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O Level & A Level Mathematics (Cambridge)

A Level Maths (Pure 1) — Trigonometry, Series and Calculus, Set 1

Take the A Level Maths (Pure 1) — Trigonometry, Series and Calculus, Set 1: 20 multiple-choice questions (30 minutes) with instant scoring and a full…

What this test covers

Sample questions with answers

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

  1. Q1. Simplify (1 − cos²θ)/sin θ.

    • A. 1
    • B. sin θ ✓
    • C. tan θ
    • D. cos θ

    Answer: B. sin θ

    Explanation: 1 − cos²θ = sin²θ, and sin²θ/sin θ = sin θ.

  2. Q2. Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

    • A. 30° and 210°
    • B. 60° and 120°
    • C. 30° and 330°
    • D. 30° and 150° ✓

    Answer: D. 30° and 150°

    Explanation: sin is positive in the first and second quadrants: 30° and 180° − 30°.

  3. Q3. Solve cos x = −0.5 for 0° ≤ x ≤ 360°.

    • A. 120° and 240° ✓
    • B. 120° and 300°
    • C. 60° and 300°
    • D. 150° and 210°

    Answer: A. 120° and 240°

    Explanation: cos is negative in the second and third quadrants: 180° ∓ 60°.

  4. Q4. What is the exact value of sin 60° × tan 30°?

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

    Answer: B. 1/2

    Explanation: sin 60° = √3/2 and tan 30° = 1/√3, so the product is 1/2.

  5. Q5. What is the period of y = sin 2x?

    • A. 90° (π/2 radians)
    • B. 720° (4π radians)
    • C. 360° (2π radians)
    • D. 180° (π radians) ✓

    Answer: D. 180° (π radians)

    Explanation: The period of sin kx is 360°/k = 180°.

The other 15 questions

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  1. Q6. Find the coefficient of x² in the expansion of (1 + 2x)⁵.

    • A. 80
    • B. 10
    • C. 40
    • D. 20

  2. Q7. Find the term independent of x in (x + 2/x)⁴.

    • A. 6
    • B. 8
    • C. 24
    • D. 16

  3. Q8. An arithmetic progression has first term 3 and common difference 4. Find the 20th term.

    • A. 80
    • B. 83
    • C. 76
    • D. 79

  4. Q9. Find the sum of the first 20 terms of that progression (a = 3, d = 4).

    • A. 1640
    • B. 800
    • C. 820
    • D. 790

  5. Q10. A geometric progression has first term 81 and common ratio 1/3. Find its sum to infinity.

    • A. 243
    • B. 54
    • C. 121.5
    • D. 108

  6. Q11. For which values of the common ratio r does a geometric progression have a sum to infinity?

    • A. r < 0
    • B. Any r ≠ 0
    • C. r > 1
    • D. |r| < 1

  7. Q12. Differentiate y = 4x³ − 2x + 5.

    • A. 12x² − 2
    • B. 4x² − 2
    • C. 12x² + 5
    • D. 12x³ − 2

  8. Q13. Differentiate y = 1/x².

    • A. −2/x³
    • B. 2/x³
    • C. −1/x
    • D. −2/x

  9. Q14. Find the gradient of y = x² − 3x at x = 4.

    • A. 4
    • B. 13
    • C. 8
    • D. 5

  10. Q15. Find the stationary point of y = x² − 6x + 1.

    • A. (3, −8)
    • B. (3, 1)
    • C. (6, 1)
    • D. (−3, 28)

  11. Q16. For y = x² − 6x + 1, what is the nature of the stationary point, and why?

    • A. A maximum, because d²y/dx² = 2 > 0
    • B. A minimum, because d²y/dx² = 2 > 0
    • C. A minimum, because dy/dx = 0
    • D. A point of inflection, because d²y/dx² is constant

  12. Q17. Differentiate (2x + 1)⁵.

    • A. 2(2x + 1)⁴
    • B. 10(2x + 1)⁴
    • C. 10(2x + 1)⁵
    • D. 5(2x + 1)⁴

  13. Q18. Find ∫ (6x² + 2x) dx.

    • A. 3x³ + x² + c
    • B. 2x³ + 2x² + c
    • C. 2x³ + x² + c
    • D. 12x + 2 + c

  14. Q19. Evaluate ∫₁³ 2x dx.

    • A. 9
    • B. 8
    • C. 6
    • D. 4

  15. Q20. Find the area between y = x², the x-axis and the lines x = 0 and x = 3.

    • A. 9
    • B. 27
    • C. 6
    • D. 3