SmartPrep

ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Functions and Limits (Set 6)

Take the ECAT Mathematics — Functions and Limits (Set 6): 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. The range of f(x) = √(x − 1) + √(5 − x) is:

    • A. [0, 2√2]
    • B. [2, 4]
    • C. [2, 2√2] ✓
    • D. [√2, 2]

    Answer: C. [2, 2√2]

    Explanation: Domain is [1, 5]. f(1) = f(5) = 2 and the maximum occurs at x = 3 where f = 2√2. So the range is [2, 2√2].

  2. Q2. The range of f(x) = x/(1 + |x|), x ∈ ℝ, is:

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

    Answer: A. (−1, 1)

    Explanation: For x ≥ 0, f = x/(1 + x) takes values in [0, 1); for x < 0, f = x/(1 − x) takes values in (−1, 0). The values ±1 are never reached, so the range is (−1, 1).

  3. Q3. If f(2x + 1) = 4x² + 2x for all real x, then f(5) equals:

    • A. 22
    • B. 20 ✓
    • C. 12
    • D. 10

    Answer: B. 20

    Explanation: Put t = 2x + 1, so x = (t − 1)/2 and f(t) = (t − 1)² + (t − 1) = t² − t. Then f(5) = 25 − 5 = 20.

  4. Q4. If f(x) + 2f(1/x) = 3x for all x ≠ 0, then f(2) equals:

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

    Answer: A. −1

    Explanation: Put x = 2: f(2) + 2f(½) = 6. Put x = ½: f(½) + 2f(2) = 3/2. Eliminating f(½): f(2) + 2(3/2 − 2f(2)) = 6 ⇒ −3f(2) = 3 ⇒ f(2) = −1.

  5. Q5. If f(x) = x/(x + 1), then the expression for (f∘f∘f)(x) is:

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

    Answer: C. x/(3x + 1)

    Explanation: f(f(x)) = x/(2x + 1) and f(f(f(x))) = [x/(2x + 1)]/[x/(2x + 1) + 1] = x/(3x + 1).

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. The inverse of f(x) = (3x − 1)/(x + 2) is f⁻¹(x) equal to:

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

  2. Q7. The function f(x) = x² − 4x is restricted to the domain [a, ∞) so that it becomes one-to-one. The smallest possible value of a is:

    • A. 4
    • B. 0
    • C. −2
    • D. 2

  3. Q8. lim (x→0) (1 − cos x cos 2x)/x² equals:

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

  4. Q9. lim (x→∞) x[√(x² + 1) − x] equals:

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

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

    • A. e²
    • B. 1
    • C. e
    • D. e³

  6. Q11. lim (x→π/2) (1 − sin x)/(π/2 − x)² equals:

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

  7. Q12. lim (x→0) (sin x − x cos x)/x³ equals:

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

  8. Q13. The function f(x) = (1 − cos 4x)/x² for x ≠ 0 is made continuous at x = 0 by defining f(0) equal to:

    • A. 2
    • B. 16
    • C. 8
    • D. 4

  9. Q14. The number of points in the open interval (−1.5, 2.5) at which f(x) = ⌊x⌋ (greatest integer function) is discontinuous is:

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

  10. Q15. For f(x) = (x² − x − 6)/(x² − 9), the behaviour at x = 3 and x = −3 respectively is:

    • A. removable, infinite
    • B. infinite, infinite
    • C. infinite, removable
    • D. removable, removable

  11. Q16. Which statement about g(x) = ln(x + √(x² + 1)) is correct?

    • A. It is even only for x > 0
    • B. It is neither even nor odd
    • C. It is even, since g(−x) = g(x)
    • D. It is odd, since g(−x) = −g(x)

  12. Q17. For f(x) = x³ − 3x, how many distinct real values of x satisfy f(x) = f(1)?

    • A. 2
    • B. 3
    • C. 1
    • D. 0

  13. Q18. If cosh x = 5/3 and x > 0, then eˣ equals:

    • A. 3
    • B. 9/4
    • C. 2
    • D. 5

  14. Q19. lim (x→0) (cosh x − 1)/x² equals:

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

  15. Q20. The domain of f(x) = 1/√(|x| − 2) is:

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

  16. Q21. If f(x) = √x and g(x) = x² − 4, the domain of (f∘g)(x) is:

    • A. [2, ∞)
    • B. (−∞, −2] ∪ [2, ∞)
    • C. [0, ∞)
    • D. [−2, 2]

  17. Q22. lim (x→1) (∛x − 1)/(√x − 1) equals:

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

  18. Q23. lim (x→0) x² sin(1/x) equals:

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

  19. Q24. lim (x→2⁻) ⌊x⌋ + lim (x→2⁺) ⌊x⌋ equals:

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

  20. Q25. The domain of f(x) = √(log₂(x − 1)) is:

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

  21. Q26. lim (x→−∞) [√(x² + x) + x] equals:

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

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

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

  23. Q28. lim (x→0) (x − tan⁻¹x)/x³ equals:

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

  24. Q29. lim (x→∞) x²(1 − cos(1/x)) equals:

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

  25. Q30. The period of f(x) = |sin x| + |cos x| is:

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