1. 1
   A firework's height above ground in metres, $h$, at time $t$ seconds after two different launches is modelled by $h=-x^2-2x+15$ type expressions rearranged into equations. The first launch satisfies $x^2+2x-15=0$ for a related horizontal distance $x$. The second satisfies $2x^2-3x+1=0$. A third scenario uses the quadratic formula on $3x^2+5x-2=0$.

   (a)

   Solve $x^2+2x-15=0$ by factorisation.

   [1 mark]

   • A$x=3$ or $x=-5$
   • B$x=-3$ or $x=5$
   • C$x=5$ or $x=-15$
   • D$x=3$ or $x=15$

   (b)

   Solve $2x^2-3x+1=0$ by factorisation.

   [1 mark]

   • A$x=1$ or $x=2$
   • B$x=-1$ or $x=-0.5$
   • C$x=1$ or $x=0.5$
   • D$x=0.5$ or $x=-1$

   (c)

   Using the quadratic formula on $3x^2+5x-2=0$, what is the positive solution for $x$?

   [1 mark]

   • A$2$
   • B$\frac{1}{3}$
   • C$\frac{2}{3}$
   • D$1$

   Total for question 1: 3 marks

2. 2
   A ball is thrown into the air. Its height above the ground satisfies $t^2-6t+8=0$ at ground level, where $t$ is time in seconds. A second throw's ground-level times satisfy $2t^2-7t+3=0$.

   (a)

   A ball is thrown so that its height in metres above the ground, $H$, after $t$ seconds satisfies $H=0$ when $t^2-6t+8=0$. Solve this equation by factorisation to find the two times at which the ball is at ground level.

   [2 marks]

   (b)

   For a second throw, solve $2t^2-7t+3=0$ by factorisation to find the two times the ball is at ground level.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangular garden has area 40 square metres, with length 3 metres more than its width, $x$. A second, unrelated garden has an equation for its area that cannot be factorised: $x^2+4x-7=0$.

   (a)

   A rectangular garden has area 40 square metres. Its length is 3 metres more than its width, $x$. Form a quadratic equation in $x$ and solve it by factorisation to find the width, rejecting any invalid solution.

   [3 marks]

   (b)

   A second garden's area equation is $x^2+4x-7=0$, which does not factorise using integers. Use the quadratic formula to solve for $x$, giving your answer to 2 decimal places.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A company's weekly profit in thousands of dollars from selling $x$ hundred units of a product is modelled by $P=-2x^2+10x-8$. The company wants to find the break-even points where $P=0$, and also wants to know the maximum profit and the number of units at which it occurs.

   (a)

   A company's weekly profit in thousands of dollars, $P$, from selling $x$ hundred units is modelled by $P=-2x^2+10x-8$. Form the equation for the break-even points (where $P=0$) and rearrange it into the form $2x^2-10x+8=0$, then simplify by dividing through by 2.

   [4 marks]

   (b)

   Solve $x^2-5x+4=0$ by factorisation to find the break-even values of $x$.

   [4 marks]

   (c)

   By using the fact that the maximum profit occurs midway between the two break-even values of $x$, find the value of $x$ at maximum profit, then calculate the maximum weekly profit in thousands of dollars using $P=-2x^2+10x-8$.

   [5 marks]

   Total for question 4: 13 marks

End of questions