Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ given:
  1. $y = 10x^{-5}$
  2. $y = \sqrt[4]{x}$
  3. $y = x(x+3)(1-5x)$
  1. $-50x^{-6}$
  2. $\dfrac{1}{4}x^{-\frac{3}{4}}$
  3. $3 - 28x - 15x^2$
Q2
Answer
  1. Find the first three terms, in ascending powers of $x$, in the expansion of $(1+2x)^7$.
  2. Hence find the coefficient of $x^2$ in the expansion of $(2-5x)(1+2x)^7$.
  1. $1 + 14x + 84x^2$
  2. $98$
Q3
Answer
Given $A(6,1)$ and $B(-2,7)$,
  1. Find the length of $AB$.
  2. Find the gradient of the line segment $AB$.
  3. Determine whether the line $4x-3y-10=0$ is perpendicular to $AB$.
  1. $10$
  2. $-\dfrac{3}{4}$
  3. Yes
Q4
Answer
Solve $$\log_{10}(x^2-10) - \log_{10}x = 2\log_{10}3$$
$10$
Q5
Answer
Prove that no number of the form $3^n$ has $5$ as its final digit.
$3^n$ is not divisible by $5$, every number which ends in a $5$ is a multiple of $5$.