1. 1
   A phone repair shop charges a call-out fee of £8 plus $c$ pounds for every minute of labour, where a repair takes $m$ minutes.

   (a)

   Which expression gives the cost, in pounds, of a repair taking $m$ minutes?

   [1 mark]

   • A$8 + cm$
   • B$8c + m$
   • C$8cm$
   • D$(8 + c)m$

   (b)

   Which of the following best describes the relationship $8 + cm = \text{Total Cost}$, given that it is true only for one specific value of $m$ when the total cost is fixed at $32$?

   [1 mark]

   • AIt is an identity
   • BIt is a formula with no variables
   • CIt is an equation
   • DIt is an expression

   (c)

   If $c = 3$, simplify the expression for the total cost, $8 + 3m$, by collecting like terms with another repair that costs $8 + 3m + 2m$ for a longer job with extra labour minutes $m$ added twice.

   [1 mark]

   • A$16 + 5m$
   • B$10 + 5m$
   • C$8 + 6m$
   • D$8 + 5m$

   Total for question 1: 3 marks

2. 2
   A gardener is designing a rectangular flower bed. One side has length $x + 4$ metres and the perpendicular side has length $3x$ metres, so the area, $A$, in square metres, is given by $A = 3x(x + 4)$.

   (a)

   Write down the term in $x$ that results from expanding $3x(x + 4)$, and simplify the full expansion.

   [2 marks]

   (b)

   The gardener increases both dimensions so the area becomes $A = (x + 4)(2x + 3)$. Expand and fully simplify this expression for $A$ in terms of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory manager models the number of faulty items produced per batch, $F$, using algebraic expressions in terms of the batch size, $n$, measured in hundreds of items.

   (a)

   A factory manager writes the number of faulty items produced per batch as $n^2 - 5n + 6$, where $n$ is the batch size in hundreds. Factorise this expression fully.

   [3 marks]

   (b)

   For a different production line, the manager models faults using $4n^2 - 9$. Show that this expression can be written as $(2n-3)(2n+3)$, and state which factorising technique you have used.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A maths teacher sets a worksheet on expressions, equations and identities to check whether her class understands the difference between simplifying, solving and proving.

   (a)

   A teacher asks two students to simplify the expression $5(2x - 3) - 2(x - 4)$. Show full working to fully expand and simplify this expression, giving your answer in the form $ax + b$.

   [4 marks]

   (b)

   One student claims that $3(x + 2) + 2(x - 1) = 5x + 4$ is an identity, true for every value of $x$. Verify this claim by expanding and simplifying the left-hand side, and state whether the student is correct, explaining your reasoning.

   [4 marks]

   (c)

   The teacher then asks the class to prove that the sum of any three consecutive integers is always a multiple of $3$. Using $n$, $n + 1$ and $n + 2$ to represent the three consecutive integers, prove this result using algebra, showing every step clearly.

   [5 marks]

   Total for question 4: 13 marks

End of questions