ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Trigonometry (Set 5): 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 value of sin 105° sin 15° is:
Answer: C. 1/4
Explanation: sin A sin B = ½[cos(A − B) − cos(A + B)] = ½(cos 90° − cos 120°) = ½(0 + ½) = 1/4.
Q2. The value of cos 36° − cos 72° is:
Answer: A. 1/2
Explanation: cos 36° = (1 + √5)/4 and cos 72° = (√5 − 1)/4, so the difference is 2/4 = 1/2.
Q3. The value of sin 10° sin 50° sin 70° is:
Answer: D. 1/8
Explanation: sin 10° sin 50° sin 70° = cos 80° cos 40° cos 20°. Multiplying and dividing by sin 20° and applying sin 2θ repeatedly gives sin 160°/(8 sin 20°) = 1/8.
Q4. The value of sin(2 tan⁻¹(1/3)) is:
Answer: A. 3/5
Explanation: With t = tan α = 1/3, sin 2α = 2t/(1 + t²) = (2/3)/(10/9) = 3/5.
Q5. If sin x + cos x = 1/√2, then sin 2x equals:
Answer: B. −1/2
Explanation: Squaring: 1 + sin 2x = 1/2, so sin 2x = −1/2.
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Q6. The minimum value of 7 + 3 sin x − 4 cos x is:
Q7. The period of f(x) = sin(x/2) + cos(x/3) is:
Q8. The sum of all solutions of 2 sin²θ + 3 cos θ = 0 in the interval [0, 2π) is:
Q9. The number of solutions of sin 3θ = sin θ in the interval [0, π] is:
Q10. If tan θ = 2, then (2 sin θ + cos θ)/(sin θ − cos θ) equals:
Q11. A circular sector has radius 10 cm and area 50 cm². Its perimeter is:
Q12. At 3:40 the smaller angle between the hour and minute hands of a clock is:
Q13. In a triangle ABC, tan A + tan B + tan C always equals:
Q14. In triangle ABC, a = 10, A = 30° and B = 45°. The side b is:
Q15. In a triangle with sides 7, 8 and 9, the cosine of the angle opposite the side 9 is:
Q16. The area of a triangle with sides 7, 8 and 9 is:
Q17. The in-radius of the triangle with sides 9, 10 and 17 is:
Q18. The angle of elevation of the top of a tower from a point on level ground is 30°. After walking 20 m towards the tower the angle becomes 45°. The height of the tower is:
Q19. The value of cos(2 sin⁻¹(1/3)) is:
Q20. tan⁻¹(1/4) + tan⁻¹(2/9) is equal to tan⁻¹ of:
Q21. The principal value of cos⁻¹(cos 7π/6) is:
Q22. The number of zeros of f(x) = sin 2x in the closed interval [0, 2π] is:
Q23. The value of sin 3θ/sin θ − cos 3θ/cos θ is:
Q24. The minimum value of sec²θ + csc²θ is:
Q25. If cos 2θ = 1/3, then sin²θ equals:
Q26. If cos θ = 7/25 with θ acute, then cos(θ/2) equals:
Q27. If cos θ = 1/4, then cos 3θ equals:
Q28. The value of cos 105° + cos 15° is:
Q29. If sin θ = −3/5 and θ lies in the third quadrant, then sec θ + tan θ equals:
Q30. If sin A = 8/17 and cos B = 4/5 with A and B acute, then cos(A − B) equals: