MAT-style geometry and logic questions

Coordinate geometry of lines and circles, plus logic: negating statements, truth-tellers and liars, and games where you work backwards. 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 chord on the axis · multiple choice

The circle with equation \(x^2+y^2=10x-4y\) cuts the \(x\)-axis twice. What is the length of the chord between those two points?

  1. \(5\)
  2. \(8\)
  3. \(10\)
  4. \(12\)
Hint 1

Points on the \(x\)-axis have \(y=0\).

Hint 2

Notice that the origin satisfies the equation.

Full solution

Answer: C: \(10\)

With \(y=0\): \(x^2=10x\), so \(x=0\) or \(x=10\). The chord has length \(10\).

Question 2: Negating a statement · multiple choice

Which statement is the negation of “every student in the class solved at least one problem”?

  1. Every student in the class solved no problems.
  2. Some student in the class solved no problems.
  3. Some student in the class solved at least one problem.
  4. No student in the class solved every problem.
Hint 1

The negation of “for all \(x\), \(P(x)\)” is “there exists \(x\) with not \(P(x)\)”.

Hint 2

“Not at least one” means “none”.

Full solution

Answer: B: Some student in the class solved no problems.

Negating “for every student, at least one problem solved” gives “for some student, no problem solved”.

Question 3: Two touching circles · longer question

The circles \(C_1: x^2+y^2=1\) and \(C_2: (x-3)^2+y^2=4\) are given.

  1. Show that the circles touch, and find the point where they touch.
  2. How many lines are tangent to both circles?
  3. Find the equations of the common tangents that do not pass through the point of contact, and the point where they meet.
Hint 1

Compare the distance between the centres with the sum of the radii.

Hint 2

Write a tangent as \(y=mx+c\): its distance from each centre equals that circle’s radius.

Full solution

(i) The centres are \(3\) apart and \(1+2=3\), so they touch externally, at \((1,0)\).

(ii) Three: two outer tangents and the common tangent \(x=1\) at the point of contact.

(iii) \(\dfrac{|c|}{\sqrt{1+m^2}}=1\) and \(\dfrac{|3m+c|}{\sqrt{1+m^2}}=2\). For an outer tangent both centres are on the same side, so \(3m+c=2c\), \(c=3m\). Then \(9m^2=1+m^2\): \(m=\pm\tfrac{\sqrt2}4\), \(c=\pm\tfrac{3\sqrt2}4\). The tangents \(y=\pm\tfrac{\sqrt2}4(x+3)\) meet at \((-3,0)\).

Results: Touch at \((1,0)\); three common tangents; \(y=\pm\frac{\sqrt2}4(x+3)\), meeting at \((-3,0)\).

2 more multiple-choice and 1 more longer question

Distance to a line; Who is telling the truth?; A take-away game. Each with two hints and a full solution.

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

Next steps

Other areas: Algebra and functions · Calculus and graphs · Counting, sequences and number. 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.