Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
Solve the simultaneous equations $$\log_{10}x + \log_{10}y = \log_{10}3$$ $$\log_{10}(3x+y) = 1$$
$x = \dfrac{1}{3}$, $y = 9$ or $x = 3$, $y = 1$
Q2
Answer
The price of a car, $P$, is modelled by $$P = A\mathrm{e}^{-kt}$$ where $t$ is the age of the car in years. When new, the car was worth $10000$ and it was worth $6000$ after 3 years. Find the age of the car when it is worth $2000$.
$9.45$ years
Q3
Answer
$\mathrm{f}(x) = \dfrac{1}{x} - \sqrt{x} + 3$
  1. $\mathrm{f}'(x)$
  2. $\mathrm{f}''(4)$
  1. $-x^{-2} - \dfrac{1}{2}x^{-\frac{1}{2}}$
  2. $\dfrac{1}{16}$
Q4
Answer
  1. The gradient between $(-2, 7)$ and $(-4, p)$ is $4$. Find $p$.
  2. The midpoint of $(-2, 7)$ and $(6, q)$ is $(m, 5)$. Find $m$ and $q$.
  3. The distance between $(-2, 7)$ and $(r, 3)$ is $2\sqrt{13}$. Find the possible values of $r$.
  1. $-1$
  2. $m = 2$, $q = 3$
  3. $-8, 4$