Proof and mathematical argumentAQA A-Level Maths: Topic test

1. 1
   The function $g$ is defined by $g(x)=\dfrac{1}{x-2}$ for $x\in\mathbb{R}$, $x\neq2$.

   (a)

   Which statement gives the range of $g$?

   [1 mark]

   • A$g(x)\in\mathbb{R}$
   • B$g(x)\neq0$
   • C$g(x)\neq2$
   • D$g(x)>0$

   (b)

   Which is the solution of $g(x)=-\dfrac12$?

   [1 mark]

   • A$x=4$
   • B$x=-2$
   • C$x=-4$
   • D$x=0$

   (c)

   The function $h$ is defined by $h(x)=\sqrt{g(x)}$. Find the largest possible domain of $h$, using set notation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For real numbers $a$ and $b$, statement $P$ is '$a^2=b^2$' and statement $Q$ is '$a=b$'.

   (a)

   Which of the following is correct?

   [1 mark]

   • A$Q\Rightarrow P$, but $P\not\Rightarrow Q$
   • B$P\Rightarrow Q$, but $Q\not\Rightarrow P$
   • C$P\Leftrightarrow Q$
   • DNeither $P\Rightarrow Q$ nor $Q\Rightarrow P$

   (b)

   Which pair of values is a counter-example to the statement $P\Rightarrow Q$?

   [1 mark]

   • A$a=2,\ b=2$
   • B$a=3,\ b=5$
   • C$a=-4,\ b=4$
   • D$a=0,\ b=0$

   (c)

   Write down a statement $R$ about $a$ and $b$ such that $P\Leftrightarrow R$, and show by algebra that $P\Rightarrow R$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Throughout this question, $k$ is an integer, so $2k+1$ represents an odd integer.

   (a)

   Prove that the sum of any three consecutive odd integers is a multiple of 3.

   [3 marks]

   (b)

   Prove that the sum of the squares of any two consecutive odd integers is 2 more than a multiple of 8.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student makes two claims about positive integers $n$. Claim (i): $n^2+n+1$ is odd for every positive integer $n$. Claim (ii): $n^2+n+1$ is never divisible by 3.

   (a)

   Prove claim (i) by considering cases.

   [6 marks]

   (b)

   Show that claim (ii) is false. Then, by considering cases, prove that $n^2+n+1$ is divisible by 3 only when $n$ leaves remainder 1 on division by 3.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The universal set is $\mathbb{Z}^+$. Set $S$ is the set of prime numbers less than 20, and set $F$ is the set of factors of 30.

   (a)

   How many elements are in $S\cap F$?

   [1 mark]

   • A2
   • B4
   • C3
   • D8

   (b)

   What is $n(S\cup F)$?

   [1 mark]

   • A13
   • B16
   • C10
   • D3

   (c)

   List the elements of $S'\cap F$, where $S'$ is the complement of $S$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A student wants to show that the square of any integer leaves a remainder of 0 or 1 when divided by 4.

   (a)

   Which method gives a valid proof?

   [1 mark]

   • ASquare the integers 1, 2, 3 and 4 and observe the remainders
   • BSquare the odd integer $2k+1$ and show that the remainder is 1
   • CAssume the remainder is 2 and find a value that does not fit
   • DSquare the even integer $2k$ and the odd integer $2k+1$, and show that each gives remainder 0 or 1

   (b)

   The student then claims that the sum of the squares of two integers always leaves a remainder of 0 or 1 when divided by 4. Which pair of integers is a counter-example?

   [1 mark]

   • A2 and 4
   • B1 and 3
   • C3 and 4
   • D2 and 6

   (c)

   Use the student's result to show that there are no integers $x$ and $y$ such that $x^2+y^2=2023$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A student writes the following 'proof' that $2=1$. Let $a$ and $b$ be non-zero real numbers with $a=b$. Line 1: $a^2=ab$. Line 2: $a^2-b^2=ab-b^2$. Line 3: $(a-b)(a+b)=b(a-b)$. Line 4: $a+b=b$. Line 5: $2b=b$. Line 6: $2=1$.

   (a)

   Identify the error in the 'proof' and explain why it makes the argument invalid.

   [3 marks]

   (b)

   Now let $a$ and $b$ be any real numbers. Prove that $a^2+b^2\ge2ab$, and state the condition on $a$ and $b$ for equality to hold.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Consider the expression $n^2-1$, where $n$ is an integer greater than 1.

   (a)

   Prove that, for any prime number $p$ greater than 3, $p^2-1$ is divisible by 24.

   [6 marks]

   (b)

   A student makes two claims. Claim 1: if $n^2-1$ is divisible by 24, then $n$ is prime. Claim 2: for every odd integer $n>1$, $n^2-1$ is divisible by 24. Disprove each claim by a counter-example, and prove that $n^2-1$ is divisible by 8 for every odd integer $n$.

   [6 marks]

   Total for question 8: 12 marks

End of questions