This question asks the student to find the inversed composite function finding the inverses of two functions then composite of these; and by finding the composite of two functions then finding the inverse. The question then concludes by asking students to compare their two answers.

This exam covers 

• laws of indices
• using surds and rationalising the denominator
• expanding brackets
• simplifying expressions
• solving linear inequalities
• finding common factors
• dividing a polynomial with remainders, using algebraic division
• factor theorem
• remainder theorem
• inverse and composite functions

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Applied questions that could be done with modulo arithmetic.

Credits to : Ben Brown.

used under a CC-BY 4.0 licence. https://creativecommons.org/licenses/by/4.0/

Several quadratics are given and students are asked to complete the square.

Solving quadratic equations using a formula,

Modular arithmetic. Find the following numbers modulo the given number $n$. Three examples to do.

This question tests the students ability to use the logarithm equivalence law to make x the subject of a given equation and to check which of a list of logarithmic expressions are equivalent to x.

Solve a quadratic equation by completing the square. The roots are not pretty!

Given sequences with missing terms, find the common difference between terms.

This question tests the student's understanding of what is and is not a surd, and on their simplification of surds.

Draws a right angled triangle based on a length and an angle.

Draws a right angled triangle based on a length and an angle.

Shows how to define variables to stop degenerate examples.

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.

Two questions on solving systems of simultaneous equations.

Rearrange equations to make $x$ the subject.

Question assessing the students understanding of linear sequences.

Students are assessed on their ability to find the common difference and first term in a linear sequence and then find the nth term of the sequence using the arithmetic formula.

Some clever variable-substitution trickery to randomly pick two sides of a right-angled triangle to give to a student, and ask for the other.

The sides are set up so they're always Pythagorean triples, and the opposite side is always odd.

As ever, most of the tricky stuff is in the advice. 

Because this was created quickly to show how to set up the randomisation, there's no diagram. It would benefit greatly from a diagram.

Some clever variable-substitution trickery to randomly pick two sides of a right-angled triangle to give to a student, and ask for the other.

The sides are set up so they're always Pythagorean triples, and the opposite side is always odd.

As ever, most of the tricky stuff is in the advice. 

Because this was created quickly to show how to set up the randomisation, there's no diagram. It would benefit greatly from a diagram.

Introductory exercise about set equality