Simplifying indices

Basic solving of linear equations

Solving linear equations.

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Find $\displaystyle\int \frac{ax+b}{(x+c)(x+d)}\;dx,\;a\neq 0,\;c \neq d $.

GIve a random linear program and ask the students to convert it to canonical form.

This question is a variation on another version which asks the student to find the height of a bridge arch and width of the river given a formula for the arch of the bridge (also available in the Numbas database)

Introduction to Laplace Transforms

An application of quadratic functions based on the Gladesville Bridge in Sydney, Australia. Student is given an equation representing the arch of the bridge and asked to find the height of the arch and the width of the river. Requires and understanding of the quadratic function and where and how to apply correct formulae.

An application of quadratic functions based on the Gateshead Bridge in the UK city of Newcastle. Student is given an equation representing the arch of the bridge and asked to find the height of the arch and the width of the river. Requires and understanding of the quadratic function and where and how to apply correct formulae.

Student is given a rational function, h(x), with randomised coefficients, and a linear function, k(x), also with randomised coeffieients and asked to find:

1. h(k(x)) or k(h(x)) (randomly selected) for a randomised value of x
2. The domain of h(x) - multiple choice part
3. A general expresion for k(h(x)) or h(k(x)) - opposite combination to first part.

Variables are constrained so that h(x) is not a degenerate form and that when evaluating h(x) denomiator is not 0.

This question gives the student the formula for the charge on a capacitor as a function of time then asks them to find the value of k, the exponential constant given other values and hence write out the formula for the given case. A custom function (in Extensions & scripts) extracts the student's formula and plots it on a JSXGraph object in the question. The student is then asked to evaluate the function at a given point using the plot or other methods. The value of the capacitor (Q_0), time (t) and charge at time t are randomised as is the value at which the formula is to be evaluated.

Calculation of the length and alternative form of the parameteric representation of a curve, involving trigonometric functions.

Students are asked to rearrane the formula giving the voltage drop across a resistor in terms of emf of battery, resistance of resistor and internal resistance of battery to make the internal resistance the subject. They are then asked to calculate the internal resistance given values for the other variables (randomised).

Given a circle equation in the form (x-a)^2+(y-b)^2=r^2, write down the radius and centre coordinates.

In this question the students have to solve a linear recurrence of order 2. The sequence is asked in recurrence form and the goal is to find its closed form.

Students are presented with an AI generated solution to rerrange the quadratic equation where the AI has made errors, they are asked to rewrite the solution correctly. No variables but this is version 5 of 5 versions of the question. This version uses a much more mangled AI generated solution that the other 4 versions and does not ask for the line with the first error, just for the student to rewrite the solution correctly.

Students are presented with an AI generated solution to rerrange the quadratic equation where the AI has made errors, they are asked to identify on which line the first error occurs, then rewrite the solution correctly. No variables but this is version 4 of 5 versions of the question.

Students are presented with an AI generated solution to rerrange the quadratic equation where the AI has made errors, they are asked to identify on which line the first error occurs, then rewrite the solution correctly. No variables but this is version 2 of 5 versions of the question.

Students are presented with an AI generated solution to rerrange the quadratic equation where the AI has made errors, they are asked to identify on which line the first error occurs, then rewrite the solution correctly. No variables but this is 1 of 5 versions of the question.

Determine the radius and centre coordinates of a circle with equation expressed in expanded form.

Rearrange a homogeneous circle equation into the form (x-xc)^2+(y-yc)^2=r^2 to find the centre and radius.

Two part question, student has to rearrange the parallel resistors formula (stated in the question) to make L_1 or L_2 the subject (variable is chosen randomly), then find the value of this variable when values of the other variables in the formula are given. These values are randomly chosen.

Note that the advice for this question has two versions, the one displayed to the student depends on which variable is selected by the question.

Two part question, student has to rearrange the parallel inductors formula (stated in the question) to make L_1 or L_2 the subject (variable is chosen randomly), then find the value of this variable when values of the other variables in the formula are given. These values are randomly chosen.

Note that the advice for this question has two versions, the one displayed to the student depends on which variable is selected by the question.

Two part question, student has to rearrange the heat flow formula (stated in the question) to make T_1 or T_2 the subject (variable is chosen randomly), then find the value of this variable when values of the other variables in the formula are given. These values are randomly chosen.

Note that the advice for this question has two versions, the one displayed to the student depends on which variable is selected by the question.

Find the equation of a line passing through two points in the form y=mx+b. The question is set up so that m and b will be integers.

State the general form of the equation of a straight line explaining the role of each of the terms in your answer.

The student is shown a plot of a mystery function. They can enter values of $x$ check, within the bounds of the plot.

They're asked to give the formula for the function, and then asked for its value at a very large value of $x$.

A plot of the student's function updates automatically as they type. Adaptive marking is used for the final part to award credit if the student gives the right value for their incorrect function.

Students need to solve a quadratic equation and recognise that only the positive root has physical significance. Roots are randomised with one always negative and one positive. Equation can be factorised fairly easily or the quadratic formula can be used to find the solution. Advice gives solution by factorisation.

Given some random finite subsets of the natural numbers, perform set operations $\cap,\;\cup$ and complement.