Proof by Contradiction (Edexcel A Level Maths: Pure): Exam Questions

Given that $p$ and $q$ are integers such that

$pq$ is even

use algebra to prove by contradiction that at least one of $p$ or $q$ is even.

Given that $m^3+5$ is odd, use proof by contradiction to show, using algebra, that $m$ is even.

Prove by contradiction that there are an infinite number of positive even numbers.

A student is attempting to answer the following exam question:

“Prove by contradiction that $\sqrt{2}$ is an irrational number. You may use without proof the fact that if a number $n^2$ is even, then $n$ must also be even.”

The student’s proof is as follows:

There is an error within the first three lines of the proof.

Find the error and write down the correct line of the proof.

Line 4 of the proof is missing.

Complete this line of the proof.

Given that $x^3$ is odd, use proof by contradiction to show, using algebra, that $x$ is odd.

$x$ and $y$ are two real numbers such that $x+y$ is an irrational number.

Prove by contradiction that $x$ or $y$ or both are irrational numbers.

Prove by contradiction that a triangle cannot have more than one obtuse angle.

Use proof by contradiction to show that $\sqrt{3}$ is an irrational number.

A student attempts to answer the following question:

The student starts the proof with:

Complete the proof.

Prove by contradiction that $\sqrt{11}$ is an irrational number. 

You may use without proof the fact that if $n^2$ is a multiple of 11, then $n$ is a multiple of 11.

Given that $x^n$ is odd where $n$ is a positive integer, use proof by contradiction to show, using algebra, that $x$ is odd.

Prove by contradiction that there are an infinite number of positive powers of 2.

Below is a proof by contradiction that ${\log }_{2}7$ is irrational.

Lines 4 is missing.

Complete this line of the proof.

Use proof by contradiction to show that, given a rational number $x$ and an irrational number $y$, $x+y$ is irrational.

Prove by contradiction that there are no positive integers $x$ and $y$ such that

$9x^2-y^2=13$

Prove by contradiction that there are no positive integers $p$ and $q$ such that

$4p^2-q^2=25$

Prove by contradiction that there are an infinite number of prime numbers.

Without solving the equation directly, use algebra to prove by contradiction that the solutions to the equation 

$3x^2+10x-8=0$

cannot be written in the form $x=\frac{a}{b}$ where $a$ and $b$ are both odd integers.

A composite number, $N$, has the following properties:

• It is a positive integer greater than 1

• It is not a prime number

• It has at least two prime factors

• It can be written as a product of its prime factors, $p_1$, $p_2$, ..., $p_k$:

$N=p_1\times p_2\times ...\times p_k$

where $k\ge 2$.

Prove by contradiction that any composite number, $N$, must have at least one prime factor that is less than or equal to $\sqrt{N}$.