1. 1
   A landscaper is tiling a garden path. The total tile area in square centimetres is given by $8xy+12y^2$. Separately, a second contractor's foundation area is modelled by $x^2+7x+10$, and a third uses $6x^2-5x-6$.

   (a)

   Factorise fully $8xy+12y^2$.

   [1 mark]

   • A$2y(4x+6y)$
   • B$4(2xy+3y^2)$
   • C$4y(2x+3)$
   • D$4y(2x+3y)$

   (b)

   Factorise $x^2+7x+10$, representing the second contractor's foundation area.

   [1 mark]

   • A$(x+7)(x+10)$
   • B$(x+1)(x+10)$
   • C$(x-2)(x-5)$
   • D$(x+2)(x+5)$

   (c)

   Factorise the third contractor's expression $6x^2-5x-6$.

   [1 mark]

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

   Total for question 1: 3 marks

2. 2
   An engineer is designing metal brackets. The material used for one batch, in square millimetres, is $15p^2q-20pq^2$. A second batch requires an amount modelled by $x^2-9x+20$.

   (a)

   Factorise fully $15p^2q - 20pq^2$.

   [2 marks]

   (b)

   Factorise $x^2-9x+20$ for the second batch.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A charity is planning a fundraising campaign. Its target amount in dollars is modelled by $2x^2+x-15$. A second stage of the campaign models leftover costs using the algebraic fraction $\frac{x^2-4}{x^2+5x+6}$.

   (a)

   A charity's fundraising target in dollars is modelled by $2x^2+x-15$. Factorise this expression fully.

   [3 marks]

   (b)

   Simplify the leftover-costs fraction $\frac{x^2-4}{x^2+5x+6}$ fully by factorising the numerator and denominator.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A construction firm is planning a rectangular plot of land with area, in square metres, modelled by $3x^2+11x-4$, and length $(3x-1)$ metres. The firm also compares this to a second, larger plot modelled by $4x^2-25$, and to a third irregular plot requiring an algebraic fraction to be simplified: $\frac{2x^2+7x+3}{4x^2-1}$.

   (a)

   A construction firm models the area of a rectangular plot, in square metres, as $3x^2+11x-4$. Factorise this expression fully and use it to state the width if the length is $(3x-1)$.

   [4 marks]

   (b)

   The second, larger plot has area $4x^2-25$ square metres. Factorise this fully and explain, with reasoning, why this expression can be factorised using the difference of two squares.

   [4 marks]

   (c)

   For the third irregular plot, simplify $\frac{2x^2+7x+3}{4x^2-1}$ fully, showing all factorisation steps.

   [5 marks]

   Total for question 4: 13 marks

End of questions