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A Level Further Maths MCQs with answers

Practise A Level Further Maths (9231) with 811 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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A Level Further Maths MCQs with answers

Practise A Level Further Maths (9231) with 811 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.

811 questions · 4 chapters

What each chapter covers 4 chapters

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

  1. Further Pure Mathematics 1 250 questionsRoots of polynomial equations, Rational functions and graphs, Summation of series, Matrices, Polar coordinates, Vectors, Proof by induction
  2. Further Pure Mathematics 2 263 questionsHyperbolic functions, Matrices, Differentiation, Integration, Complex numbers, Differential equations
  3. Further Mechanics 155 questionsMotion of a projectile, Equilibrium of a rigid body, Circular motion, Hooke's law, Linear motion under a variable force, Momentum
  4. Further Probability & Statistics 143 questionsContinuous random variables, Inference using normal and t-distributions, χ²-tests, Non-parametric tests, Probability generating functions
Sample A Level Further Maths MCQs with answers 12 questions

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

1. A, B and X are 2 × 2 matrices with A and B non-singular, and AXB = C. Find X.

  1. A
    X = CA−1B−1
  2. B
    X = A−1B−1C
  3. C
    X = A−1CB−1
  4. D
    X = C(AB)−1
Show answer

Answer: C. Pre-multiply by A−1 and post-multiply by B−1: A−1AXBB−1 = A−1CB−1, so X = A−1CB−1. The inverses must go on the same side as the matrices they cancel.

2. The curve y = x2, 0 ≤ x ≤ 1, is rotated through 2π about the y-axis. Find the area of the surface formed.

  1. A
    (π/6)(5√5 − 1)
  2. B
    (π/6)(5√5 + 1)
  3. C
    (π/12)(5√5 − 1)
  4. D
    (π/6)(3√5 − 1)
Show answer

Answer: A. Rotating about the y-axis, the radius of each band is x, so S = 2π∫01 x√(1 + 4x2) dx = 2π × (1/12)[(1 + 4x2)3/2]01 = (π/6)(5√5 − 1) ≈ 5.33. Using y as the radius would give a different (rotation about the x-axis) integral.

3. A ball is projected from level ground with speed 24 m s−1 at 35° above the horizontal. Ignoring air resistance, the velocity of the ball as it returns to ground level is

  1. A
    19.7 m s−1 at 35° below the horizontal
  2. B
    24 m s−1 at 35° below the horizontal
  3. C
    24 m s−1 at 55° below the horizontal
  4. D
    19.7 m s−1 horizontally
Show answer

Answer: B. Energy is conserved, so the speed at the same level equals the launch speed, 24 m s−1. The horizontal component is unchanged and the vertical component is reversed, so the direction is 35° below the horizontal.

4. X has probability generating function G(t). Which expression gives Var(X)?

  1. A
    G″(1) − [G′(1)]2
  2. B
    G″(1) + [G′(1)]2
  3. C
    G″(1) − G′(1)
  4. D
    G″(1) + G′(1) − [G′(1)]2
Show answer

Answer: D. G″(1) = E(X(X − 1)) = E(X2) − E(X), so E(X2) = G″(1) + G′(1). Then Var(X) = E(X2) − [E(X)]2 = G″(1) + G′(1) − [G′(1)]2.

5. A, B and C are non-singular matrices of the same order. Which expression equals (ABC)−1?

  1. A
    A−1B−1C−1
  2. B
    B−1A−1C−1
  3. C
    C−1A−1B−1
  4. D
    C−1B−1A−1
Show answer

Answer: D. Inverses are taken in reverse order: (ABC)−1 = C−1B−1A−1. Check: ABC C−1B−1A−1 = I by cancelling from the middle outwards.

6. The area under the curve y = √x between x = 0 and x = 1 is estimated using four rectangles of equal width, each with its height taken at the right-hand end of its strip. Which statement is correct?

  1. A
    The estimate is 0.768, an over-estimate of the area
  2. B
    The estimate is 0.518, an under-estimate of the area
  3. C
    The estimate is 0.768, an under-estimate of the area
  4. D
    The estimate is 0.667, which equals the exact area
Show answer

Answer: A. Widths are 0.25 and the right-hand heights are √0.25, √0.5, √0.75, √1, so the total is 0.25 × (0.5 + 0.7071 + 0.8660 + 1) = 0.768. Because √x is increasing, each rectangle sticks out above the curve, so this exceeds the true area 2/3. The figure 0.518 uses left-hand heights.

7. A particle moves in a circle of radius 0.40 m with a constant speed of 12 m s−1. Find its angular speed.

  1. A
    4.8 rad s−1
  2. B
    0.033 rad s−1
  3. C
    75.4 rad s−1
  4. D
    30 rad s−1
Show answer

Answer: D. v = rω gives ω = v/r = 12/0.40 = 30 rad s−1. The value 4.8 comes from multiplying instead of dividing.

8. X has probability generating function G(t) = e3(t − 1). Find Var(X).

  1. A
    3
  2. B
    9
  3. C
    12
  4. D
    6
Show answer

Answer: A. G′(t) = 3e3(t − 1), so G′(1) = 3, and G″(t) = 9e3(t − 1), so G″(1) = 9. Var(X) = 9 + 3 − 9 = 3. The value 9 is G″(1) and 12 is G″(1) + G′(1), each missing the last step.

9. Find the acute angle between the planes x + y + z = 3 and x − y + 2z = 1.

  1. A
    28.1°
  2. B
    61.9°
  3. C
    118.1°
  4. D
    76.4°
Show answer

Answer: B. The angle between planes equals the angle between their normals: cos θ = |(1, 1, 1)·(1, −1, 2)|/(√3 × √6) = 2/√18 = 0.4714, so θ = 61.9°. 28.1° is 90° − θ, and 118.1° is the obtuse angle.

10. Solve y′′ − 3y′ + 2y = 0, given that y = 1 and dy/dx = 0 when x = 0.

  1. A
    y = 2e2x − ex
  2. B
    y = ex
  3. C
    y = 3ex − 2e2x
  4. D
    y = 2ex − e2x
Show answer

Answer: D. The CF is y = Aex + Be2x. Then A + B = 1 and y′(0) = A + 2B = 0, so B = −1 and A = 2. The swapped answer gives y′(0) = −1 + 4 = 3, not 0.

11. A light elastic string of natural length 0.6 m stores 1.5 J of elastic potential energy when its extension is 0.3 m. Find its modulus of elasticity.

  1. A
    10 N
  2. B
    40 N
  3. C
    20 N
  4. D
    5 N
Show answer

Answer: C. E = λx2/(2l) gives λ = 2El/x2 = 2 × 1.5 × 0.6/0.09 = 20 N. Missing the 2 gives 10 N.

12. The probability generating function of X is G(t) = 0.2 + 0.5t + 0.3t3. Find Var(X).

  1. A
    3.2
  2. B
    0.4
  3. C
    1.8
  4. D
    1.24
Show answer

Answer: D. G′(t) = 0.5 + 0.9t2, so G′(1) = 1.4. G″(t) = 1.8t, so G″(1) = 1.8. Var(X) = G″(1) + G′(1) − [G′(1)]2 = 1.8 + 1.4 − 1.96 = 1.24. Omitting the last term gives 3.2.

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