ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Trigonometry (Set 7)

Take the ECAT Mathematics — Trigonometry (Set 7): 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. At 5:10 the smaller angle between the hour hand and the minute hand of a clock is

   • A. 105°
   • B. 65°
   • C. 85°
   • D. 95° ✓

   Answer: D. 95°

   Explanation: At 5:10 the hour hand has moved 5·30° + 10·0.5° = 155° from 12 and the minute hand is at 10·6° = 60°. The difference is 95°, which is less than 180°.

2. Q2. A wheel of radius 35 cm rolls without slipping along a straight road for 220 m. Taking π = 22/7, the number of complete turns it makes is

   • A. 200
   • B. 100 ✓
   • C. 50
   • D. 110

   Answer: B. 100

   Explanation: The circumference is 2·(22/7)·35 = 220 cm. The distance 220 m = 22000 cm, so the number of turns is 22000/220 = 100.

3. Q3. If A + B = 45°, the value of (1 + tan A)(1 + tan B) is

   • A. 1
   • B. √2
   • C. 4
   • D. 2 ✓

   Answer: D. 2

   Explanation: tan 45° = 1 = (tan A + tan B)/(1 − tan A tan B), so tan A + tan B = 1 − tan A tan B. Then (1 + tan A)(1 + tan B) = 1 + (tan A + tan B) + tan A tan B = 1 + 1 = 2.

4. Q4. The value of sin 10° + sin 50° − sin 70° is

   • A. 0 ✓
   • B. ½
   • C. −½
   • D. 1

   Answer: A. 0

   Explanation: sin 50° − sin 70° = 2cos 60° sin(−10°) = −sin 10°. So the expression equals sin 10° − sin 10° = 0.

5. Q5. The value of cos 20° + cos 100° + cos 140° is

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

   Answer: B. 0

   Explanation: cos 100° + cos 140° = 2cos 120° cos 20° = −cos 20°. Adding cos 20° gives 0.

The other 25 questions

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

6. Q6. In a triangle with sides 4, 5 and 6, the cosine of the smallest angle is

   • A. 9/16
   • B. 5/8
   • C. 3/4
   • D. 1/8

7. Q7. In any triangle ABC, the expression b cos C + c cos B equals

   • A. b + c
   • B. a
   • C. 2a
   • D. a/2

8. Q8. In triangle ABC, a = √3, b = 1 and ∠A = 60°. Then ∠B equals

   • A. 30°
   • B. 45°
   • C. 60°
   • D. 150°

9. Q9. For the right-angled triangle with sides 5, 12 and 13, the ratio circum-radius : in-radius is

   • A. 5 : 2
   • B. 13 : 2
   • C. 13 : 4
   • D. 6 : 1

10. Q10. From the top of a 100 m lighthouse the angles of depression of two boats in line with its foot, on the same side, are 30° and 45°. The distance between the boats is

    • A. 100(√3 − 1) m
    • B. 50(√3 − 1) m
    • C. 100√3 m
    • D. 100(√3 + 1) m

11. Q11. The value of sin(2 cos⁻¹(−3/5)) is

    • A. 24/25
    • B. −7/25
    • C. 7/25
    • D. −24/25

12. Q12. The principal value of cos⁻¹(cos(−5π/3)) is

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

13. Q13. The sum of all solutions of sin x + sin 3x = 0 in the interval [0, π] is

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

14. Q14. The number of solutions of tan x + cot x = 2 in the interval [0, 2π) is

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

15. Q15. The sum of all solutions of √3 cos x − sin x = 1 in [0, 2π) is

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

16. Q16. The greatest value attained by g(x) = sin x + sin(x + 2π/3) over all real x is

    • A. ½
    • B. √3
    • C. 2
    • D. 1

17. Q17. The smallest positive T with f(x + T) = f(x) for all x, where f(x) = sin 2x · cos 3x, is

    • A. π
    • B. 2π/5
    • C. π/2
    • D. 2π

18. Q18. The range of f(x) = cos²x − sin x is

    • A. [−5/4, 1]
    • B. [−1, 5/4]
    • C. [−1, 3/2]
    • D. [−1, 1]

19. Q19. The value of cos 2010° is

    • A. ½
    • B. √3/2
    • C. −½
    • D. −√3/2

20. Q20. A is acute with sin A = 4/5, and B lies in the second quadrant with cos B = −5/13. Then sin(A + B) equals

    • A. 56/65
    • B. 16/65
    • C. −56/65
    • D. −16/65

21. Q21. If tan θ = 3/4, then cos 4θ equals

    • A. 527/625
    • B. 7/25
    • C. −527/625
    • D. −24/25

22. Q22. The minimum value of 9 tan²x + 4 cot²x is

    • A. 6
    • B. 36
    • C. 13
    • D. 12

23. Q23. The value of sin²10° + sin²20° + sin²30° + … + sin²80° is

    • A. 8
    • B. 4.5
    • C. 4
    • D. 3

24. Q24. The value of tan 1° · tan 2° · tan 3° · … · tan 89° is

    • A. 1
    • B. ½
    • C. 0
    • D. 89

25. Q25. In triangle ABC, sin A = 2 sin B cos C. The triangle must be

    • A. Isosceles with A = B
    • B. Right-angled at A
    • C. Isosceles with B = C
    • D. Equilateral

26. Q26. A triangle has sides 2, 3 and 4. The triangle is

    • A. Equilateral
    • B. Acute-angled
    • C. Obtuse-angled
    • D. Right-angled

27. Q27. If tan θ + cot θ = 3, then tan³θ + cot³θ equals

    • A. 9
    • B. 18
    • C. 27
    • D. 24

28. Q28. If cos θ + sec θ = 5/2, then cos²θ + sec²θ equals

    • A. 9/4
    • B. 21/4
    • C. 25/4
    • D. 17/4

29. Q29. If tan θ = 1/3, then tan 3θ equals

    • A. 13/9
    • B. 1
    • C. 13/27
    • D. 9/13

30. Q30. In triangle ABC, b = 6 and ∠B = 30°. The circum-radius R of the triangle is

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