1. 1
   A landscape gardener is designing a rectangular flower bed with a path.

   (a)

   The area of the bed in square metres is given by $3x(2x - 4y)$. Which expression is the fully expanded form?

   [1 mark]

   • A$6x^2 - 12xy$
   • B$6x^2 - 4xy$
   • C$5x^2 - 12xy$
   • D$6x - 12xy$

   (b)

   The gardener also calculates the area of a second bed given by $(2x+1)(x-4)$. Which is the fully expanded form?

   [1 mark]

   • A$2x^2 - 8x - 4$
   • B$2x^2 + 7x - 4$
   • C$2x^2 - 7x - 4$
   • D$2x^2 - 7x + 4$

   (c)

   The gardener writes the total path area as $9x^2 + 15xy$ and wants it fully factorised. Which is correct?

   [1 mark]

   • A$3x(3x + 15y)$
   • B$3x(3x + 5y)$
   • C$9x(x + 15y)$
   • D$x(9x + 15y)$

   Total for question 1: 3 marks

2. 2
   A picture framer cuts a rectangular mount with length $(x+5)$ cm and width $(x-2)$ cm.

   (a)

   Expand and simplify $(x+5)(x-2)$ to find an expression for the area.

   [2 marks]

   (b)

   The framer's offcut area is given by $6x^2 + 10x$ cm². Factorise this expression fully.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A carpenter models the volume of a box as the product of three linear brackets.

   (a)

   Expand $(x-2)(x+3)(2x+1)$ fully and simplify, showing your working.

   [3 marks]

   (b)

   The carpenter finds that the surface area of a related panel is given by $a^2x^2 - b^2y^2$ where $a=3$ and $b=2$, giving $9x^2 - 4y^2$. Factorise this expression as a difference of two squares.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   An architect models the shape of a bridge cable and support structure using quadratic and cubic expressions.

   (a)

   An architect models the height of a bridge cable using $ax^2 + bx + c$ where $a=2$, $b=7$, $c=3$, giving $2x^2 + 7x + 3$. Factorise this quadratic expression fully.

   [4 marks]

   (b)

   The architect also needs the expression $ax^3 + bx^2 + cx$ where $a=3$, $b=-6$, $c=-24$, giving $3x^3 - 6x^2 - 24x$. Factorise this expression fully.

   [4 marks]

   (c)

   To find the lowest point of the cable, the architect must complete the square for $2x^2 + 7x + 3$, writing it in the form $p(x+q)^2 + r$. Show full working and state $p$, $q$ and $r$.

   [5 marks]

   Total for question 4: 13 marks

End of questions