1. 1
   A carpenter is designing a rectangular tabletop. Its length is represented by the expression $2x+5$ and its width is $3x$, both measured in centimetres. Two other tabletops have side lengths $x+8$, $x-5$, and $x+2$, $x+3$, $x-1$.

   (a)

   Expand $3x(2x+5)$.

   [1 mark]

   • A$6x+15$
   • B$6x^2+5x$
   • C$5x^2+8x$
   • D$6x^2+15x$

   (b)

   The area of a second tabletop with sides $(x+8)$ and $(x-5)$ is found by expanding $(x+8)(x-5)$. Which expression gives the simplified area?

   [1 mark]

   • A$x^2+3x-40$
   • B$x^2-3x-40$
   • C$x^2+13x-40$
   • D$x^2+3x+40$

   (c)

   A third tabletop's volume (for a solid block) is found from $(x+2)(x+3)(x-1)$. What is the coefficient of $x^2$ in the fully expanded expression?

   [1 mark]

   • A$2$
   • B$4$
   • C$5$
   • D$6$

   Total for question 1: 3 marks

2. 2
   A gardener is planning two triangular flower beds side by side. The first bed's area (before subtracting a path) is modelled by $4y(3y-2)$. The combined perimeter expression for both beds is $2(3y-2)+2(5y+1)$.

   (a)

   A gardener is fencing a plot. Expand and simplify $4y(3y-2)$.

   [2 marks]

   (b)

   The combined perimeter of both beds is $2(3y-2)+2(5y+1)$. Expand and simplify this expression fully.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bakery models its weekly profit in dollars as $(2n-3)(n+7)$, where $n$ is the number of cakes sold, in hundreds. The bakery also compares this to a rival formula $(n-3)^2$ representing a simpler stall's profit.

   (a)

   A bakery models weekly profit in dollars as $(2n-3)(n+7)$, where $n$ is the number of cakes sold, in hundreds. Expand and fully simplify this expression.

   [3 marks]

   (b)

   The rival stall's profit is modelled as $(n-3)^2$. Expand this fully and show that it can be written as $n^2-6n+9$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A packaging designer creates an open box by cutting a square of side $x$ from each corner of a rectangular sheet measuring 20 cm by 15 cm, then folding up the sides. She also designs a second box using the expression $(x+4)(x-4)(2x+1)$ to model a different volume, and a third comparison uses $(3x-2)^2 - (2x+1)^2$.

   (a)

   A packaging designer models the volume of an open box, formed by cutting squares of side $x$ from the corners of a rectangular sheet, as $x(20-2x)(15-2x)$. Expand this expression fully to show it can be written as $4x^3-70x^2+300x$.

   [4 marks]

   (b)

   For a second box design, expand and fully simplify $(x+4)(x-4)(2x+1)$.

   [4 marks]

   (c)

   For a third comparison, expand $(3x-2)^2 - (2x+1)^2$ fully and simplify your answer as much as possible.

   [5 marks]

   Total for question 4: 13 marks

End of questions