Newcastle University Mathematics and Statistics

Differentiate

\[ \sqrt{a x^m+b})\]

Differentiate $\displaystyle (ax^m+bx^2+c)^{n}$.

Differentiate $\displaystyle \ln((ax+b)^{m})$

Questions used in a university course titled "Group theory".

The questions deal with permutations written in cycle or two-line notation.

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Find $\displaystyle I=\int \frac{2 a x + b} {a x ^ 2 + b x + c}\;dx$ by substitution or otherwise.

Given a generator matrix for a linear code, write a parity check matrix, compute the syndrome of a word, and decide if it's a codeword.

4 questions on using partial fractions to solve indefinite integrals.

Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by=0$. Complex roots.

Linear program described in words. Student must write out constraints as equations in standard form, then identify the optimal solution in a finished simplex tableau.

Express $\displaystyle a \pm  \frac{c}{x + d}$ as an algebraic single fraction.

Given  $P(A)$, $P(A\cup B)$, $P(B^c)$ find $P(A \cap B)$, $P(A^c \cap B^c)$, $P(A^c \cup B^c)$ etc..

Compute the information rate and minimum distance of some binary codes.

Curl and divergence of a vector field.  Determine whether the vector field is irrotational or solenoidal.

Correct an error in a received message which was encoded using Hamming's square code

Decode a message of two codewords encoded using Hamming's [7,4] code, with at most one error per codeword.

Find upper and lower bounds on the number of codewords in three maximal codes given their codeword lengths and minimum distances.

Uses Hamming, Singleton and Gilbert-Varshamov bounds.

List all vectors in a spanning set.

(repeated 3 times)

Write down a lexicographic parity check matrix for a Hamming code and correct two received codewords.

Write down the lexicographic parity check matrix for a Hamming code, and correct two received codewords.

Compute the word length, minimum distance and dimension of some given Hamming codes.

Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.

A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.

There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.

Find the Cartesian form $ax+by+cz=d$ of the equation of the plane $\boldsymbol{r=r_0+\lambda a+\mu b}$.

The solution is not unique. The constant on right hand side could be given to ensure that the left hand side is unique.

When are vectors $\boldsymbol{v,\;w}$ orthogonal?

Part b) is not answered in Advice, the given solution is for a different question.