Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
A circle has centre $C$ and equation $x^2 + y^2 - 8x - 2y - 3 = 0$
  1. Find the centre and radius of the circle.
  2. Find the values of $k$ for which $y = k$ is tangent to the curcle, giving your answers in surd form.
  3. The points $S$ and $T$ lie on the circumference of the circle. $M$ is the midpoint of $ST$. Given that $CM$ has length $2$, calculate the length of $ST$.
  4. Find the coordinates of the point where the circle meets the line $x - 2y - 12 = 0$.
  1. $(4, 1)$ radius $\sqrt{20}$
  2. $1 \pm 2\sqrt{5}$
  3. $8$
  4. $(6,-3)$
Q2
Answer
  1. Find the binomial expansion of $(2x+5)^4$
  2. Hence find $(2x+5)^4 - (2x-5)^4$
  3. Verify that $x = 2$ is a root of the equation $(2x+5)^4 - (2x-5)^4 = 3680x - 800$ and find the other possible values of $x$.
  1. $625 + 1000x + 600x^2 + 160x^3 + 16x^4$
  2. $320x^3 + 2000x$
  3. $2, \dfrac{1}{2}, -\dfrac{5}{2}$
Q3
Answer
Show that $x = 1$ is a root of the equation $$x^3 - x^2 - 3x + 3 = 0$$ and hence find the other two solutions.
Use factor theorem and polynomial division, $-\sqrt{3}, \sqrt{3}$