A Level Further Maths chapter 1: Further Pure Mathematics 1 MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

250 exam-style questions on Further Pure Mathematics 1, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.

250 questions · 7 topics · 10 per test

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What chapter 1 covers

Ten Further Pure Mathematics 1 questions with answers

Try each one, then open the answer. These ten stay the same; the test draws new ones every time.

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. 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.

3. 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.

4. The roots of x2 − 3x + 1 = 0 are α and β. Find the quadratic equation whose roots are α3 and β3.

  1. A
    x2 − 18x + 1 = 0
  2. B
    x2 − 27x + 1 = 0
  3. C
    x2 − 9x + 1 = 0
  4. D
    x2 − 18x + 3 = 0
Show answer

Answer: A. α3 + β3 = (α + β)3 − 3αβ(α + β) = 27 − 9 = 18 and α3β3 = 1. So x2 − 18x + 1 = 0. Cubing the sum alone gives 27.

5. Find the perpendicular distance from the point (2, −1, 3) to the plane x + 2y − 2z = 4.

  1. A
    10
  2. B
    10/3
  3. C
    2/3
  4. D
    10/9
Show answer

Answer: B. Distance = |n·P − d|/|n| = |2 − 2 − 6 − 4|/√(1 + 4 + 4) = 10/3. Forgetting to divide by |n| gives 10; dividing by |n|² gives 10/9; 2/3 comes from a sign slip in d.

6. Two of the roots of x3 + px2 + qx − 12 = 0 are 1 and 2. Find q.

  1. A
    −20
  2. B
    8
  3. C
    12
  4. D
    20
Show answer

Answer: D. The product of the three roots is −(−12) = 12, so the third root is 12/(1 × 2) = 6. Then q = Σαβ = 1·2 + 1·6 + 2·6 = 20. Using the product of the roots as q or the sign convention for the sum gives the wrong values.

7. For the curve r = 2 + cos θ, 0 ≤ θ < 2π, which statement is correct?

  1. A
    The curve does not pass through the pole, and the least value of r is 1
  2. B
    The curve passes through the pole when θ = π
  3. C
    The curve does not pass through the pole, and the least value of r is 2
  4. D
    The curve passes through the pole when θ = π/2
Show answer

Answer: A. r = 2 + cos θ ≥ 2 − 1 = 1 > 0 for all θ, so r is never zero and the curve avoids the pole. The least value 1 occurs at θ = π, the greatest value 3 at θ = 0.

8. A sequence has u1 = 1 and un+1 = un + 2n + 1. Calculating the first few terms suggests a formula for un. Which one?

  1. A
    un = 2n − 1
  2. B
    un = n(n + 1)/2
  3. C
    un = 2n − 1
  4. D
    un = n2
Show answer

Answer: D. The terms are 1, 4, 9, 16, …, the square numbers, so un = n2. This can be proved by induction: k2 + 2k + 1 = (k + 1)2. The formula 2n − 1 fits u1 = 1 but not u2 = 4.

9. The roots of x2 − 6x + 4 = 0 are α and β. Find the value of 1/α + 1/β.

  1. A
    2/3
  2. B
    −3/2
  3. C
    6
  4. D
    3/2
Show answer

Answer: D. 1/α + 1/β = (α + β)/(αβ) = 6/4 = 3/2. Inverting the fraction (giving 2/3) or forgetting to divide by the product are the usual slips.

10. f(x) = x2 − 4x + 7. Which statement about the graph of y = 1/f(x) is correct?

  1. A
    It has a maximum at (2, 1/3) and no vertical asymptotes
  2. B
    It has a minimum at (2, 1/3) and no vertical asymptotes
  3. C
    It has a maximum at (2, 3) and no vertical asymptotes
  4. D
    It has a maximum at (2, 1/3) and a vertical asymptote at x = 2
Show answer

Answer: A. f(x) = (x − 2)2 + 3 has minimum value 3 at x = 2. Taking reciprocals turns the minimum into a maximum of 1/3, and since f never equals 0 there are no vertical asymptotes; y → 0 as x → ±∞.

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Further Pure Mathematics 1: questions students ask

How many Further Pure Mathematics 1 MCQs are there?

250, each with the answer and a short explanation, across 7 topics: Roots of polynomial equations, Rational functions and graphs, Summation of series, Matrices, Polar coordinates, Vectors, Proof by induction. Every chapter test draws ten of them, so you can take it many times.

Is the A Level Further Maths chapter 1 test free?

Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.

How is the chapter test marked?

The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.

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