MAT-style algebra and functions questions

Quadratics, polynomials, functions and their graphs: core material for MAT-style questions and for the TMUA. Fast, accurate algebra matters more than clever tricks. These are original problems in the style of the MAT. The MAT is no longer used for Oxford maths (from 2026 Oxford uses the TMUA), so treat them as problem-solving practice for the TMUA, STEP and interviews.

Practice questions

Try each one before you open a hint. The multiple-choice questions should take a few minutes each; give the longer ones 20 minutes or more.

Question 1: A quartic in disguise · multiple choice

What is the sum of the real solutions of \((x^2-4x)^2-2(x^2-4x)-15=0\)?

  1. \(4\)
  2. \(6\)
  3. \(8\)
  4. \(10\)
Hint 1

Let \(u=x^2-4x\).

Hint 2

Check that both values of \(u\) give real \(x\).

Full solution

Answer: C: \(8\)

With \(u=x^2-4x\): \(u^2-2u-15=0\), so \(u=5\) or \(u=-3\). \(x^2-4x-5=0\) gives \(x=5,-1\); \(x^2-4x+3=0\) gives \(x=1,3\). All four are real and their sum is \(8\).

Question 2: Two negative roots · multiple choice

For which values of \(k\) does \(x^2+kx+k+3=0\) have two distinct real roots that are both negative?

  1. \(k>6\)
  2. \(k<-2\)
  3. \(-3
  4. \(k>-3\)
Hint 1

Distinct real roots: discriminant \(>0\).

Hint 2

Both negative: sum of roots \(<0\) and product \(>0\).

Full solution

Answer: A: \(k>6\)

Discriminant \(k^2-4k-12=(k-6)(k+2)>0\): \(k>6\) or \(k<-2\). Sum \(-k<0\) needs \(k>0\); product \(k+3>0\). Together: \(k>6\).

Question 3: When does a cubic have three roots? · longer question

Let \(g(x)=x^3-3px+q\), where \(p>0\) and \(q\) are real.

  1. Find the stationary points of \(g\) and show that \(g(x)=0\) has three distinct real roots if and only if \(q^2<4p^3\).
  2. For \(p=1\), find the values of \(q\) for which there are three distinct real roots.
  3. For \(p=3\), find the positive value of \(q\) for which \(g\) has a repeated root, and find all the roots in that case.
Hint 1

\(g'(x)=3x^2-3p\), so the turning points are at \(x=\pm\sqrt p\).

Hint 2

Three distinct roots iff the two turning values have opposite signs: \(g(\sqrt p)\,g(-\sqrt p)<0\).

Full solution

(i) \(g'(x)=0\) at \(x=\pm\sqrt p\). \(g(\pm\sqrt p)=q\mp2p\sqrt p\). Three distinct roots iff these have opposite signs: \((q-2p\sqrt p)(q+2p\sqrt p)<0\), that is \(q^2<4p^3\).

(ii) \(q^2<4\): \(-2

(iii) A repeated root needs \(q^2=4\cdot27=108\), so \(q=6\sqrt3\). Then \(g(\sqrt3)=3\sqrt3-9\sqrt3+6\sqrt3=0\), so \(\sqrt3\) is the double root, and since the roots add to \(0\) the third is \(-2\sqrt3\).

Results: \(-2

2 more multiple-choice and 1 more longer question

A remainder from two values; A function of a function; A functional equation. Each with two hints and a full solution.

Included with A Level plans, and with IB, IGCSE or CBSE plans.

Next steps

Other areas: Calculus and graphs · Counting, sequences and number · Geometry and logic. Then: TMUA practice · STEP topic guides · Strategies

The MAT was set by the University of Oxford. A Level Math Revision is independent: it is not affiliated with or endorsed by the University of Oxford or any university. Every problem here is our own, written in the style of the MAT; for real papers use Oxford's own past papers page.