1. 1
   Let $n$ be an integer. Three consecutive integers can be written as $n-1$, $n$ and $n+1$.

   (a)

   Find a simplified expression for the sum of the three consecutive integers.

   [1 mark]

   • A$3n$
   • B$3n+3$
   • C$3n-1$
   • D$n^3-n$

   (b)

   Find the expanded form of the product $(n-1)n(n+1)$.

   [1 mark]

   • A$n^3+n$
   • B$n^3-n$
   • C$n^3-1$
   • D$n^3-3n$

   (c)

   Show that the sum of the squares of the three consecutive integers is always 2 more than a multiple of 3.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the expression $f(x) = (2x-5)^2 - 4(x-1)(x-4)$, where $x\in\mathbb{R}$.

   (a)

   Find the value of $f(0)$.

   [1 mark]

   • A$41$
   • B$25$
   • C$9$
   • D$-16$

   (b)

   Which of the following statements about $f(x)$ is correct?

   [1 mark]

   • A$f(x) = 9$ only when $x = 0$
   • B$f(x) \equiv 8x^2 - 40x + 41$
   • C$f(x) \equiv -9$
   • D$f(x) \equiv 9$

   (c)

   Prove that $f(x)\equiv 9$, setting out your working as a left-hand side to right-hand side proof.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A magician asks an audience member to think of any number $x$, then to add 3, multiply the result by 4, subtract 8, divide by 2, and finally subtract twice the number they first thought of. The magician then announces the final result without being told the starting number.

   (a)

   Show that the final result is always 2.

   [3 marks]

   (b)

   The magician changes the instruction “subtract 8” to “subtract $k$”, where $k$ is a constant, and keeps every other step the same. Find the value of $k$ for which the final result is always 0, and check your answer using a starting number of your choice.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $m$ be an integer, so that $2m+1$ and $2m+3$ are consecutive odd integers. Define $D = (2m+3)^2 - (2m+1)^2$ and $P = (2m+1)(2m+3)$.

   (a)

   (i) Show that $D = 8(m+1)$.
   (ii) Hence prove that $D$ is always a multiple of 8, and deduce that $D$ is a multiple of 16 whenever $m$ is odd.

   [6 marks]

   (b)

   (i) Show that $P + 1 \equiv (2m+2)^2$.
   (ii) Hence deduce that the product of any two consecutive odd integers is one less than a multiple of 4.
   (iii) Check the identity in (i) when $m = 4$.

   [6 marks]

   Total for question 4: 12 marks

End of questions