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A Level Maths: Pure 3 MCQs with answers

Practise A Level Maths: Pure 3 (9709 Paper 3) with 250 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 Maths: Pure 3 MCQs with answers

Practise A Level Maths: Pure 3 (9709 Paper 3) with 250 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.

250 questions · 9 chapters

What each chapter covers 9 chapters

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

  1. Algebra 30 questions
  2. Logarithmic and exponential functions 29 questions
  3. Trigonometry 30 questions
  4. Differentiation 32 questions
  5. Integration 27 questions
  6. Numerical solution of equations 23 questions
  7. Vectors 26 questions
  8. Differential equations 24 questions
  9. Complex numbers 29 questions
Sample A Level Maths: Pure 3 MCQs with answers 12 questions

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

1. Expand (1 + x)−2 in ascending powers of x, up to and including the term in x3.

  1. A
    1 + 2x + 3x2 + 4x3
  2. B
    1 − 2x + 6x2 − 24x3
  3. C
    1 − 2x + 3x2 − 4x3
  4. D
    1 − 2x − 3x2 − 4x3
Show answer

Answer: C. With n = −2: nx = −2x; n(n − 1)/2! = (−2)(−3)/2 = 3; n(n − 1)(n − 2)/3! = (−2)(−3)(−4)/6 = −4. Leaving out the factorials gives 6x2 and −24x3.

2. Given that 2 log5 y − log5 x = 1, where x and y are positive, express y in terms of x.

  1. A
    y = 5x/2
  2. B
    y = √(x + 5)
  3. C
    y = 5√x
  4. D
    y = √(5x)
Show answer

Answer: D. log5(y2/x) = 1, so y2/x = 5 and y = √(5x). Writing 2 log5 y as log5 2y gives the wrong 5x/2.

3. Find the exact value of cot(π/6).

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

Answer: A. tan(π/6) = 1/√3, so cot(π/6) = 1/tan(π/6) = √3. The value 2 is cosec(π/6).

4. Differentiate x2e3x with respect to x.

  1. A
    6xe3x
  2. B
    xe3x(x + 2)
  3. C
    3x2e3x
  4. D
    xe3x(3x + 2)
Show answer

Answer: D. Product rule: 2x × e3x + x2 × 3e3x = xe3x(2 + 3x). The derivative of a product is not the product of the derivatives.

5. Find the exact value of ∫26 6/(3x − 2) dx.

  1. A
    12 ln 2
  2. B
    4 ln 2
  3. C
    36 ln 2
  4. D
    2 ln 12
Show answer

Answer: B. ∫ 6/(3x − 2) dx = 6 × ⅓ ln(3x − 2) = 2 ln(3x − 2). So the value is 2 ln 16 − 2 ln 4 = 2 ln 4 = 4 ln 2. Forgetting to divide by the 3 gives 6 ln 4 = 12 ln 2; note that ln 16 − ln 4 = ln 4, not ln 12.

6. The equation ln x = 3 − x has one root, α. Which of these intervals contains α?

  1. A
    2.1 < α < 2.2
  2. B
    2.3 < α < 2.4
  3. C
    2.0 < α < 2.1
  4. D
    2.2 < α < 2.3
Show answer

Answer: D. Let f(x) = ln x + x − 3. f(2.2) = −0.012 and f(2.3) = 0.133, so f changes sign between 2.2 and 2.3. f is negative at 2.0, 2.1 and 2.2 and positive at 2.3 and 2.4, so there is no change of sign in the other intervals.

7. Find the magnitude of the vector 2i − 3j + 6k.

  1. A
    7
  2. B
    5
  3. C
    49
  4. D
    √31
Show answer

Answer: A. |2i − 3j + 6k| = √(22 + (−3)2 + 62) = √49 = 7. (−3)2 is +9, not −9 (which would give √31), and adding the components (to get 5) does not give the length.

8. Find the general solution of the differential equation dy/dx = y2 sin x.

  1. A
    y = 1/(A − cos x)
  2. B
    y = 1/(A + cos x)
  3. C
    y = Ae−cos x
  4. D
    y = ∛(A − 3 cos x)
Show answer

Answer: B. ∫ y−2 dy = ∫ sin x dx gives −1/y = −cos x + c, so 1/y = cos x + A and y = 1/(A + cos x). Writing ∫ sin x dx = cos x gives the wrong sign; ∛(A − 3 cos x) comes from multiplying by y2 instead of dividing.

9. A cubic equation with real coefficients has roots 2 and 1 − 3i. What is its third root?

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

Answer: B. Non-real roots of a polynomial equation with real coefficients occur in conjugate pairs, so the conjugate 1 + 3i must also be a root. The conjugate changes the sign of the imaginary part only.

10. Express (5x2 + 3x + 7)/[(x + 2)(x2 + 3)] in partial fractions.

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

Answer: B. 5x2 + 3x + 7 ≡ A(x2 + 3) + (Bx + C)(x + 2). x = −2: 21 = 7A, A = 3. x2 terms: 5 = A + B, B = 2. Constants: 7 = 3A + 2C, C = −1.

11. Solve the equation ln(2x − 1) = 3, giving the exact answer.

  1. A
    x = (e3 − 1)/2
  2. B
    x = e3/2 + 1
  3. C
    x = (e3 + 1)/2
  4. D
    x = (ln 3 + 1)/2
Show answer

Answer: C. ln and e are inverse functions, so 2x − 1 = e3, 2x = e3 + 1 and x = (e3 + 1)/2 ≈ 10.5. Both terms must be divided by 2.

12. Find the greatest value of 10/(7 + 3 sin x − 4 cos x).

  1. A
    5/6
  2. B
    10/7
  3. C
    2
  4. D
    5
Show answer

Answer: D. 3 sin x − 4 cos x = 5 sin(x − α), which varies between −5 and 5, so the denominator varies between 2 and 12. The fraction is greatest when the denominator is least: 10/2 = 5. The value 5/6 is the least value.

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