ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Trigonometry (Set 6): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30 ECAT…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. The exact value of cos 36° − cos 72° is:
Answer: C. 1/2
Explanation: cos 36° = (√5 + 1)/4 and cos 72° = (√5 − 1)/4, so the difference is 2/4 = 1/2.
Q2. The value of sin 10° · sin 30° · sin 50° · sin 70° is:
Answer: B. 1/16
Explanation: Using sin x sin(60° − x) sin(60° + x) = ¼ sin 3x with x = 10°: sin 10° sin 50° sin 70° = ¼ sin 30° = 1/8. Multiplying by sin 30° = ½ gives 1/16.
Q3. The value of sin 20° · sin 40° · sin 60° · sin 80° is:
Answer: B. 3/16
Explanation: Using sin x sin(60° − x) sin(60° + x) = ¼ sin 3x with x = 20°: sin 20° sin 40° sin 80° = ¼ sin 60° = √3/8. Multiplying by sin 60° = √3/2 gives 3/16.
Q4. The value of tan 20° + tan 40° + √3 tan 20° tan 40° is:
Answer: A. √3
Explanation: tan(20° + 40°) = √3 gives (tan 20° + tan 40°)/(1 − tan 20° tan 40°) = √3, so tan 20° + tan 40° = √3 − √3 tan 20° tan 40°. The required sum is √3.
Q5. In triangle ABC with a = 7, b = 8 and c = 9, cos A equals:
Answer: A. 2/3
Explanation: cos A = (b² + c² − a²)/(2bc) = (64 + 81 − 49)/(2 · 8 · 9) = 96/144 = 2/3.
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Q6. The circum-radius of the triangle with sides 13, 14 and 15 is:
Q7. The value of tan(sin⁻¹(3/5) + sin⁻¹(5/13)) is:
Q8. The value of 2 tan⁻¹(1/3) + tan⁻¹(1/7) is:
Q9. The number of solutions of sin x + cos x = 1 in the interval [0, 2π) is:
Q10. The sum of all solutions of 2 sin²x = 3 cos x in the interval [0, 2π) is:
Q11. The number of solutions of sin 3x = 1/2 in the interval [0, π] is:
Q12. The maximum value of 3 sin²x + 8 sin x cos x − 3 cos²x is:
Q13. The fundamental period of f(x) = |sin x| + |cos x| is:
Q14. The graph of y = 2 sin(3x − π) is the graph of y = 2 sin 3x shifted:
Q15. A sector of a circle of radius 5 cm has a perimeter of 20 cm. Its area is:
Q16. A wheel of radius 0.5 m turns at 120 revolutions per minute. The speed of a point on its rim is:
Q17. From a point A on level ground the angle of elevation of the top of a tower is 30°. From a point B 20 m nearer to the tower, on the same line, the elevation is 45°. The height of the tower is:
Q18. In triangle ABC, A = 45°, B = 60° and a = √2. The side b equals:
Q19. The largest angle of the triangle with sides 3, 5 and 7 is:
Q20. The area of a triangle with sides 6 and 8 enclosing an angle of 30° is:
Q21. The value of cos 36° · cos 72° is:
Q22. If sec θ + tan θ = 2, then sin θ equals:
Q23. The value of cos 75° − cos 15° is:
Q24. For an acute angle θ with tan θ = 3/4, the value of sin 3θ is:
Q25. The value of tan 75° − tan 15° is:
Q26. The value of sin(2 cos⁻¹(3/5)) is:
Q27. Using principal values, cos⁻¹(−1/2) − sin⁻¹(−1/2) equals:
Q28. If tan θ = −5/12 and θ lies in the second quadrant, then cos θ − sin θ equals:
Q29. The number of real solutions of sin x = x/10 is:
Q30. If cos x + cos²x = 1, then sin²x + sin⁴x equals: