The subtraction algortihm using the borrow and pay back method with integers.

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 $\phi$.

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 $\phi$.

Given a matrix in row reduced form use this to find bases for the null, column and row spaces of the matrix.

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

A ball is thrown vertically upwards from a position above ground level. Find its greatest height, and the total time its in the air.

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

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

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 $(c,d)$.

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

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

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..

Inverse and division of complex numbers.  Four parts.

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 $(c,d)$.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

A ball is thrown vertically upwards from a position above ground level. Find its greatest height, and the total time its in the air.

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Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Questions on powers, the laws of indices, and exponential growth.

Questions on powers, the laws of indices, and exponential growth.

Questions on powers, the laws of indices, and exponential growth.

Inverse and division of complex numbers.  Four parts.

Rearrange some expressions involving logarithms by applying the relation $\log_b(a) = c \iff a = b^c$.

Questions on powers, the laws of indices, and exponential growth.

This question asks learners to use row operations to find the inverse of a 3x3 matrix.

This question asks learners to use row operations to find the inverse of a 3x3 matrix.

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

Modular arithmetic. Find the following numbers modulo the given number $n$. Three examples to do.

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

When throwing a dice once what is the probability of ....

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