Q1
Answer
A geometric series has first term $20480$ and its sum to infinity is $81920$.
  1. Show that the common ratio of the series is $\frac{3}{4}$.
  2. Calculate the difference between the fifth and the sixth term of the series.
  3. Determine the smallest number of terms that should be added so that their total exceeds $80000$.
  1. $81920 = \dfrac{20480}{1-r}$
  2. $1620$
  3. $14$
Q2
Answer
Amy and Ben are planning to save money for the next 48 months. Amy plans to save $£300$ this month, increasing the amount by $£5$ each successive month. Ben plans to save $£a$ this month, increasing the amount by $£15$ each successive month.
  1. Find the amount Amy will save on the twelfth month.
  2. Find the total amount Amy will in 48 months.
  3. Ben saves the same amount as Amy. Find the value of $a$.
  1. $355$
  2. $20040$
  3. $130$
Q3
Answer
A circle $C$ has equation $4x^2+4y^2-8x+24y-5=0$.
  1. Find the coordinates of the centre of the circle.
  2. Find the radius of the circle, giving your answer in the form $k\sqrt{5}$.
  3. A straight line passes through the point $P(8,11)$ and touches the circle at the point $Q$. Find the distance $PQ$.
  1. $(1, -3)$
  2. $\dfrac{3}{2}\sqrt{5}$
  3. $\sqrt{\dfrac{935}{4}} = 15.3$