311 results authored by Simon Thomas - search across all users.

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• Question
Using the mid-ordinate rule with 5 ordinates to approximate $\int_a^b{c^x}$
• Question
Using the trapezium rule with 6 ordinates to approximate $\int_a^b{c^x}$
• Question
Using the trapezium rule with 5 ordinates to approximate $\int_a^b{1/(x^2+c)}$
• Question
Calculate d2y/dx2 for a curve defined parametrically
• Question
Find the equation of the tangent to a curve defined parametrically
• Question
Calculate dy/dx for a curve defined parametrically
• Question in Calculus

Find $\displaystyle \frac{d}{dx}\left(\frac{m\sin(ax)+n\cos(ax)}{b\sin(ax)+c\cos(ax)}\right)$. Three part question.

• Exam (2 questions)

Find the first few terms of the Maclaurin and Taylor series of given functions.

• Question

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 $\phi$.

• Question

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 $\phi$.

• Question

Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.

• Question

Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.

• Question

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 $p,\;q$.

• Question

Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.

• Question

Real numbers $a,\;b,\;c$ and $d$ are such that $a+b+c+d=1$ and for the given vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4$ $a\textbf{v}_1+b\textbf{v}_2+c\textbf{v}_3+d\textbf{v}_4=\textbf{0}$. Find $a,\;b,\;c,\;d$.

• Question

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

• Question

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

• Question

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

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• Question

Rationalise the denominator with increasingly difficult examples involving compound denominators.

• Question

This question tests the student's understanding of what is and is not a surd, and on their simplification of surds.

• Question

Manipulate surds and rationalise the denominator of a fraction when it is a surd.

• Question in Stats

The human resources department of a large finance company is attempting to determine if an employee’s performance is influenced by their undergraduate degree subject. Personnel ratings are used to judge performance and the task is to use expected frequencies and the chi-squared statistic to test the null hypothesis that there is no association.

• Question

Given two sets of data, sample mean and sample standard deviation, on performance on the same task, make a decision as to whether or not the mean times differ. Population variance not given, so the t test has to be used in conjunction with the pooled sample standard deviation.

Link to use of t tables and p-values in Show steps.