Pick four numbers from $1900\dots 2015$ and ask the student to factorise them.

Custom marking scripts make sure the student has entered a complete factorisation.

Factorising 5 to 7 digit numbers into a product of prime powers.

Uses the marking algorithms from question 1 of this CBA

Number Theory. 

Given $n \in \mathbb{N}$ find $\mu(n),\;\tau(n),\;\sigma(n),\;\phi(n).$

Given $\frac{a}{b} \in \mathbb{Q}$ for suitable choices of $a$ and $b$, find all $n \in \mathbb{N}$ such that $\phi(n)=\frac{a}{b}n$.

Given $m \in \mathbb{N}$, find values of $n\in \mathbb{N}$ such that $\sigma(n)=m$.

There are at most two such solutions in this question.

Given $m \in \mathbb{N}$, find the smallest natural number $n \in \mathbb{N}$ with $\tau(n)=m$ divisors.

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.

Multiple correlation question. Given the correlation coefficent of $Y$ with $X_1$ is $r_{01}$, the correlation coefficent of $Y$ with $X_2$ is $r_{02}$ and the correlation coefficent of $X_1$ with $X_2$ is $r_{12}$ then explain the proportion of variablity of $Y$. Also find the partial corr coeff between $Y$ and $X_2$ after fitting $X_1$ and find how much of the remaining variability in $Y$ is explained by $X_2$ after fitting $X_1$.

Multiplication of $2 \times 2$ matrices.

A box contains $n$ balls, $m$ of these are red the rest white.

$r$ are drawn without replacement.

What is the probability that at least one of the $r$ is red?

Two numbers are drawn at random without replacement from the numbers m to n.

Find the probability that both are odd given their sum is even.

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.

Given a large number of gambles, find the expected profit.

Find the first 3 terms in the MacLaurin series for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

Integrate $f(x) = ae ^ {bx} + c\sin(dx) + px^q$. Must input $C$ as the constant of integration.

Find $\displaystyle \int ax ^ m+ bx^{c/n}\;dx$.

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$.

Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

Details on inputting numbers into Numbas.

Entering numbers and algebraic symbols  in Numbas.

Determine if the following describes a probability mass function.

$P(X=x) = \frac{ax+b}{c},\;\;x \in S=\{n_1,\;n_2,\;n_3,\;n_4\}\subset \mathbb{R}$.

Given subset $T \subset S$ of $m$ objects in $n$ find the probability of choosing without replacement $r\lt n-m$ from $S$ and not choosing any element in $T$.

Reducing fractions to their lowest form by cancelling common factors in the numerator and denominator. There are 4 questions. 

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