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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Trigonometry (Set 1)

Take the ECAT Mathematics — Trigonometry (Set 1): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30 ECAT…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. The radian measure of 150° is:

    • A. 5π/6 ✓
    • B. 5π/3
    • C. 3π/4
    • D. 2π/3

    Answer: A. 5π/6

    Explanation: 150° × π/180° = 5π/6.

  2. Q2. The area of a sector of a circle of radius 6 cm with central angle π/3 radians is:

    • A. 2π cm²
    • B. 6π cm² ✓
    • C. 12π cm²
    • D. 3π cm²

    Answer: B. 6π cm²

    Explanation: Area = ½r²θ = ½ × 36 × π/3 = 6π cm².

  3. Q3. The value of sin 330° is:

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

    Answer: D. −1/2

    Explanation: 330° = 360° − 30° lies in quadrant IV where sine is negative: sin 330° = −sin 30° = −1/2.

  4. Q4. From a point 30 m away from the foot of a vertical tower on level ground, the angle of elevation of its top is 60°. The height of the tower is:

    • A. 10√3 m
    • B. 60 m
    • C. 30√3 m ✓
    • D. 15 m

    Answer: C. 30√3 m

    Explanation: tan 60° = h/30 ⇒ h = 30√3 m.

  5. Q5. The value of tan 225° is:

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

    Answer: C. 1

    Explanation: 225° = 180° + 45° lies in quadrant III where tangent is positive: tan 225° = tan 45° = 1.

The other 25 questions

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  1. Q6. The expression (sec θ − tan θ)(sec θ + tan θ) simplifies to:

    • A. 0
    • B. 1
    • C. 2 sec θ
    • D. sec²θ + tan²θ

  2. Q7. The exact value of sin 75° is:

    • A. (√6 + √2)/4
    • B. (√6 − √2)/4
    • C. (√3 + 1)/2
    • D. √3/4

  3. Q8. The value of cos 15° − sin 15° is:

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

  4. Q9. tan(45° + θ) is equal to:

    • A. (1 − tan θ)/(1 + tan θ)
    • B. (1 + tan θ)/(1 − tan θ)
    • C. 1 + tan θ
    • D. (tan θ − 1)/(tan θ + 1)

  5. Q10. If sin θ = 3/5 and θ is acute, then sin 2θ equals:

    • A. 12/25
    • B. 7/25
    • C. 24/25
    • D. 6/5

  6. Q11. If tan θ = 1/2, then tan 2θ equals:

    • A. 4/3
    • B. 2/3
    • C. 1
    • D. 3/4

  7. Q12. The triple-angle formula for sin 3θ is:

    • A. 3 sin θ + 4 sin³θ
    • B. 4 sin³θ − 3 sin θ
    • C. 3 sin θ − 4 sin³θ
    • D. 3 cos θ − 4 cos³θ

  8. Q13. The value of sin 75° cos 15° is:

    • A. (1 + √3)/4
    • B. (2 + √3)/4
    • C. (2 − √3)/4
    • D. 1/4

  9. Q14. sin 50° + sin 10° is equal to:

    • A. 2 cos 20°
    • B. sin 60°
    • C. cos 20°
    • D. sin 20°

  10. Q15. The period of the function f(x) = sin 3x is:

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

  11. Q16. The period of the function f(x) = tan(x/2) is:

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

  12. Q17. The maximum value of 3 sin x + 4 cos x is:

    • A. 5
    • B. 12
    • C. 7
    • D. 4

  13. Q18. The range of the principal value function y = tan⁻¹x is:

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

  14. Q19. The principal value of sin⁻¹(−1/2) is:

    • A. 5π/6
    • B. 11π/6
    • C. 7π/6
    • D. −π/6

  15. Q20. The value of tan⁻¹(1/2) + tan⁻¹(1/3) is:

    • A. π/3
    • B. π/6
    • C. π/4
    • D. π/2

  16. Q21. The value of sin(cos⁻¹(5/13)) is:

    • A. 5/12
    • B. 8/13
    • C. 12/13
    • D. 13/12

  17. Q22. The general solution of the equation sin x = 0 is (n ∈ ℤ):

    • A. x = nπ + π/2
    • B. x = nπ/2
    • C. x = nπ
    • D. x = 2nπ

  18. Q23. The solutions of tan x = √3 in the interval [0, 2π) are:

    • A. π/3 and 5π/3
    • B. π/3 and 2π/3
    • C. π/3 and 4π/3
    • D. π/6 and 7π/6

  19. Q24. The solutions of 2cos²x − cos x − 1 = 0 in the interval [0, 2π) are:

    • A. 0, 2π/3, 4π/3
    • B. π, π/3, 5π/3
    • C. 0, 2π/3
    • D. 0, π/3, 5π/3

  20. Q25. In triangle ABC, a = 10, ∠A = 30° and ∠B = 45°. By the law of sines, side b equals:

    • A. 5√2
    • B. 10√3
    • C. 20
    • D. 10√2

  21. Q26. A triangle has sides 5, 7 and 8. The angle opposite the side of length 7 is:

    • A. 60°
    • B. 45°
    • C. 90°
    • D. 120°

  22. Q27. The area of triangle ABC with a = 8, b = 5 and ∠C = 30° is:

    • A. 10√3
    • B. 20
    • C. 40
    • D. 10

  23. Q28. Using Hero's formula, the area of a triangle with sides 13, 14 and 15 is:

    • A. 84
    • B. 91
    • C. 168
    • D. 42

  24. Q29. The radius of the inscribed circle (in-radius) of the triangle with sides 13, 14 and 15 is:

    • A. 65/8
    • B. 3.5
    • C. 4
    • D. 6

  25. Q30. If tan θ = −3/4 and θ lies in the second quadrant, then cos θ equals:

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