Proof, algebra and functions practice questions (A-Level Maths)

Sample questions

All 3 free A-Level Maths questions on this topic, from the free A-Level Maths Practice Paper. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 14 marks · Indices & surds
   Indices and surds.
   (a)

   Simplify $\frac{8^{\frac{2}{3}} \times 4^{-\frac{1}{2}}}{2^{-3}}$, giving your answer as an integer.

   [2]
   (b)

   Rationalise the denominator of $\frac{14}{3 - \sqrt{2}}$, giving your answer in the form $a + b\sqrt{2}$, where $a$ and $b$ are integers.

   [2]
   Show answerFrom the free A-Level Maths Practice Paper →
2. Question 25 marks · Quadratics & the discriminant
   The function $f$ is defined by $f(x) = 2x^2 - 12x + 23$.
   $f(x) = 2x^2 - 12x + 23$
   (a)

   Express $f(x)$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are integers to be found.

   [2]
   (b)

   Hence write down the coordinates of the vertex of the curve $y = f(x)$.

   [1]
   (c)

   Using the discriminant, determine the number of real roots of the equation $f(x) = 0$.

   [2]
   Show answerFrom the free A-Level Maths Practice Paper →
3. Question 34 marks · Proof
   Prove that, for every integer $n$, $(2n + 1)^2 - 1$ is divisible by 8.
   Show answerFrom the free A-Level Maths Practice Paper →

Key terms

The definitions and formulas A-Level Maths expects for proof, algebra and functions, typeset as they appear on the paper.

Laws of indices

The rules for multiplying, dividing and raising powers, with negative and fractional indices as reciprocals and roots. Rewrite every term as a power before differentiating or integrating.

$$a^m a^n = a^{m+n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = \frac{1}{a^n}, \quad a^{\frac{1}{n}} = \sqrt[n]{a}$$

Surds and rationalising the denominator

Exact roots left unevaluated. Multiply numerator and denominator by the conjugate to clear a surd from the denominator.

$$\frac{1}{a + \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}$$

Discriminant

Decides how many real roots a quadratic has: two distinct if positive, one repeated if zero, none if negative. "The line is a tangent to the curve" means the discriminant of the combined equation is zero.

$$b^2 - 4ac$$

Completing the square

Writing a quadratic as $a(x + p)^2 + q$ gives the vertex $(-p, q)$ and the line of symmetry $x = -p$, and solves equations when factorising fails.

$$ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}$$

Factor theorem

If $f(a) = 0$ then $(x - a)$ is a factor of the polynomial $f(x)$, and conversely. Use it to find a first factor, then divide to factorise a cubic fully.

$$f(a) = 0 \iff (x - a) \text{ is a factor of } f(x)$$

Partial fractions

Splitting a rational expression into simpler fractions over its linear factors (including a repeated factor), needed before integrating or expanding as a binomial series.

$$\frac{px + q}{(ax + b)(cx + d)} = \frac{A}{ax + b} + \frac{B}{cx + d}$$

Methods of proof

Proof by deduction (algebraic argument from definitions), by exhaustion (checking every case), disproof by counter-example, and proof by contradiction, such as showing $\sqrt{2}$ is irrational or that there are infinitely many primes.

Frequently asked questions

Other A-Level Maths topics

• Coordinate geometry
• Sequences and series
• Trigonometry
• Exponentials and logarithms
• Differentiation
• Integration
• Numerical methods and vectors
• Sampling and data presentation
• Probability and distributions
• Hypothesis testing
• Kinematics
• Forces, Newton's laws and moments

All A-Level Maths topics · A-Level Maths exam guide · All free practice exams