Pure: ProofEdexcel A-Level Maths: Topic test

1. 1
   Four consecutive integers are written as $n$, $n+1$, $n+2$ and $n+3$, where $n$ is an integer.

   (a)

   Which expression gives the sum of the four integers?

   [1 mark]

   • A$4n+3$
   • B$4n+6$
   • C$4n+10$
   • D$4n^2+6$

   (b)

   Which statement about the sum is true for every integer $n$?

   [1 mark]

   • AThe sum is always a multiple of 4.
   • BThe sum is always odd.
   • CThe sum is always even but never a multiple of 4.
   • DThe sum is always a multiple of 3.

   (c)

   Prove that $n^2+(n+1)^2+(n+2)^2+(n+3)^2$ is even for every integer $n$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Every integer $m$ is either even or odd, so it can be written as $m=2k$ or $m=2k+1$ for some integer $k$.

   (a)

   Which expression is equal to $(2k+1)^2$?

   [1 mark]

   • A$4k^2+4k+1$
   • B$4k^2+1$
   • C$2k^2+2k+1$
   • D$4k^2+2k+1$

   (b)

   Why is it enough to consider the two cases $m=2k$ and $m=2k+1$ when proving a result about all integers $m$?

   [1 mark]

   • ABecause each case is a typical example that is likely to work for all integers.
   • BBecause the two cases are the only values that $k$ can take.
   • CBecause a result that works for $m=2$ and $m=3$ works for every integer.
   • DBecause every integer falls into exactly one of the two cases, so all possible cases are covered.

   (c)

   Prove that the square of any integer leaves a remainder of 0 or 1 when divided by 4.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student wants to prove that there are no integers $a$ and $b$ such that $a^2-b^2=10$.

   (a)

   The student begins a proof by contradiction by assuming that such integers exist. Show that $a+b$ and $a-b$ are either both odd or both even.

   [3 marks]

   (b)

   Hence complete the proof by contradiction.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $n$ be a positive integer and let $T=n^3-n$.

   (a)

   Prove that $T$ is divisible by 6 for every positive integer $n$.

   [6 marks]

   (b)

   Use the result in part (a) to prove by contradiction that $T$ is never a prime number.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A student makes the claim: 'For all real numbers $x$ and $y$, if $x^2=y^2$ then $x=y$.'

   (a)

   Which pair of values is a counter-example to the claim?

   [1 mark]

   • A$x=3,\ y=3$
   • B$x=0,\ y=0$
   • C$x=2,\ y=-2$
   • D$x=2,\ y=4$

   (b)

   Which statement is the negation of the claim?

   [1 mark]

   • AFor all real $x$ and $y$, $x^2=y^2$ and $x\neq y$.
   • BThere exist real $x$ and $y$ such that $x^2=y^2$ and $x\neq y$.
   • CThere exist real $x$ and $y$ such that $x^2\neq y^2$ and $x=y$.
   • DFor all real $x$ and $y$, if $x^2\neq y^2$ then $x\neq y$.

   (c)

   Prove that the claim is true when $x$ and $y$ are both positive.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   It is known that $\sqrt{2}$ is irrational. A student attempts to prove, by contradiction, that $3+\sqrt{2}$ is irrational.

   (a)

   Which statement should the proof begin with?

   [1 mark]

   • AAssume that $3+\sqrt{2}$ is rational.
   • BAssume that $3+\sqrt{2}$ is irrational.
   • CAssume that $\sqrt{2}$ is rational and that $3$ is irrational.
   • DAssume that $3$ is irrational.

   (b)

   The student writes $3+\sqrt{2}=\frac{a}{b}$, where $a$ and $b$ are integers and $b\neq0$. Which expression for $\sqrt{2}$ follows?

   [1 mark]

   • A$\frac{a-3}{b}$
   • B$\frac{a+3b}{b}$
   • C$\frac{a}{3b}$
   • D$\frac{a-3b}{b}$

   (c)

   Complete the proof.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A student claims that $p^2+2$ is a multiple of 3 for every prime number $p$ with $p>3$.

   (a)

   Use proof by exhaustion to show that the claim is true for every prime $p$ with $3<p<20$.

   [3 marks]

   (b)

   Prove that the claim is true for all primes $p>3$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A Pythagorean triple is a set of three positive integers $a$, $b$ and $c$ such that $a^2+b^2=c^2$.

   (a)

   Let $m$ and $n$ be integers with $m>n>0$. Prove by deduction that $m^2-n^2$, $2mn$ and $m^2+n^2$ form a Pythagorean triple.

   [6 marks]

   (b)

   Prove by contradiction that $a$, $b$ and $c$ cannot all be odd.

   [6 marks]

   Total for question 8: 12 marks

End of questions