Proof by contradictionAQA A-Level Maths: Subtopic test

1. 1
   A student wishes to prove by contradiction that $\sqrt3$ is irrational.

   (a)

   Which of these is the correct first line of the proof?

   [1 mark]

   • AAssume that $\sqrt3$ is irrational.
   • BAssume that $\sqrt3=\frac{p}{q}$, where $p$ and $q$ are integers with $q\neq0$ and no common factor greater than $1$.
   • CAssume that $\sqrt3$ is an integer.
   • DAssume that $\sqrt3=\frac{p}{q}$, where $p$ and $q$ are both multiples of $3$.

   (b)

   Squaring $\sqrt3=\frac{p}{q}$ gives $p^2=3q^2$. What can be deduced about $p$?

   [1 mark]

   • A$p$ is a multiple of $3$.
   • B$q$ is a multiple of $3$.
   • C$p$ is even.
   • D$p=3$.

   (c)

   Given that $p^2=3q^2$ and $p$ is a multiple of $3$, complete the proof that $\sqrt3$ is irrational.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Claim: there are no positive integers $a$ and $b$ such that $a^2-b^2=1$.

   (a)

   Which is the correct first line of a proof by contradiction of the claim?

   [1 mark]

   • AAssume that $a^2-b^2\neq1$ for all positive integers $a$ and $b$.
   • BAssume that $a=b+1$.
   • CAssume that $a$ and $b$ are both even.
   • DAssume that there exist positive integers $a$ and $b$ such that $a^2-b^2=1$.

   (b)

   Given that $(a-b)(a+b)=1$ with $a$ and $b$ positive integers, which statement must be true?

   [1 mark]

   • A$a-b=-1$ and $a+b=-1$
   • B$a-b=2$ and $a+b=\frac12$
   • C$a-b=1$ and $a+b=1$
   • D$a-b=0$ and $a+b=1$

   (c)

   Complete the proof that no such integers exist.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Euclid's proof that there are infinitely many prime numbers begins by assuming that there are only finitely many, say $p_1,p_2,\ldots,p_n$, and then considers the number $N=p_1p_2\cdots p_n+1$.

   (a)

   Show that $N$ leaves remainder $1$ when divided by each of $p_1,p_2,\ldots,p_n$, and state what this means for the divisibility of $N$.

   [3 marks]

   (b)

   Hence complete the proof by contradiction that there are infinitely many primes.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question a rational number is one that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b\neq0$; any other real number is irrational. You may quote that $\sqrt2$ is irrational.

   (a)

   Prove by contradiction that $\log_2 3$ is irrational.

   [6 marks]

   (b)

   Prove by contradiction that $3+2\sqrt2$ is irrational.

   [6 marks]

   Total for question 4: 12 marks

End of questions