1. 1
   A gardener designs rectangular plots whose dimensions satisfy various quadratic equations.

   (a)

   A gardener designs a rectangular plot whose area in square metres satisfies $x^2 - 5x - 24 = 0$, where $x$ is the plot's width. Solve by factorisation to find the positive value of $x$.

   [1 mark]

   • A$x = 8$
   • B$x = 6$
   • C$x = 3$
   • D$x = 24$

   (b)

   A second plot satisfies $2x^2 + x - 6 = 0$. Solve by factorisation to find the positive value of $x$.

   [1 mark]

   • A$x = 0.5$
   • B$x = 2$
   • C$x = 3$
   • D$x = 1.5$

   (c)

   A third plot satisfies $x^2 - 9 = 0$. Which pair of solutions is correct?

   [1 mark]

   • A$x = 9$ or $x = -9$
   • B$x = 3$ or $x = -3$
   • C$x = 3$ only
   • D$x = -3$ only

   Total for question 1: 3 marks

2. 2
   An architect models arch heights using quadratic equations that must be solved by completing the square.

   (a)

   An architect models the height of an arch using $x^2 + 6x + 5 = 0$. Solve this equation by completing the square, showing full working.

   [2 marks]

   (b)

   A second arch design satisfies $x^2 - 4x - 1 = 0$. Solve this equation by completing the square, giving your answer in surd form.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An engineer analyses a component's trajectory using a quadratic model combined with a linear constraint.

   (a)

   An engineer models the trajectory of a component using $3x^2 - 6x - 2 = 0$. Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ to solve this equation, giving your answers in surd form.

   [3 marks]

   (b)

   The engineer must also solve the pair of equations $y = x + 1$ and $y = x^2 - 5$ simultaneously to find where the trajectory meets a support line. Find all solutions.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A ball is thrown into the air, and its path is modelled by a quadratic height function; its horizontal position also follows a linear relationship.

   (a)

   A ball is thrown so that its height in metres above the ground after $t$ seconds is $h = -5t^2 + 20t + 1$. Find the two values of $t$ at which the ball is at a height of 16 metres, giving your answers to 2 decimal places where necessary.

   [4 marks]

   (b)

   The ball's horizontal distance from the thrower, in metres, is given by $d = 6t$. Find the horizontal distance travelled by the ball at each of the two times found in part (a).

   [4 marks]

   (c)

   A second ball is thrown from the same point, following $y = x^2 - 4x + 2$ (height $y$ against horizontal distance $x$, both in metres), and travels in a straight line described by $y = x - 2$ until they meet. Find the coordinates of the point(s) where the two paths intersect, showing full working.

   [5 marks]

   Total for question 4: 13 marks

End of questions