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Approximate the integral of a quadratic by Riemann sums,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Approximating integral of a quadratic by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.
Make decision based on expected monetary value
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given data on probabilities of three levels of success of three options and projections of the profits that the options will accrue depending on the level of success, find the expected monetary value (EMV) for each option and choose the one with the greatest EMV.
Factorise numbers into products of prime powers
Doesn't work
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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

Uses the marking algorithms from question 1 of this CBA
Factorise four numbers
Doesn't work
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

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.
Apply extended Euclidean algorithm to find gcd
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Find gcd g of two positive integers x, y and also find integers a, b such that ax+by=g with prescribed intervals for a and b.
Find all solutions of a quadratic in Z_n, given one
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given one solution of the quadratic equation in $\mathbb{Z}_n$ where $n=pq$ is a product of two primes find the other 3 solutions.
Perform arithmetic in Z_n
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Calculations in $\mathbb{Z_n}$ for three values of $n$ .
Simplify integers and rationals in Z_n
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Express various integers and rationals mod $\mathbb{Z}_3, \;\mathbb{Z}_5,\;\mathbb{Z}_7,\;\mathbb{Z}_{11}$
True/false statements on modular arithmetic
Has some problems
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Modular arithmetic
Solve a pair of congruences
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solving a pair of congruences of the form $\begin{align}x &\equiv b_1\;\textrm{mod} \;n_1\\x &\equiv b_2\;\textrm{mod}\;n_2 \end{align}$ where $n_1,\;n_2$ are coprime.
Solve a pair of congruences
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solving two simultaneous congruences:

$\begin{eqnarray*} c_1x\;&\equiv&\;b_1\;&\mod&\;n_1\\ c_2x\;&\equiv&\;b_2\;&\mod&\;n_2\\ \end{eqnarray*}$ where $\operatorname{gcd}(c_1,n_1)=1,\;\operatorname{gcd}(c_2,n_2)=1,\;\operatorname{gcd}(n_1,n_2)=1$
Solve congruence in given range
Has some problems
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solving an equation of the form $ax \equiv\;b\;\textrm{mod}\;n$ where $\operatorname{gcd}(a,n)|r$ . In this case we can find all solutions. The user is asked for the two greatest.
Solve three congruences using Chinese Remainder Theorem
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solving three simultaneous congruences using the Chinese Remainder Theorem:

$\begin{eqnarray*} x\;&\equiv&\;b_1\;&\mod&\;n_1\\ x\;&\equiv&\;b_2\;&\mod&\;n_2\\x\;&\equiv&\;b_3\;&\mod&\;n_3 \end{eqnarray*}$ where $\operatorname{gcd}(n_1,n_2,n_3)=1$
Solve three congruences with the Chinese Remainder Theorem
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solving three simultaneous congruences using the Chinese Remainder Theorem:

$\begin{eqnarray*} x\;&\equiv&\;b_1\;&\mod&\;n_1\\ x\;&\equiv&\;b_2\;&\mod&\;n_2\\x\;&\equiv&\;b_3\;&\mod&\;n_3 \end{eqnarray*}$ where $\operatorname{gcd}(n_1,n_2,n_3)=1$
Find $\mu(n)$ , $\sigma(n)$ , $\tau(n)$ , $\phi(n)$
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Number Theory.

Given $n \in \mathbb{N}$ find $\mu(n),\;\tau(n),\;\sigma(n),\;\phi(n).$
Find all numbers with given value of $\phi(n)$
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $\frac{a}{b} \in \mathbb{Q}$ for suitable choices of $a$ and $b$ , find all $n \in \mathbb{N}$ such that $\phi(n)=\frac{a}{b}n$ .
Find n such that $\sigma(n)$ is the given number
Needs to be tested
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

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.
Find the smallest $n$ with given number of divisors
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $m \in \mathbb{N}$ , find the smallest natural number $n \in \mathbb{N}$ with $\tau(n)=m$ divisors.
Find n such that $\phi(n)$ is the given value
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $m \in \mathbb{N}$ , find all $n \in \mathbb{N}$ such that $\phi(n)=m$ and enter the largest and second largest if they exist.
Perform a one-way ANOVA
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

One-way ANOVA example
Classify singular points of a second order ODE
Doesn't work
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Trying out something: get the student to enter a set for each of "regular singular points" and "essential singular points".

Find and classify singular points of a second-order ordinary differential equation. One equation is chosen from a selection of 10.
Find sample standard deviations of two samples
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.
Multiply 2x2 matrices
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Multiplication of $2 \times 2$ matrices.
Solve a separable first order ODE with trig functions,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$ .
Solve a separable first order ODE,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Find the solution of a first order separable differential equation of the form $axyy'=b+y^2$ .
Solve a separable first order ODE,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$ .
Solve a separable first order ODE,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find the solution of a first order separable differential equation of the form $(a+x)y'=b+y$ .
Solve a constant coefficient second order ODE,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by'+cy=0$ .
Multiple and partial correlation(old)
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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 $X_1$ .
Construct a probability distribution function, then find CDF and expectation
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

The random variable $X$ has a PDF which involves a parameter $c$ . Find the value of $c$ . Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$ .

Also find the expectation $\displaystyle \operatorname{E}[X]=\int_{-\infty}^{\infty}xf_X(x)\;dx$ .

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