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Algebra and functions practice problems A level Maths, Topic 1

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 exam-style question

Indices and surds, quadratics and the discriminant, the factor theorem, composite and inverse functions, and partial fractions. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Exam question practice
  5. Common mistakes
  6. Keep going

Key equations for algebra and functions

b2 − 4acdiscriminant: > 0 two roots, = 0 one, < 0 none
f(a) = 0 ⇒ (x − a) is a factorfactor theorem
fg(x) = f(g(x))composite function
am/n = (n√a)mfractional indices

A formula booklet is provided in the exam. Answers within 2.5% are marked correct, and you can type fractions such as 1/6. What the booklet gives you, and what it leaves out, is in our A level Maths formula booklet guide.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
Find the discriminant of 2x2 − 3x + 5.

Hint

b2 − 4ac with a = 2, b = −3, c = 5.

Worked solution
  1. (−3)2 − 4(2)(5) = 9 − 40.
  2. = −31, so there are no real roots.
Q2Warm-up
Evaluate 82/3.

Hint

Take the cube root first, then square.

Worked solution
  1. ∛8 = 2.
  2. 22 = 4.
Q3Warm-up
f(x) = x3 − 2x2 − 5x + 6. Which of these is a factor of f(x)?

Hint

Use the factor theorem: test f(a) for each.

Worked solution
  1. f(1) = 1 − 2 − 5 + 6 = 0.
  2. So (x − 1) is a factor: B. (f(−1) = 8, f(2) = −4, f(−3) = −24.)

Exam-style questions

Questions at the level of the A level Maths exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
Rationalise 6/(3 − √3), writing it as a + √3. Find a.

Hint

Multiply top and bottom by (3 + √3).

Worked solution
  1. 6(3 + √3)/((3 − √3)(3 + √3)) = 6(3 + √3)/6.
  2. = 3 + √3, so a = 3.
Q5Exam-style
x2 + kx + 9 = 0 has equal roots, and k > 0. Find k.

Hint

Equal roots: the discriminant is zero.

Worked solution
  1. k2 − 36 = 0.
  2. k = 6 (taking k > 0).
Q6Exam-style
f(x) = 2x + 1 and g(x) = x2. Find fg(3).

Hint

Apply g first, then f.

Worked solution
  1. g(3) = 9.
  2. f(9) = 2(9) + 1 = 19.
Q7Exam-style
f(x) = (x + 4)/3. Find f−1(5).

Hint

Find the inverse, or solve f(x) = 5.

Worked solution
  1. (x + 4)/3 = 5 gives x + 4 = 15.
  2. f−1(5) = 11.
Q8ChallengeYear 2
(5x + 1)/((x + 1)(x − 1)) = A/(x + 1) + B/(x − 1). Find B.

Hint

Multiply through, then substitute x = 1.

Worked solution
  1. 5x + 1 = A(x − 1) + B(x + 1).
  2. x = 1: 6 = 2B.
  3. B = 3 (and A = 2, from x = −1).

Exam question practice

One original exam-style question in the style of an A level paper. Answer it in full, with working, before you open the mark scheme: M marks are for method, A marks for accurate answers that follow from it, and B marks for correct statements on their own.

Exam-style question style · 6 points · about 8 minutes

f(x) = x3 − 7x − 6.

  1. (a) Use the factor theorem to show that (x + 1) is a factor of f(x).
  2. (b) Factorise f(x) completely.
  3. (c) Hence solve f(x) = 0.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

f(−1) = −1 + 7 − 6 = 0, so (x + 1) is a factor.

Part (b)

f(x) = (x + 1)(x2 − x − 6) = (x + 1)(x + 2)(x − 3).

Part (c)

x = −1, x = −2 or x = 3.

Your mark: 0 of 6

Common mistakes in algebra and functions

Keep going

Revise the ideas behind these questions:

Then try real exam questions: A level Maths past papers, with official scoring guidelines. · All A level Maths units