Simple procedures are given and student is asked to carry them out or un-do them.

Version 1: i and ii have the same answer. iii and iv both have two answers.

Version 2: i and ii have different answers. iii has two answers,biv has one answer.

Version 3: i and ii have different answer. iii has one answer, iv has two answers.

Version 4: i and ii have the same answer. iii has one answer, iv has two answers.

Simple procedures are given and student is asked to carry them out or un-do them.

Version 1: i and ii have the same answer. iii and iv both have two answers.

Version 2: i and ii have different answers. iii has two answers,biv has one answer.

Version 3: i and ii have different answer. iii has one answer, iv has two answers.

Version 4: i and ii have the same answer. iii has one answer, iv has two answers.

Three graphs are given with areas underneath them shaded. The student is asked to calculate their areas, using integration.  Q1 has a polynomial. Q2 has exponentials and fractional functions. Q3 requires solving a trig equation and integration by parts.

Three graphs are given with areas underneath them shaded. Student is asked to determine the minimum and maximum $x$-values of the regions. This will involve solving a linear equation and two trigonmetric equations.

Graphs are given and students are required to match them with their equation.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

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

A quadratic is and a graph of it is given. A tangent is also sketch. The equation of the tangent line is asked for.

Recognising the equation of a circle.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$.

Find a regression equation.

Find a regression equation.

Factorise three quadratic equations of the form $x^2+bx+c$.

The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.

Factorise three quadratic equations of the form $x^2+bx+c$.

The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.

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

Solve for $x$ and $y$:  \[ \begin{eqnarray} a_1x+b_1y&=&c_1\\   a_2x+b_2y&=&c_2 \end{eqnarray} \]

The included video describes a more direct method of solving when, for example, one of the equations gives a variable directly in terms of the other variable.

Couting application where repitition is allowed and order does not matter.

The student is asked to factorise a quadratic $x^2 + ax + b$. A custom marking script uses pattern matching to ensure that the student's answer is of the form $(x+a)(x+b)$, $(x+a)^2$, or $x(x+a)$.

To find the script, look in the Scripts tab of part a.

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Some questions demonstrating new features in Numbas v4.0: pattern-matching, inference of variable types in mathematical expression parts, and marking equations.

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

Solving an equation of the form $ax \equiv b\;\textrm{mod}\;n$ where $a$ and $n$ are coprime.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

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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 $x^2+y^2+dxy +ax+by=c$ 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.

Questions involving various techniques for rearranging and solving quadratic expressions and equations

This exercise will help you solve equations of type ax-b = c.

Complete the square for a quadratic polynomial $q(x)$ by writing it in the form $a(x+b)^2+c$.  Find both roots of the equation $q(x)=0$.