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Practice · Oxbridge maths · Timed test

Oxbridge maths practice test

Ten MAT multiple-choice questions, a full 20-question TMUA paper, or four STEP problems, against the clock. Worked solutions for every question at the end.

Choose a test

Opens in its own window, laid out like a computer-based admissions test: one timer, previous and next, flag for review, no calculator. STEP answers accept exact forms such as pi/4, ln(2)/2 or sqrt(3).

What this test contains

Ten MAT multiple-choice questions, a full 20-question TMUA paper, or four STEP problems, against the clock. Worked solutions for every question at the end.

Oxford MAT

Multiple choice at the level of the Mathematics Admissions Test: algebra, calculus, sequences, trigonometry and counting.

40 questions · 10 questions · 50 min →

TMUA

Both papers: applied mathematical knowledge, and logic and proof, including contrapositives, counterexamples and necessary and sufficient conditions.

30 questions · 20 questions · 75 min →

STEP

Longer problems at STEP 2 and STEP 3 level, each with a single final answer and a full method.

20 questions · 4 questions · 60 min →

Three questions from this test

Taken from the banks above, with the worked answers, so you can see the level before you start the clock.

1 Easy MAT
What is the minimum value of x2−6x+13x^2-6x+13 for real xx?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 9
Show answer
Answer: C. 4
Complete the square: (x−3)2+4(x-3)^2+4. The square is at least 0, so the minimum is 4, at x=3x=3.
31 Medium MAT
How many real solutions does 2x=x+22^{x}=x+2 have?
  1. 0
  2. 1
  3. 2
  4. 3
  5. Infinitely many
Show answer
Answer: C. 2
Let f(x)=2x−x−2f(x)=2^x-x-2. f(−2)=0.25>0f(-2)=0.25>0, f(0)=−1<0f(0)=-1<0 and f(2)=0f(2)=0, and ff is convex, so the curve meets the line exactly twice: once between −2-2 and 0, and again at x=2x=2.
21 Medium TMUA Paper 2
Which statement is logically equivalent to "PP only if QQ"?
  1. Q⇒PQ\Rightarrow P
  2. P⇒QP\Rightarrow Q
  3. P⇔QP\Leftrightarrow Q
  4. (not PP) ⇒\Rightarrow (not QQ)
  5. QQ only if PP
Show answer
Answer: B. P⇒QP\Rightarrow Q
"PP only if QQ" says PP cannot hold without QQ, which is P⇒QP\Rightarrow Q. "PP if QQ" is the other direction.

How it is scored

Every question is marked right or wrong as you go; there is no partial credit, exactly as in the real thing. At the end you get your score, the time you used against the time allowed, your average per question and your longest run of correct answers. Every question then comes back with your answer, the correct one and a full worked solution.

Browse every question with its worked solution on the Oxbridge maths practice page. Preparing for the real thing? Oxbridge and US STEM admissions.