Precision | Oxford Maths Admissions Test Livestream

Precision

Part of the Oxford Maths Admissions Test Livestream 2026

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General Advice

We can use logic and proof to make an idea precise.

  • The TMUA Content Specification has sections on logic and proof, reproduced below.
  • For TMUA, if statements A and B are both true, then the statement "A or B" is considered to be true.
  • The statement "A if B" can be rewritten as "if B then A".
  • The statement "A only if B" can be rewritten as "if A then B".
  • The statement "A if and only B" can be rewritten as "if A then B, and if B then A".
  • The statement "A is necessary for B" can be rewritten as "if B then A".
  • The statement "A is sufficient for B" can be rewritten as "if A then B".
  • The statement "A is necessary and sufficient for B" can be rewritten as "if A then B, and if B then A".

 

TMUA Specification (April 2025, Section 2 § "The Logic of Arguments")

  • Understand and be able to use mathematical logic in simple situations:

    • The terms true and false;
    • The terms and, or (meaning inclusive or), not;
    • Statements of the form:
      if A then B,
      A if B,
      A only if B,
      A if and only if B.
    • The converse of a statement;
    • The contrapositive of a statement;
    • The relationship between the truth of a statement and its converse and its contrapositive.

    Note: candidates will not be expected to recognise or use symbolic notation for any of these terms, nor will they be expected to complete formal truth tables.

  • Understand and use the terms necessary and sufficient.
  • Understand and use the terms for all, for some (meaning for at least one), and there exists.
  • Be able to negate statements that use any of the above terms.

 

TMUA Specification (April 2025, Section 2 § "Mathematical Proof")

  • Follow a proof of the following types, and in simple cases know how to construct such a proof:
    • Direct deductive proof ("Since A, therefore B, therefore C, ... , therefore Z, which is what we wanted to prove.");
    • Proof by cases (for example, by considering even and odd cases separately);
    • Proof by contradiction;
    • Disproof by counterexample.
  • Deduce implications from given statements.
  • Make conjectures based on small cases, and then justify these conjectures.
  • Rearrange a sequence of statements into the correct order to give a proof for a statement.
  • Problems requiring a sophisticated chain of reasoning to solve.

 

TMUA Specification (April 2025, Section 2 § "Identifying Errors in Proofs")

  • Identifying errors in purported proofs.
  • Be aware of common mathematical errors in purported proofs; for example, claiming "if $ab=ac$, then $b=c$", or assuming "if $\sin A = \sin B$, then $A=B$" neither of which are valid deductions.

 

Warm-up

What's wrong with the following claims?

  • if $ab=ac$, then $b=c$.
  • if $\sin A = \sin B$, then $A=B$.
  • if $\sqrt{x^2}=\sqrt{y^2}$ then $x=y$.
  • for all real $x$, if $x^2\gt x$ then $x\gt 1$.
  • the equation $a(x-p)(x-q)=0$ has two real solutions for $x$.
  • if $b^2-4ac\gt 0$ then $ax^2+bx+c=0$ has two real solutions for $x$.
  • for all integers $N$, if $N^2$ is a multiple of 100 then $N$ is a multiple of 100.
  • for all integers $n$, the integer $n^2+n+41$ is prime.
  • for all positive real numbers $a$ and for all $x$ and $y$, if $a^x=a^y$ then $x=y$.

 

Questions

TMUA 2021 Paper 2 Question 10

The first seven terms of a sequence of positive integers are: \[ u_1 = 15,\qquad u_2 = 21,\qquad u_3 = 30,\qquad u_4 = 37,\qquad u_5 = 44,\qquad u_6 = 51,\qquad u_7 = 59. \] Consider the following statement about this sequence:

    $(*)$ If $n$ is a prime number, then $u_n$ is a multiple of 3 or $u_n$ is a multiple of 5.

What is the smallest value of $n$ that provides a counterexample to $(*)$?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
(F) 6
(G) 7

 

TMUA 2022 Paper 2 Question 5

A straight line $L$ passes through $(1, 2)$.

Let P be the statement

    if the $y$-intercept of $L$ is negative, then the $x$-intercept of $L$ is positive.

Which of the following statements must be true?

I P
II the converse of P
III the contrapositive of P

(A) none of them
(B) I only
(C) II only
(D) III only
(E) I and II only
(F) I and III only
(G) II and III only
(H) I, II and III

 

TMUA 2021 Paper 2 Question 5

On which line is the first error in the following argument?

(A) $\sin^2 x + \cos^2 x = 1$ for all values of $x$.
(B) Therefore $\cos x = \sqrt{1 - \sin^2 x}$ for all values of $x$.
(C) Hence $1 + \cos x = 1 + \sqrt{1 - \sin^2 x}$ for all values of $x$.
(D) Thus $(1 + \cos x)^2 = \left(1 + \sqrt{1 - \sin^2 x}\right)^2$ for all values of $x$.
(E) Substituting $x = \pi$ gives $0 = 4$.

 

TMUA 2022 Paper 2 Question 6

A list consists of $n$ integers. Consider the following statements:

    P: $n$ is odd.
    Q: The median of the list is one of the numbers in the list.

Which one of the following is true?

(A) P is necessary and sufficient for Q.
(B) P is necessary but not sufficient for Q.
(C) P is sufficient but not necessary for Q.
(D) P is not necessary and not sufficient for Q.

 

MAT 2021 Q1J

Four distinct real numbers $a$, $b$, $c$, and $d$ are used to define four points \[ A=(a,b), \quad B=(b,c),\quad C=(c,d),\quad D=(d,a). \] The quadrilateral $ABCD$ has all four sides the same length

(a) if and only if $(a-b)^2=(c-d)^2$,
(b) if and only if $(a-c)^2=(b-d)^2$,
(c) if and only if $(a-d)^2=(b-c)^2$,
(d) if and only if $a-b+c-d=0$,
(e) for no values of $a$, $b$, $c$, $d$.

 

Part of an Interview

Adapted from an interview question used by James Munro for Maths interviews at Oxford. Reproduced here with permission.

I'd like my computer to calculate the sum $1+2+3+4+5.$

My computer can only add two numbers at a time, and it's expensive! The cost for replacing $a+b$ with $c$ is £$a\times b \times c$.

For example, The sum 2 + 3 is being replaced with the number 5. costs £30, and I'm not done; my sum is now $1+5+4+5$.

The calculation ends when there's only a single number.

There are lots of ways that we could order the operations.

What's the minimum total cost?

 

 

(If you have a theory but you can't prove it, it can help to look at cases that are simpler but more general; try experimenting with the different ways to get my computer to calculate the sum $a+b+c$.)

(It's also a good idea to try to understand $a^2 b + ab^2$. Where have you seen terms like that?)

 

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