Given data on probabilities of three levels of success of three options and projections of the profits that the options will accrue depending on the level of success, find the expected monetary value (EMV) for each option and choose the one with the greatest EMV.

This question assesses the students ability to find the expected number of times an event occurs given the probability of the event occurring for a single trial and the total number of trials.

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.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

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

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

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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.

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Find angle and side in a right angled triangle.

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Two masses in a box which is attached to a string. Finding acceleration by modelling the two masses as the whole system. Find forces exerted by the two masses.

Find roots and the area under a parabola

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.

Finding the acceleration of a particle on an inclined plane which is at an angle to the horizontal. Smooth then with value for coefficient of friction.

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.

Find the $x$ and $y$ components of a force which is applied at an angle to a particle. Resolve using $F \cos \theta$. The force is applied in the negative $x$ direction but the positive $y$. 

Using $F=ma$ to find magnitudes of unknown forces acting on an accelerating body.

Find roots and the area under a parabola

Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$

Find $\displaystyle \int (ax)\ln(cx)\; dx $

Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

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).

Students will locate and write the ordered pair for the vertex of a quadratic function from a graph.

Find roots and the area under a curve