A Level Further Maths chapter 2: Further Pure Mathematics 2 MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

263 exam-style questions on Further Pure Mathematics 2, 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.

263 questions · 6 topics · 10 per test

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What chapter 2 covers

Ten Further Pure Mathematics 2 questions with answers

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

1. 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.

2. 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.

3. 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.

4. How many real solutions does the equation cosh x = sinh x + 1/2 have?

  1. A
    1
  2. B
    0
  3. C
    2
  4. D
    infinitely many
Show answer

Answer: A. cosh x − sinh x = e−x, so the equation becomes e−x = ½, giving the single solution x = ln 2. On a graph, sinh x + ½ is increasing while cosh x is U-shaped, and the curves cross once.

5. Find ∫ cosh2 x dx.

  1. A
    ½x − ¼ sinh 2x + c
  2. B
    ⅓ cosh3 x + c
  3. C
    ½ sinh 2x + c
  4. D
    ½x + ¼ sinh 2x + c
Show answer

Answer: D. Use cosh 2x = 2cosh2 x − 1, so cosh2 x = ½(1 + cosh 2x). Integrating gives ½x + ¼ sinh 2x + c. The minus sign belongs to ∫ cos2 x dx, not the hyperbolic case.

6. Find limn→∞ Σr=1n 1/(n + r).

  1. A
    1
  2. B
    ln 2
  3. C
    ln(3/2)
  4. D
    1/2
Show answer

Answer: B. Write 1/(n + r) = (1/n) × 1/(1 + r/n). This is a Riemann sum for ∫01 1/(1 + x) dx = ln 2.

7. Differentiate y = cos−1(x/2), for −2 < x < 2.

  1. A
    −2/√(4 − x2)
  2. B
    1/√(4 − x2)
  3. C
    −1/(2√(4 − x2))
  4. D
    −1/√(4 − x2)
Show answer

Answer: D. dy/dx = −(1/√(1 − x2/4)) × ½ = −1/(2√((4 − x2)/4)) = −1/√(4 − x2). Forgetting the factor ½ from the chain rule gives −2/√(4 − x2), and a positive sign is the derivative of sin−1(x/2).

8. The constant term in the expansion of (z + z−1)6 is 20. Use this to find the mean value of cos6 θ over one full period.

  1. A
    5/8
  2. B
    15/64
  3. C
    5/16
  4. D
    1/64
Show answer

Answer: C. With z = cos θ + i sin θ, (2cos θ)6 = (z + z−1)6 = 2cos 6θ + 12cos 4θ + 30cos 2θ + 20. Every cosine term averages to 0 over a period, so the mean value of 64cos6 θ is 20 and that of cos6 θ is 20/64 = 5/16.

9. On an Argand diagram, which pair of geometrical effects does multiplication by 1 − i produce?

  1. A
    an enlargement, scale factor √2, and a rotation of 45° clockwise about the origin
  2. B
    an enlargement, scale factor √2, and a rotation of 45° anticlockwise about the origin
  3. C
    an enlargement, scale factor 2, and a rotation of 45° clockwise about the origin
  4. D
    an enlargement, scale factor √2, and a rotation of 135° clockwise about the origin
Show answer

Answer: A. |1 − i| = √2 and arg(1 − i) = −π/4. Multiplying by a complex number enlarges by its modulus and rotates by its argument, so the effect is an enlargement of scale factor √2 with a 45° clockwise rotation.

10. Find the coefficient of x3 in the Maclaurin series for ln(1 + sin x).

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

Answer: C. Let u = sin x = x − x3/6 + …. Then ln(1 + u) = u − u2/2 + u3/3 − …, and the x3 terms are −1/6 (from u) and 1/3 (from u3/3), giving 1/6. Leaving out the u3 term gives −1/6.

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Further Pure Mathematics 2: questions students ask

How many Further Pure Mathematics 2 MCQs are there?

263, each with the answer and a short explanation, across 6 topics: Hyperbolic functions, Matrices, Differentiation, Integration, Complex numbers, Differential equations. Every chapter test draws ten of them, so you can take it many times.

Is the A Level Further Maths chapter 2 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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