Graph the function y=x^2+2 on [-3,3] by plotting points. State the domain and range. This is part of HELM Book 2.2.1.

This question models an experiment: the student must collect some data and enter it at the start of the question, and the expected answers to subsequent parts are marked based on that data.

The student's values of the variables width, depth and height are stored once they move on from the first part.

This question models an experiment: the student must collect some data and enter it at the start of the question, and the expected answers to subsequent parts are marked based on that data.

A downside of working this way is that you have to set up the variable replacements on each part of the question. You could avoid this by using explore mode.

Evaluate a composition of functions for a randomised numerical input. The functions are 3t+2 and t+3. This is part of HELM Book 2.1.3.

HELM book 2.1.3 exercises

Given f(x)=(x+a)/(x+b) and g(x) = 1/x, compute f(g(x)) and g(f(x)).

a and b are randomised integers.

Given 2 randomised functions f(x) (linear) and g(x) (quadratic), find one of f(f), f(g), g(f) or g(g) at a randomised integer x-value

Given 2 randomised functions f (linear) and g (quadratic), find one of f(f), f(g), g(f) or g(g)

An economic dispatch problem with three generators. The steps help the students to solve this using lagrangian multipliers. This question is designed to allow students studying economic dispatch to practice solving the problem.

The student has to enter `diff(y,x,2)`, equivalent to $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$, as their answer. It's marked by pattern matching, using a custom marking algorithm.

Simultaneous equations question. values for the coefficients are generated to be small numbers, random values are generated for the weights and the resultant energies are calculated for the question. Student needs to solve equations to find coefficients. Advice gives solution using method of elimination.

Express $\displaystyle \frac{a}{x + b} \pm  \frac{c}{x + d}$ as an algebraic single fraction over a common denominator. 

Reading gradient and intercept from y=mx+c.

Solve linear equations with unknowns on one. Including brackets and fractions.

Solve linear equations with unknowns on one. Including brackets and fractions. Solutions may require rounding.

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Student is asked to find the distance from a given point, A, to a house, given the distance between A and another point B, and the angles at A and B. Requires use of the sine rule. Distance and angles are randomised.

Student is asked to find the distance from a given point, B, to a house, given the distance between B and another point A, and the angles at A and B. Requires use of the sine rule. Distance and angles are randomised.

Question about use of trig identities, student has to use identities to find exact value of \(\sin \frac{\pi}{12}\). Question is used in exam where student has to write out the solution and upload it for grading.

Question about use of trig identities, student has to use identities to find exact value of \(\cos \frac{7\pi}{12}\). Question is used in exam where student has to write out the solution and upload it for grading.

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.

Simultaneous equation problem as circuit analysis to find unknown currents. Students need to solve the equations and type in the solutions for each variable. Advice is given in terms of solution by elimination.

Simultaneous equation problem as circuit analysis to find unknown voltages. Students need to solve the equations and type in the solutions for each variable. Advice is given in terms of solution by elimination.

Student needs to solve a quadratic equation to calculate time taken for a diver to hit the water after diving from a diving board. Height of the board and initial upward velocity of the diver are randomly generated values. student needs to know that surface of the water is height 0, and only positive root of quadratic has physical meaning. Question is set to always give one positieve and one negative root.

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

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Question requires students to determine if the smallest angle of a triangle is smaller than a given value. Answer is Yes/No but students need to use cosine rule to find the smallest angle and to know that smallest angle is oppositeshortest side (otherwise they will need to find all angles of the triangle). Designed for a test where students upload handwritten working for each question as a check against guessing. Also designed to make it difficult for students to google or use AI to find the answer.

Question requires students to determine if the largest angle of a triangle is smaller than a given value. Answer is Yes/No but students need to use cosine rule to find the largest angle and to know that largest angle is opposite longest side (otherwise they will need to find all angles of the triangle). Designed for a test where students upload handwritten working for each question as a check against guessing. Also designed to make it difficult for students to google or use AI to find the answer.

A two part question. Students are first given the formula for the time for a ball to come to rest after being dropped on a block. Part a) asks the students to rearrange the formula to make e, the coefficient of restitution, the subject of the formula. Part b) gives students realistic values for variables in the formula and asks them to calculate the coefficient of restitution using the formula derived in part a). 

Students are given lengths of 3 sides of a triangle (all randomised) and asked to find one of the angles in degrees. Requires use of the cosine rule.