Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
$$\mathrm{f}(x) = x^3 - 9x^2 + 7x + 33$$
  1. Show that $x-3$ is a factor of $\mathrm{f}(x)$.
  2. Find the quotient when $\mathrm{f}(x)$ is divided by $x-3$.
  3. Solve the equation $\mathrm{f}(x) = 0$ giving exact values as your answer.
  1. Factor theorem
  2. $x^2 - 6x - 11$
  3. $x = 3, 3\pm2\sqrt{5}$
Q2
Answer
For the points $A(2,7)$ and $B(-1,-2)$
  1. Find the equation of the line through $A$ parallel to the line $y = 4x - 5$.
  2. Calculate the length of $AB$ giving your answer in simplified surd form.
  3. Find the equation of the perpendicular bisector of $AB$. Give your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers
  1. $y = 4x - 1$
  2. $3\sqrt{10}$
  3. $x + 3y - 8 = 0$
Q3
Answer
  1. Find the first four terms in ascending powers of $x$ in the expansion of $(1+4x)^7$
  2. The coefficient of $x^2$ in the expansion of $(3+ax)(1+4x)^7$ is $1001$. Find $a$.
  1. $1 + 28x + 336x^2 + 2240x^3$
  2. $-\dfrac{1}{4}$
Q4
Answer
Given $y = 3x^5 - \sqrt{x} + 15$, find $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$
$60x^3 + \dfrac{1}{4}x^{-\frac{3}{2}}$