1. 1
   A landscape gardener is at a garden centre, working out algebraic expressions (in metres) for the side lengths of several rectangular flower beds before ordering edging stones.

   (a)

   Simplify fully: $5x + 3 - 2x + 7$

   [1 mark]

   • A$7x + 10$
   • B$3x + 10$
   • C$3x + 4$
   • D$7x + 4$

   (b)

   Expand and simplify: $4(2x - 3) + 5$

   [1 mark]

   • A$8x + 2$
   • B$8x - 17$
   • C$8x - 7$
   • D$8x + 17$

   (c)

   Factorise fully: $6x^2 + 9x$

   [1 mark]

   • A$x(6x + 9)$
   • B$3(2x^2 + 3x)$
   • C$3x(2x + 9)$
   • D$3x(2x + 3)$

   Total for question 1: 3 marks

2. 2
   Two builders, Builder A and Builder B, use algebraic expressions to represent their charges in pounds for small repair jobs, where $x$ represents the number of hours or hundreds of tiles involved.

   (a)

   Builder A's total charge, in pounds, is given by $3(x + 8) - 2(x - 5)$. Expand and simplify this expression fully.

   [2 marks]

   (b)

   Builder B's charge for materials, in pounds, is represented by $4x^2 - 9$, where $x$ is the number of tiles bought, in hundreds. Factorise this expression fully.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A framer is comparing the areas of square photo frames, using algebraic expressions for side lengths measured in centimetres, before cutting mount board to size.

   (a)

   One frame has side length $(x + 5)$ cm and another has side length $(x - 2)$ cm. Show that the difference between the two frames' areas, $(x+5)^2 - (x-2)^2$, simplifies to $14x + 21$.

   [3 marks]

   (b)

   A third frame has an area, in cm², given by $2x^2 + 7x - 15$. Factorise this expression fully.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   An engineer is modelling the cross-sectional area of a metal bracket using algebraic expressions in $x$, measured in millimetres, before the design is submitted for manufacture.

   (a)

   The bracket's profile has three sections with lengths $(x+2)$, $(x-3)$ and $(x+1)$. Expand and simplify $(x+2)(x-3)(x+1)$ fully.

   [4 marks]

   (b)

   Show that $3(2x-1)^2 - (6x-1)(2x+3)$ simplifies to an expression of the form $ax + b$, stating the values of $a$ and $b$.

   [4 marks]

   (c)

   The cross-sectional area, in mm², of the bracket is given by $4x^3 - 4x^2 - 24x$. Factorise this expression fully, then use your answer to find the value of $x$, where $x > 0$, for which the area is zero.

   [5 marks]

   Total for question 4: 13 marks

End of questions