Q1
Answer
The line with equation $mx-y-2=0$ touches the circle with equation $x^2+6x+y^2-8y=4$ exactly once. Find the two possible values of $m$ in exact form.
$\dfrac{9\pm2\sqrt{29}}{10}$
Q2
Answer
A stone is thrown from the top of a cliff. The height, $h$ metres, of the stone above the ground after $t$ seconds is modelled by $$h(t) = 115 + 12.25t-4.9t^2$$
  1. Give a physical interpretation of the meaning of the constant term 115 in the model.
  2. Write $h(t)$ in the form $A-B(t-C)^2$, where $A$, $B$ and $C$ are constants to be found.
  3. Find, with justification, the time taken after the stone is thrown for it to reach the ground.
  4. Find the maximum height of the stone above the ground and the time at which this maximum height is reached.
  1. Height of cliff
  2. $\dfrac{3925}{32} - \dfrac{49}{10}\left(t - \frac{5}{4}\right)^2$
  3. $t = \dfrac{5}{4} + \sqrt{\dfrac{39250}{1568}} = 6.25$
  4. $h = \dfrac{3925}{32}$ and $t = 1.25$
Q3
Answer
The equations of two circles are $x^2 + 10x + y^2 - 12y = 3$ and $x^2 - 6x + y^2 - 2qy = 9$.
  1. Find the centre and radius of each circle, giving your answers in terms of $q$ where necessary.
  2. Given that the distance between the centres of the circles is $\sqrt{80}$, find the two possible values of $q$.
  1. $(-5, 6),\ r = 8$ and $(3,q)\ r = \sqrt{18 + q^2}$
  2. $2, 10$