Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
Solve $$\mathrm{e}^x - 1 = 2\mathrm{e}^{-x}$$
$\ln 2$
Q2
Answer
The pressure, $P$, of a gas is related to its volume, $V$, by $$P = aV^b$$ A plot of $\log_{10} P$ against $\log_{10} V$ has a gradient of $-1.4$ and vertical intercept $1.97$.
  1. Find the constants $a$ and $b$.
  2. Estimate the pressure when the volume is $2$.
  1. $a = 93.3$ and $b = -1.4$
  2. $35.4$
Q3
Answer
$y = \dfrac{5}{x^2} - \dfrac{1}{4x} + x$
  1. Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$.
  2. Find $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$.
  1. $-10x^{-3} + \dfrac{1}{4}x^{-2} + 1$
  2. $30x^{-4} - \dfrac{1}{2}x^{-3}$
Q4
Answer
The equation $kx^2 - 30x + 25k = 0$ has equal roots. Find $k$.
$3$ or $-3$
Q5
Answer
  1. Solve the simultaneous equations $$y = 2x^2 - 3x - 5$$ $$10x + 2y + 11 = 0$$
  2. What can you deduce about the line $10x + 2y + 11 = 0$ and the curve $y = 2x^2 - 3x - 5$
  1. $x = -\dfrac{1}{2}$ and $y = -3$
  2. The line is tangent to the curve