Proof by contradictionEdexcel A-Level Maths: Subtopic test

1. 1
   A student begins a proof that $\sqrt{2}$ is irrational by assuming that $\sqrt{2}=\frac{a}{b}$, where $a$ and $b$ are integers and $b\neq0$.

   (a)

   What further assumption must be made at the start of the proof?

   [1 mark]

   • A$a$ and $b$ are both even.
   • B$b=1$.
   • C$a>b$.
   • D$a$ and $b$ have no common factor other than 1, so $\frac{a}{b}$ is in its lowest terms.

   (b)

   Squaring gives $a^2=2b^2$. What can be deduced immediately?

   [1 mark]

   • A$a^2$ is even, so $a$ is even.
   • B$b$ is even.
   • C$a=2b$.
   • D$a$ is odd.

   (c)

   Given that $a^2=2b^2$ and that $a$ is even, complete the proof that $\sqrt{2}$ is irrational.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Euclid's proof that there are infinitely many primes begins by assuming that there is a finite number of primes, $p_1,p_2,\ldots,p_n$, and then considers $N=p_1p_2\cdots p_n+1$.

   (a)

   What is the remainder when $N$ is divided by any one of the primes $p_i$?

   [1 mark]

   • A$0$
   • B$1$
   • C$p_i$
   • D$p_i-1$

   (b)

   Which is the correct deduction about $N$?

   [1 mark]

   • A$N$ is divisible by $p_1$.
   • B$N$ is a multiple of every prime in the list.
   • CEither $N$ is prime or it has a prime factor that is not in the list.
   • D$N$ must be a square number.

   (c)

   Complete the proof that there are infinitely many primes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Assume, for a contradiction, that $\log_2 3=\frac{p}{q}$, where $p$ and $q$ are positive integers.

   (a)

   Show that $3^q=2^p$.

   [3 marks]

   (b)

   Hence complete the proof that $\log_2 3$ is irrational.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   You may use the fact that a rational number can be written as $\frac{m}{n}$ where $m$ and $n$ are integers and $n\neq0$.

   (a)

   Prove by contradiction that $\sqrt{3}$ is irrational.

   [6 marks]

   (b)

   Prove by contradiction that the sum of a rational number and an irrational number is irrational. Hence show that $2+\sqrt{3}$ is irrational.

   [6 marks]

   Total for question 4: 12 marks

End of questions