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AS Level Mathematics MCQs with answers

Practise AS Level Mathematics (9709) with 167 exam-style MCQs, each with the answer and a short explanation. Every test is marked the moment you finish and shows your score chapter by chapter, so you know what to revise next. It is free and needs no sign-up.

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AS Level Mathematics MCQs with answers

Practise AS Level Mathematics (9709) with 167 exam-style MCQs, each with the answer and a short explanation. Every test is marked the moment you finish and shows your score chapter by chapter, so you know what to revise next. It is free and needs no sign-up.

167 questions · 3 chapters

What each chapter covers 3 chapters

The questions follow the syllabus chapter by chapter. You can test the whole subject or one chapter at a time.

  1. Pure Mathematics 1 87 questionsQuadratics, Functions, Coordinate geometry, Circular measure, Trigonometry, Series, Differentiation, Integration
  2. Mechanics 40 questionsForces and equilibrium, Kinematics of motion in a straight line, Momentum, Newton’s laws of motion, Energy, work and power
  3. Probability & Statistics 1 40 questionsRepresentation of data, Permutations and combinations, Probability, Discrete random variables, The normal distribution
Sample AS Level Mathematics MCQs with answers 12 questions

12 questions from the test, one or two from each chapter. Try each one, then open the answer.

1. Find dy/dx when y = 4/(3x − 1)2.

  1. A
    −24/(3x − 1)3
  2. B
    −8/(3x − 1)3
  3. C
    24/(3x − 1)3
  4. D
    −24/(3x − 1)2
Show answer

Answer: A. y = 4(3x − 1)−2, so dy/dx = 4 × (−2)(3x − 1)−3 × 3 = −24/(3x − 1)3. Both the −2 and the chain-rule factor 3 are needed.

2. Particles of mass 3 kg and 2 kg are connected by a light inextensible string over a smooth fixed pulley and released from rest. Find the tension in the string. (g = 10 m s−2)

  1. A
    20 N
  2. B
    30 N
  3. C
    24 N
  4. D
    25 N
Show answer

Answer: C. Whole system: 30 − 20 = 5a, so a = 2 m s−2. For the 2 kg mass: T − 20 = 2 × 2, so T = 24 N. The tension lies between the two weights.

3. How many even 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 if no digit is repeated?

  1. A
    840
  2. B
    120
  3. C
    360
  4. D
    720
Show answer

Answer: C. Fix the last digit first: 3 choices (2, 4 or 6). The remaining three places can be filled in 6 × 5 × 4 = 120 ways. Total 3 × 120 = 360.

4. Find the set of values of k for which the equation kx2 + 4x + k = 0 has no real roots.

  1. A
    k > 2
  2. B
    −2 < k < 2
  3. C
    k < −2 or k > 2
  4. D
    k > 4
Show answer

Answer: C. No real roots needs b2 − 4ac < 0: 16 − 4k2 < 0, so k2 > 4, giving k < −2 or k > 2. Sketch y = k2 − 4 to get the outside region, not the inside.

5. A car of mass 1200 kg tows a trailer of mass 300 kg on a level road using a light rigid tow-bar. The driving force is 3600 N and the resistances are 300 N on the car and 300 N on the trailer. Find the tension in the tow-bar.

  1. A
    600 N
  2. B
    900 N
  3. C
    1200 N
  4. D
    3000 N
Show answer

Answer: B. Whole system: 3600 − 600 = 1500a, so a = 2 m s−2. Trailer alone: T − 300 = 300 × 2, so T = 900 N. Forgetting the trailer resistance gives 600 N.

6. Group A has 10 values with mean 20. Group B has 15 values with mean 30. Find the mean of all 25 values.

  1. A
    25
  2. B
    27
  3. C
    24
  4. D
    26
Show answer

Answer: D. Total of A = 200, total of B = 450. Combined mean = 650/25 = 26. Averaging the two means (25) ignores the different group sizes.

7. Find the equation of the tangent to the circle (x − 1)2 + (y − 2)2 = 25 at the point (4, 6).

  1. A
    4x − 3y = −2
  2. B
    3x + 4y = 12
  3. C
    3x + 4y = 36
  4. D
    3x − 4y = −12
Show answer

Answer: C. The radius from (1, 2) to (4, 6) has gradient 4/3, so the tangent (perpendicular to the radius) has gradient −3/4. y − 6 = −3/4 (x − 4) gives 4y − 24 = −3x + 12, i.e. 3x + 4y = 36.

8. A block of mass 5 kg rests in limiting equilibrium on a rough plane inclined at 30° to the horizontal. Find the coefficient of friction, to 3 significant figures. (g = 10 m s−2)

  1. A
    0.577
  2. B
    0.5
  3. C
    0.866
  4. D
    1.73
Show answer

Answer: A. Along the plane: F = 50 sin 30° = 25 N. Perpendicular: R = 50 cos 30° = 43.3 N. Limiting, so μ = F/R = 25/43.3 = 0.577 (= tan 30°). Using R = 50 N gives the wrong 0.5.

9. Find the number of arrangements of the letters of the word ORANGES that begin and end with a vowel.

  1. A
    360
  2. B
    720
  3. C
    1440
  4. D
    5040
Show answer

Answer: B. There are 3 vowels (O, A, E). The first and last letters can be chosen in 3 × 2 = 6 ways and the other 5 letters arranged in 5! = 120 ways: 6 × 120 = 720. Using C(3, 2) = 3 for the ends ignores their order and gives 360.

10. Simplify (1 − cos2θ)/(sin θ cos θ).

  1. A
    sin θ
  2. B
    sin θ cos θ
  3. C
    cos θ/sin θ
  4. D
    tan θ
Show answer

Answer: D. Use sin2θ + cos2θ = 1: the numerator is sin2θ. Then sin2θ/(sin θ cos θ) = sin θ/cos θ = tan θ.

11. A car accelerates uniformly from 5 m s−1 to 20 m s−1 in 6 s. Find its acceleration.

  1. A
    3.33 m s−2
  2. B
    2.5 m s−2
  3. C
    15 m s−2
  4. D
    0.4 m s−2
Show answer

Answer: B. v = u + at: 20 = 5 + 6a, so a = 15/6 = 2.5 m s−2. Use the change in velocity, not the final velocity.

12. For a set of 10 values, Σx = 120 and Σx2 = 1690. Find the standard deviation.

  1. A
    25
  2. B
    5
  3. C
    13
  4. D
    2.24
Show answer

Answer: B. Mean = 12. Variance = Σx2/n − mean2 = 169 − 144 = 25, so the standard deviation is √25 = 5. Do not forget to take the square root.

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Ways to practise 5 ways

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The 167 MCQs are our own, written to the syllabus in the style of the exam.

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