Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

Given the following three vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3$ Find out whether they are a linearly independent set are not. Also if linearly dependent find the relationship $\textbf{v}_{r}=p\textbf{v}_{s}+q\textbf{v}_{t}$ for suitable $r,\;s,\;t$ and integers $p,\;q$.

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.

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

Find the determinant of  a random (3x3) matrix.

$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Find the eigenfunctions of an irregular Sturm–Liouville problem and hence solve an inhomogeneous boundary value problem by writing the solution as an eigenfunction expansion.

Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots. 

This exercise will help you rearrange a linear equation to find the value of a given variable

Characterise the cosh function as continuous, many-to-one, even, and find the limit as x approaches 1. Part of HELM book 2.4.3.

Find the area, first moment of area, and coordinates of a general spandrel.  The area may be above or below the function.

Find the unit vectors in the direction of four 3-dimensional vectors. Three of the vectors are given, and the fourth is expressed as a linear combination of two of the other vectors.

Find the centroid of a shape made up of a rectangle and two triangles.

Find the centroid of a shape made from a rectangle, triangle, and circle.

series of R and L and C

find Z (mag and Phase)

step find Xl

Calculating the derivative of a function of the form $\frac{(x-a)^2}{bx}$, finding its stationary points and determining their nature.

Find the derivative of a function of the form $y=a \sin(bx+c)$ using a table of derivatives.

Given the position vectors of three 2-dimensional points $A$, $B$ and $C$, find the coordinates of a fourth point $D$ such that $ABCD$ forms a parallelogram. 

Find the unit vectors in the direction of four 2-dimensional vectors. Three of the vectors are given, and the fourth is expressed as a linear combination of two of the other vectors.

Equilibrium of a rigid body.  Find tension to raise a pole.

Homework set.  Find 2-D moments or balance moments using Verignon's theorem:  $ M = \pm F_x d_y \pm F_y d_x$

Find the determinant of a random (2x2) matrix.

Finding the resistance in a simple parallel RL circuit.

Finding the inductance in a simple series RL circuit.

Finding the inductive current in a simple parallel RL circuit.

Finding the current in a simple parallel RL circuit.

Finding the resistive voltage in a simple series RL circuit.

Find the cross product of 2D vectors (WARNING: requires scalar answer (magnitude only))