Given two numbers, find the gcd, then use Bézout's algorithm to find $s$ and $t$ such that $as+bt=\operatorname{gcd}(a,b)$.

Finally, find all solutions of an equation $\mod b$.

Implicit differentiation including finding tangents

5 questions on using substitution to find indefinite integrals.

Scale a page to some percentage of its original size, then increase/decrease by another percentage. Find the size of the final copy as a percentage of the original.

Based on question 2 from section 3 of the Maths-Aid workbook on numerical reasoning.

Given the selling price of an item both as a cash amount and as a percentage of the cost of production, find the cost of production and the profit.

Based on question 1 from section 3 of the Maths-Aid workbook on numerical reasoning.

Given percentages of males and females working on a project, and the percentage of the total staff who are male (or female), find the percentage of all staff working on the project.

Based on question 3 from section 3 of the maths-aid workbook on numerical reasoning.

3 questions. One question on limits of standard sequences. Other two on finding least $N$ such that $|a_n-L |\lt 10^{-r},\;\;n \geq N$ where $L$ is limit of $(a_n)$.

3 questions. Finding the stationary points of functions of 2 variables.

Partial differentiation.

5 questions on finding local and global maxima and minima on compact intervals and on the real line for differentiable functions.

Implicit differentiation including finding tangents

5 questions on using substitution to find indefinite integrals.

Statistics and probability. 2 questions. Both simple regression. First with 8 data points, second with 10. Find $a$ and $b$ such that $Y=a+bX$. Then find the residual value for one of the data points.

Two questions on completing the square. The first asks you to express $x^2+ax+b$ in the form $(x+c)^2+d$ for suitable numbers $c$ and $d$. The second asks you to complete the square on the quadratic of the form $ax^2+bx+c$ and then find its roots.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

Given a codeword in $w=\mathbb{Z}^4_3$ find and list all codewords at Hamming distance $1$ from $w$.

Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.

Finding the volumes of basic 3D shapes

Customised for the Numbas demo exam

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Video in Show steps.

Finding lengths of sides of triangles

Using trig identities to find solutions to equations

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Finding the modulus and argument (in radians) of four complex numbers; the arguments between $0$ and $2 \pi$ and careful with quadrants!

Find the general solution of $y''+2py'+(p^2-q^2)y=A\sin(fx)$ in the form  $A_1e^{ax}+B_1e^{bx}+y_{PI}(x),\;y_{PI}(x)$ a particular integral. Use initial conditions to find $A_1,B_1$.

Let $x_n=\frac{an+b}{cn+d},\;\;n=1,\;2\ldots$. Find  $\lim_{x \to\infty} x_n=L$ and find least $N$ such that $|x_n-L| \lt 10^{-r},\;n \geq N,\;r \in \{2,\;3,\;4\}$.

$x_n=\frac{an+b}{cn+d}$. Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| \lt 10 ^{-r},\;n\geq N$, $2\leq r \leq 6$.

Given a set in predicate form i.e. $A=\{x|P(x)\}$, find and input the elements of the set.

Enumerate elements of a set given in predicate form.

For example, find all elements of $A=\{x\in\mathbb{Z}\;|\;\;|2x-5|<4\}$.

Implicit differentiation.

Given $x^2+y^2+ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Implicit differentiation.

Given $x^2+y^2+dxy +ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

Students enter equation and turning point