A Level Maths: Pure 3 chapter 1: Algebra MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

30 exam-style questions on Algebra, 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.

30 questions · 1 topics · 10 per test

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What chapter 1 covers

Ten Algebra questions with answers

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

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. 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.

3. When x3 + ax2 − 4 is divided by (x − 2) and when it is divided by (x + 1), the remainders are equal. Find a.

  1. A
    a = 3
  2. B
    a = −7/3
  3. C
    a = −3
  4. D
    a = −9/5
Show answer

Answer: C. p(2) = 8 + 4a − 4 = 4a + 4 and p(−1) = −1 + a − 4 = a − 5. Setting them equal: 4a + 4 = a − 5, so 3a = −9 and a = −3. Using p(−2) and p(1) by mistake gives a = 3.

4. Solve the inequality |2x − 1| > |x + 4|.

  1. A
    −1 < x < 5
  2. B
    x < −5 or x > 1
  3. C
    x > 5 only
  4. D
    x < −1 or x > 5
Show answer

Answer: D. Both sides are non-negative, so square: 4x2 − 4x + 1 > x2 + 8x + 16, giving 3x2 − 12x − 15 > 0, i.e. (x − 5)(x + 1) > 0. This quadratic is positive outside its roots: x < −1 or x > 5.

5. Which is the correct form for expressing (2x2 + 1)/[(x − 3)(x2 + 4)] in partial fractions?

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

Answer: B. x2 + 4 has no real factors, so the numerator over it must be a general linear expression Bx + C; a constant alone cannot always match. x2 + 4 is not (x − 2)(x + 2): that is x2 − 4.

6. For what values of x is the expansion of (1 + 3x)−1/3 in ascending powers of x valid?

  1. A
    |x| < 1/3
  2. B
    |x| < 3
  3. C
    |x| < 1
  4. D
    x > −1/3
Show answer

Answer: A. The expansion of (1 + y)n for a non-integer n is valid for |y| < 1. Here y = 3x, so |3x| < 1, i.e. |x| < 1/3.

7. Solve the equation |2x − 5| = 7.

  1. A
    x = 6 only
  2. B
    x = 6 or x = 1
  3. C
    x = −6 or x = 1
  4. D
    x = 6 or x = −1
Show answer

Answer: D. Either 2x − 5 = 7, giving x = 6, or 2x − 5 = −7, giving x = −1. Both cases are needed; ignoring the negative case loses x = −1.

8. Express (5x + 1)/[(x − 1)(x + 2)] in partial fractions.

  1. A
    3/(x − 1) + 2/(x + 2)
  2. B
    2/(x − 1) − 3/(x + 2)
  3. C
    2/(x − 1) + 3/(x + 2)
  4. D
    6/(x − 1) − 9/(x + 2)
Show answer

Answer: C. 5x + 1 ≡ A(x + 2) + B(x − 1). x = 1: 6 = 3A, so A = 2. x = −2: −9 = −3B, so B = 3. The values 6 and −9 still have to be divided by 3 and −3.

9. Find the first three terms in the expansion of √(1 − 4x) in ascending powers of x.

  1. A
    1 − 2x − 2x2
  2. B
    1 − 2x + 2x2
  3. C
    1 − 2x − ½x2
  4. D
    1 + 2x − 2x2
Show answer

Answer: A. With n = ½ and −4x in place of x: 1 + ½(−4x) + [(½)(−½)/2](−4x)2 = 1 − 2x − ⅛ × 16x2 = 1 − 2x − 2x2. The whole of −4x must be squared, giving 16x2, not 4x2.

10. Solve the inequality |x + 3| < 2x.

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

Answer: B. 2x must be positive, so x > 0; then x + 3 > 0 and the inequality is x + 3 < 2x, i.e. x > 3. Squaring gives (x − 3)(x + 1) > 0, but the part x < −1 must be rejected because there 2x is negative and cannot exceed a modulus.

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How many Algebra MCQs are there?

30, each with the answer and a short explanation, across 1 topics: Algebra. Every chapter test draws ten of them, so you can take it many times.

Is the A Level Maths: Pure 3 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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