ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Trigonometry (Set 2)

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

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. The period of the function f(x) = 3 cos(x/3) is:

   • A. 2π/3
   • B. 2π
   • C. 6π ✓
   • D. 3π

   Answer: C. 6π

   Explanation: cos(bx) has period 2π/b; here b = 1/3, so the period is 2π ÷ (1/3) = 6π. The factor 3 only changes the amplitude.

2. Q2. The period of the function f(x) = cos 2x + sin 3x is:

   • A. 2π/3
   • B. 2π ✓
   • C. π
   • D. 6π

   Answer: B. 2π

   Explanation: cos 2x has period π and sin 3x has period 2π/3; the smallest common period is the L.C.M. of π and 2π/3, which is 2π.

3. Q3. Using a half-angle formula, the exact value of cos 22.5° is:

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

   Answer: C. √(2 + √2)/2

   Explanation: cos²(22.5°) = (1 + cos 45°)/2 = (1 + √2/2)/2 = (2 + √2)/4, and cos 22.5° > 0, so cos 22.5° = √(2 + √2)/2.

4. Q4. The exact value of tan 15° is:

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

   Answer: D. 2 − √3

   Explanation: tan(45° − 30°) = (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1) = (√3 − 1)²/2 = 2 − √3.

5. Q5. An arc of a circle of radius 10 cm subtends an angle of 1.5 radians at the centre. The length of the arc is:

   • A. 6.67 cm
   • B. 75 cm
   • C. 1.5 cm
   • D. 15 cm ✓

   Answer: D. 15 cm

   Explanation: l = rθ = 10 × 1.5 = 15 cm.

The other 25 questions

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6. Q6. A vertical pole of height 6√3 m casts a shadow 6 m long on level ground. The angle of elevation of the sun is:

   • A. 60°
   • B. 75°
   • C. 45°
   • D. 30°

7. Q7. In triangle ABC, a = 10 and ∠A = 30°. The circum-radius R of the triangle is:

   • A. 5
   • B. 20
   • C. 10
   • D. 10√3

8. Q8. If cos θ = 1/3, then cos 2θ equals:

   • A. −7/9
   • B. −1/9
   • C. 2/3
   • D. 7/9

9. Q9. The value of tan(sin⁻¹(3/5)) is:

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

10. Q10. The minimum value of 5 sin x − 12 cos x is:

    • A. −13
    • B. −17
    • C. −7
    • D. −12

11. Q11. In triangle ABC, b = 3, c = 5 and ∠A = 60°. The side a is:

    • A. √19
    • B. 4
    • C. √34
    • D. 7

12. Q12. The value of cos 300° is:

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

13. Q13. The exact value of cos 105° is:

    • A. (√6 + √2)/4
    • B. (√6 − √2)/4
    • C. −(√6 + √2)/4
    • D. (√2 − √6)/4

14. Q14. An angle of π/12 radians is equal to:

    • A. 12°
    • B. 7.5°
    • C. 15°
    • D. 30°

15. Q15. For −1 ≤ x ≤ 1, the value of sin⁻¹x + cos⁻¹x is:

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

16. Q16. The principal value of cos⁻¹(−1/2) is:

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

17. Q17. In triangle ABC, a = 6, b = 6√3 and ∠A = 30°. How many triangles satisfy these data?

    • A. 1
    • B. infinitely many
    • C. 2
    • D. 0

18. Q18. The solutions of sin 2x = sin x in the interval [0, π] are:

    • A. π/3 only
    • B. 0 and π only
    • C. 0, π/6 and π
    • D. 0, π/3 and π

19. Q19. The triple-angle formula for cos 3θ is:

    • A. 3cos θ − 4cos³θ
    • B. 4cos³θ + 3cos θ
    • C. 3cos³θ − 4cos θ
    • D. 4cos³θ − 3cos θ

20. Q20. The range of the principal value function y = cos⁻¹x is:

    • A. (−π/2, π/2)
    • B. [0, π]
    • C. [−π/2, π/2]
    • D. (0, π)

21. Q21. Using Hero's formula, the area of a triangle with sides 7, 8 and 9 is:

    • A. 12√3
    • B. 6√5
    • C. 24√5
    • D. 12√5

22. Q22. If sec θ = −2 and sin θ > 0, then tan θ equals:

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

23. Q23. The general solution of cos x = 1/2 is (n ∈ ℤ):

    • A. x = nπ ± π/3
    • B. x = 2nπ ± π/6
    • C. x = 2nπ ± π/3
    • D. x = nπ + (−1)ⁿπ/3

24. Q24. The product 2 sin 3x cos x is equal to:

    • A. sin 4x − sin 2x
    • B. cos 4x + cos 2x
    • C. sin 3x + sin x
    • D. sin 4x + sin 2x

25. Q25. The expression (1 − cos²θ)(1 + cot²θ) simplifies to:

    • A. 1
    • B. cos²θ
    • C. sin²θ
    • D. tan²θ

26. Q26. If sin θ + cos θ = 7/5, then sin θ cos θ equals:

    • A. 49/50
    • B. 24/25
    • C. 6/25
    • D. 12/25

27. Q27. The value of cot(−135°) is:

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

28. Q28. The value of tan⁻¹1 + tan⁻¹2 + tan⁻¹3 is:

    • A. π
    • B. 3π/4
    • C. π/2
    • D. 0

29. Q29. The expression sin(A + B) sin(A − B) is equal to:

    • A. sin²A − cos²B
    • B. cos²A − sin²B
    • C. sin²A + sin²B
    • D. sin²A − sin²B

30. Q30. The value of cos 75° + cos 15° is:

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