ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Functions and Limits (Set 7)

Take the ECAT Mathematics — Functions and Limits (Set 7): 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.

1. Q1. lim (x→0) (eˣ − e⁻ˣ − 2x)/(x − sin x) equals:

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

   Answer: A. 2

   Explanation: eˣ − e⁻ˣ − 2x = 2(x³/6 + …) = x³/3 + …, and x − sin x = x³/6 + …, so the ratio tends to (1/3)/(1/6) = 2.

2. Q2. lim (x→∞) [√(x² + 6x + 1) − x] equals:

   • A. 6
   • B. 0
   • C. 3 ✓
   • D. ∞

   Answer: C. 3

   Explanation: Multiply by the conjugate: (6x + 1)/(√(x² + 6x + 1) + x) → 6/2 = 3.

3. Q3. lim (x→0) (9ˣ − 1)/(27ˣ − 1) equals:

   • A. 2/3 ✓
   • B. 1/3
   • C. 1/9
   • D. 3/2

   Answer: A. 2/3

   Explanation: (aˣ − 1) ~ x ln a, so the limit is ln 9/ln 27 = 2 ln 3/(3 ln 3) = 2/3.

4. Q4. lim (x→0) (1 + x²)^(1/(1 − cos x)) equals:

   • A. e
   • B. e^(1/2)
   • C. e² ✓
   • D. e⁴

   Answer: C. e²

   Explanation: ln L = lim ln(1 + x²)/(1 − cos x) = lim x²/(x²/2) = 2, so L = e².

5. Q5. lim (x→1) sin(πx)/(x − 1) equals:

   • A. 0
   • B. −π ✓
   • C. −1
   • D. π

   Answer: B. −π

   Explanation: Put h = x − 1: sin(π + πh) = −sin(πh), so the quotient → −π.

The other 25 questions

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6. Q6. If f(x) = 1/(1 − x) for x ≠ 1, then f applied 2024 times, f(f(…f(3)…)), equals:

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

7. Q7. The domain of f(x) = √(log₁/₂ (x − 1)) is:

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

8. Q8. The domain of f(x) = √(cos x), restricted to x ∈ [0, 2π], is:

   • A. [π/2, 3π/2]
   • B. [0, π/2]
   • C. [0, π]
   • D. [0, π/2] ∪ [3π/2, 2π]

9. Q9. The range of f(x) = (x² − 1)/(x² + 1), x ∈ ℝ, is:

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

10. Q10. If f(x) = 2ˣ/(2ˣ + 1), the inverse function f⁻¹(y), for 0 < y < 1, is:

    • A. log₂[(1 − y)/y]
    • B. log₂[y/(1 − y)]
    • C. log₂[y/(1 + y)]
    • D. log₂[(1 + y)/y]

11. Q11. The least value of |x − 2| + |x − 5| + |x − 9| over real x is:

    • A. 3
    • B. 8
    • C. 6
    • D. 7

12. Q12. The function f(x) = x(2ˣ − 1)/(2ˣ + 1) is:

    • A. odd
    • B. both even and odd
    • C. even
    • D. neither even nor odd

13. Q13. The period of f(x) = sin²x + cos⁴x is:

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

14. Q14. The function f(x) = (e^(3x) − 1)/sin 2x for x ≠ 0 is made continuous at x = 0 by defining f(0) equal to:

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

15. Q15. If f(x) = ⌊x⌋ + ⌊−x⌋, where ⌊·⌋ is the greatest-integer function, the number of points in the open interval (0, 5) at which f is discontinuous is:

    • A. 1
    • B. 4
    • C. 5
    • D. 0

16. Q16. lim (x→∞) (x + sin x)/(2x − cos x) equals:

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

17. Q17. lim (n→∞) n(2^(1/n) − 1) equals:

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

18. Q18. lim (x→0) (tan x − x)/(x − sin x) equals:

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

19. Q19. lim (x→0) (sin x/x)^(1/x²) equals:

    • A. e^(1/6)
    • B. e^(−1/6)
    • C. e^(−1/3)
    • D. 1

20. Q20. lim (x→0) (sin⁻¹x − tan⁻¹x)/x³ equals:

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

21. Q21. lim (x→0) ln(1 + x²)/(sec x − 1) equals:

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

22. Q22. If f(x) = ln x and g(x) = √(1 − x), the domain of (g∘f)(x) = g(f(x)) is:

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

23. Q23. The function f(x) = (3x + 5)/(x + k) is its own inverse (f(f(x)) = x) when k equals:

    • A. 3
    • B. −5
    • C. 5
    • D. −3

24. Q24. cosh(2 ln 2) equals:

    • A. 15/8
    • B. 4
    • C. 17/8
    • D. 5/4

25. Q25. A function satisfies f(x) + 2f(−x) = x² + 3x for all real x. Then f(3) equals:

    • A. 6
    • B. −18
    • C. 18
    • D. −6

26. Q26. Eliminating t from x = t + 1/t, y = t − 1/t gives:

    • A. x² − y² = 4
    • B. y² − x² = 4
    • C. xy = 4
    • D. x² + y² = 4

27. Q27. The number of real solutions of the equation eˣ = x² is:

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

28. Q28. A linear function f satisfies f(f(x)) = 9x + 8 and f is increasing. Then f(0) equals:

    • A. −4
    • B. 4
    • C. 8/3
    • D. 2

29. Q29. lim (x→2) (x² − 2ˣ)/(xˣ − 4) equals:

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

30. Q30. If f(x) = 2x − 3 and g(x) = x², the sum of all real x satisfying f(g(x)) = g(f(x)) is:

    • A. 12
    • B. 6
    • C. −6
    • D. 3