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.

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

Independent events in probability. Choose whether given three given pairs of events are independent or not.

Given results from a survey about what people eat for breakfast, where some people eat one or both of cereal and toast. Student is asked to pick the probability of eating either one or the other from a list. Distractors pick out common errors.

Given descriptions of some pairs of random events, pick the ones which are mutually exclusive.

Simple probability question. Counting number of occurrences of an event in a sample space with given size and finding the probability of the event.

rebelmaths

Given the probabilities that each of four out of five friends will win a round of mini-golf, work out the probability that the fifth friend won't win, then use that to find the probability that they will win.

Solve for x and y on a given triangle and calculate the area

rebelmaths

several statements are given regarding trig identities. student is to select which are true and which are false

Find angle and side in a right angled triangle.

rebelmaths

Given the original formula the student enters the transformed formula.

READY TO USE

Given three vectors, arrange them in a tip to tail arrangement using geogebra, then estimate the magnitude and direction of their resultant.

A block lies on rough horizontal ground. The coefficient of friction is given and students find the maximum value of friction $F_{\text{MAX}} = \mu R$. This is then used to determine the magnitude of force that must be exceeded to achieve movement when the force is applied horizontally and at two different angles to the horizontal.

A particle is projected up a rough plane at a given speed. Given the angle of the slope and the coefficient friction, find the distance the particle travels before it comes to instantaneous rest.

A block of given mass is sliding down the plane, with given acceleration. Find the normal reaction force, the coefficient of friction, and the distance travelled before reaching a given speed.

A particle is projected up a rough plane at a given speed. Given the angle of the slope and the coefficient friction, find the distance the particle travels before it comes to instantaneous rest.

Given a random variable $X$  normally distributed as $\operatorname{N}(m,\sigma^2)$ find probabilities $P(X \gt a),\; a \gt m;\;\;P(X \lt b),\;b \lt m$.

Students will analyze a graph of a quadratic function to identify the x-intercepts (or Roots, or Zeros).

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.

Three forces act on a particle. You're given the magnitude of one, and the directions of all three. Find the magnitudes of the other two forces.

A particle is in equilibrium on an incline. Four forces are acting on it. You're given the angle of the incline and three forces. Resolve the forces to find the angle and magnitude of the other force.

Identifying y=mx+b given two points

Use $\cos^2\theta+\sin^2\theta=1$ and/or an understanding on the unit circle definitions to determine $\cos\theta$ given $\sin\theta$ and the quadrant theta is in.

Use $\cos^2\theta+\sin^2\theta=1$ and/or an understanding on the unit circle definitions to determine $\sin\theta$ given $\cos\theta$ and the quadrant theta is in.

No description given

Given one point and the gradient determine the equation of the line.

Given the graph of the line determine the equation of the line.

Given three vectors with integer components, find the corresponding magnitude and direction.

Given three vectors, arrange them in a tip to tail arrangement using geogebra, then estimate the magnitude and direction of their resultant.