Content created by Newcastle University

Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$.

Find the solution of a first order separable differential equation of the form $axyy'=b+y^2$.

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Exercise using a given uniform distribution $Y$, calculating the expectation and variance as well as asking for the CDF. Also finding $P(Y \le a)$ and $P( b \lt Y \lt c)$ for a given values $a,\;b,\;c$.

Example showing how to calculate the probability of A or B using the law $p(A \;\textrm{or}\; B)=p(A)+p(B)-p(A\;\textrm{and}\;B)$. 

Also converting percentages to probabilities.

Easily adapted to other applications.

Given three linear combinations of four i.i.d. variables, find the expectation and variance of these estimators of the mean $\mu$. Which are unbiased and efficient?

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 $\phi$.

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

Evaluate $\int_0^{\,m}e^{ax}\;dx$, $\int_0^{p}\frac{1}{bx+d}\;dx,\;\int_0^{\pi/2} \sin(qx) \;dx$. 

No solutions given in Advice to parts a and c.

Tolerance of 0.001 in answers to parts a and b. Perhaps should indicate to the student that a tolerance is set. The feedback on submitting an incorrect answer within the tolerance says that the answer is correct - perhaps there should be a different feedback in this case if possible for all such questions with tolerances.

Find  $\displaystyle \int\cosh(ax+b)\;dx,\;\;\int x\sinh(cx+d)\;dx$.

Advice tidied up.

Inputting algebraic expressions into Numbas.

Inputting ratios of algebraic expressions.

Dealing with functions in Numbas.

Details on inputting numbers into Numbas.

Entering numbers and algebraic symbols  in Numbas.

Information on inputting powers

An experiment is performed twice, each with $5$ outcomes

$x_i,\;y_i,\;i=1,\dots 5$ . Find mean and s.d. of their differences $y_i-x_i,\;i=1,\dots 5$.

$X$ is a continuous uniform random variable defined on $[a,\;b]$. Find the PDF and CDF of $X$ and find  $P(X \ge c)$.

Given subset $T \subset S$ of $m$ objects in $n$ find the probability of choosing without replacement $r\lt n-m$ from $S$ and not choosing any element in $T$.

Questions testing understanding of numerators and denominators of numerical fractions, and reduction to lowest terms.

Solve for $x$: $\displaystyle ax+b = cx+d$

Solve for $x$: $\displaystyle \frac{px+s}{ax+b} = \frac{qx+t}{cx+d}$ with $pc=qa$.

Solve for $x$: $\displaystyle \frac{s}{ax+b} = \frac{t}{cx+d}$

Find the centre and radius of a circle when given an equation in standard form.

Find the equation of the straight line parallel to the given line that passes through the given point $(a,b)$.

Find the points of intersection of a straight line and a circle.

Find the points of intersection of two circles.

Questions testing understanding of the precedence of operators using BIDMAS, applied to integers. These questions only test DMAS. That is, only Division/Multiplcation and Addition/Subtraction.

Questions testing understanding of the precedence of operators using BIDMAS. That is, they test Brackets, Indices, Division/Multiplication and Addition/Subtraction.