0) x·⌊1/x⌋ equals:

    • A. does not exist
    • B. 0
    • C. 1
    • D. −1

  13. Q18. If f(x) = (3x + 2)/(x − 3) for x ≠ 3, then f⁻¹(x) is:

    • A. (3x − 2)/(x + 3)
    • B. (3x + 2)/(x − 3)
    • C. (x − 3)/(3x + 2)
    • D. (3x + 3)/(x − 2)

  14. Q19. tanh(ln 3) equals:

    • A. 3/5
    • B. 4/5
    • C. 2/3
    • D. 5/4

  15. Q20. The parametric equations x = 3 cosh t, y = 4 sinh t represent:

    • A. an ellipse with semi-axes 3 and 4
    • B. a circle of radius 5
    • C. both branches of the hyperbola x²/9 − y²/16 = 1
    • D. the right branch of the hyperbola x²/9 − y²/16 = 1

  16. Q21. If f(x) = 2x + 3, the number of real x with f(f(x)) = x is:

    • A. none
    • B. exactly one
    • C. infinitely many
    • D. two

  17. Q22. f(x) = 1/(x − 1) and g(x) = √x. The domain of f∘g is:

    • A. x ≠ 1
    • B. x > 1
    • C. x ≥ 0, x ≠ 1
    • D. x ≥ 0

  18. Q23. The even part of f(x) = eˣ, i.e. [f(x) + f(−x)]/2, is:

    • A. cosh x
    • B. e²ˣ
    • C. eˣ
    • D. sinh x

  19. Q24. Which of the following functions is NOT periodic?

    • A. tan 3x
    • B. sin x cos x
    • C. sin(x²)
    • D. |cos x|

  20. Q25. The number of onto functions from {1, 2, 3} onto {a, b} is:

    • A. 8
    • B. 9
    • C. 2
    • D. 6

  21. Q26. lim (x→∞) (2ˣ + 3ˣ)^(1/x) equals:

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

  22. Q27. lim (x→0) [(1 + x)^(1/x) − e]/x equals:

    • A. e/2
    • B. −e
    • C. −e/2
    • D. 0

  23. Q28. lim (x→0) (sin x / x)^(1/x²) equals:

    • A. 1
    • B. e^(1/6)
    • C. e⁻¹
    • D. e^(−1/6)

  24. Q29. f(x) = (x² − 1)/(x − 1) at x = 1 has:

    • A. a removable discontinuity
    • B. an infinite discontinuity
    • C. a jump discontinuity
    • D. no discontinuity, since f(1) is defined

  25. Q30. lim (x→0) ln(1 + x²)/sin²x equals:

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