Question 16. A marmalade manufacturer produced thousands of jars of marmalade each week. The mass, $x$ grams, of marmalade in a randomly chosen jar has mean $25$g. Following a slight adjustment to the filling machine, a random sample of $50$ Read More …

## Q15. Poisson and related distributions

Question 15. In a certain country, the number of deaths due to old age in the northern and southern parts are recorded separately. It is found that, on average, there are 2 deaths in the northern parts and 1 death Read More …

## Q14. Normal and binomial distributions

Question 14. A fruit grower grows both red and green apples which have masses that are normally distributed. The mass of a randomly chosen red apple has mean $75$ g and standard deviation $12.5$ g. The mass of a randomly chosen Read More …

## Q13. Probability, approximations

Question 13. Ali has 5 different bags and 5 different wallets. They are of 5 colors: black, blue, brown, red and white. He places one wallet into one bag at random. A wallet is said to be in the correct bag Read More …

## Q12. Permuatations

Question 12. (a) 10 friends, including Amy, Betty and Charlie are to line up in a single row for a photograph. Find the number of ways they can be arranged if (i) Amy and Betty are to be together; (ii) Read More …

## Q11. Approximations of distributions

Question 11. A study of the number of male and female children in families of inhabitants on a planet very far away reveals that the probability that a baby girl is born is 0.005. 500 babies are chosen at random. Read More …

## Q10. Maclaurin Series

Question 10. It is given that $\displaystyle y = (\cos x)^\frac{1}{2}$. (a) Show that $\displaystyle 2y \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 2 \left ( \frac{\mathrm{d}y}{\mathrm{d}x} \right )^2 + y^2 = 0.$ (b) Find the Maclaurin series for $y$ up to and including the Read More …

## Q9. Inequalities

Question 9. Solve, without a calculator, the inequality $$\frac{x^2+5}{x-1} \geq 1.$$ Hints Remember: we cannot cross multiply if we don’t know the whether a term is positive or negative! (After making one side of the equation 0) If we cannot Read More …

## Q8. Functions

Question 8. The functions $f$ and $g$ are defined by \begin{align} &f : x \mapsto \left | x^2 – 6x + 5 \right | \qquad \textrm{for } x \in \mathbb{R}, 2 \leq x \leq 6,\\ &g : x \mapsto \frac{1}{2-x} Read More …

## Q7. Mathematical Induction

Question 7. Write down the value of $\displaystyle \sum_{r=1}^n r2^{-r}$ for $n = 1, 2, 3$ and $4$. By expressing your answers in the form $\displaystyle 2 – \frac{f(n)}{2^n}$ where $f(n)$ is an exprression in $n$, form a conjecture for $\displaystyle \sum_{r=1}^n Read More …