Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
The price of a stamp, $P$ is modelled by $$P = A\mathrm{e}^{kt}$$ where $t$ is the number of years after the year 2000. In 2001, the stamp was worth $10$. In 2002, the stamp was worth $16$.
  1. Find $A$.
  2. Find $k$.
  3. Predict the value of the stamp in 2020.
  4. Suggest one limitation of the model.
  1. $6.25$
  2. $\ln 1.6 = 0.470$
  3. $75500$
  4. Model predicts unlimited growth of price
Q2
Answer
  1. Find the first three terms in ascending powers of $x$ in the expansion of $(1-3x)^4$.
  2. Use this to estimate $0.994^4$. Write down all the digits in your calculator.
  3. Calculate the percentage error in your approximation.
  1. $1 - 12x + 54x^2$
  2. $0.976216$
  3. $8.84\times10^{-5}$
Q3
Answer
Express $\dfrac{15 + \sqrt{3}}{3 - \sqrt{3}}$ in the form $a + b\sqrt{3}$ where $a$ and $b$ are integers.
$8 + 3\sqrt{3}$
Q4
Answer
Differentiate the following with respect to $x$:
  1. $\dfrac{(3x)^2\times x^4}{x}$
  2. $\sqrt[3](x)$
  3. $\dfrac{1}{2x^3}$
  1. $45x^4$
  2. $\dfrac{1}{3}x^{-\frac{2}{3}}$
  3. $-\dfrac{3}{2}x^{-4}$
Q5
Answer
Given that $5x^2 + px - 8 = q(x-1)^2 + r$ for all values of $x$, find $p$, $q$ and $r$.
$p = -10$, $q = 5$, $r = -13$
Q6
Answer
Prove that no one or two digit square number ends in $2$.
Try $1^2$ up to $9^2$