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

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

Integration by susbtitution, no hint given

Multiple choice question. Given a randomised polynomial select the possible ways of writing the domain of the function.

Given two 3 dim vectors, find vector equation of line through one vector in the direction of another. Find two such lines and their point of intersection.

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

Given vectors $\boldsymbol{a,\;b}$, find $\boldsymbol{a\times b}$

rebelmaths

Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.

Calculate the hypotenuse of a right triangle, given the other two side lengths.

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

rebelmaths

rebelmaths

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

Integration by susbtitution, no hint given

Integration by susbtitution, no hint given

Integration by susbtitution, no hint given

Given a table of the number of days in which sales were between £x1000 and £(x+1)1000 find the relative percentage frequencies of these volume of sales.

rebelmaths

Scores  in 20 games and frequency given -- calculate mean.

rebelmaths

Scores (including negatives)  in 20 games and frequency given -- calculate mean.

rebelmaths

Integration by susbtitution, no hint given

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.

Differentiation question with customised feedback to catch some common errors and corresponding partial marks.

malrules

Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$, find $g(x)$. Find $h(x)$, $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$. 

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.

Integration by susbtitution, no hint given

Implicit differentiation.

Given $x^2+y^2+dxy +ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

Implicit differentiation.

Given $x^2+y^2+ax+by=c$ find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$.

Two column vectors are given, with one entry unknown. Given that the vectors are perpendicular, question is to determine the unknown entry.

Find a regression equation.

Finding the confidence interval at either 90%, 95% or 99% for the mean given the mean of a sample. The population variance is given and so the z values are used. Various scenarios are included.

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

A few quadratic equations are given, to be solved by completing the square. The number of solutions is randomised.