Use $\cos^2\theta+\sin^2\theta=1$ and/or an understanding on the unit circle definitions to determine $\cos\theta$ given $\sin\theta$ and the quadrant theta is in.

Use $\cos^2\theta+\sin^2\theta=1$ and/or an understanding on the unit circle definitions to determine $\sin\theta$ given $\cos\theta$ and the quadrant theta is in.

HELM Book 1.3.1 exercises

Given a randomised log function select the possible ways of writing the domain of the function.

Given a randomised rational function select the possible ways of writing the domain of the function.

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Curve fitting, function zeros, sequences, etc.

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Find the reactions of a rigid body (a truss) at a pin and roller.  All loads are either horizontal or vertical.

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Decimal Places and Significant Figures

rebelmaths

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Simple Percentages

rebelmaths

A measurement is performed multiple times for the same object, the student will

• calculate the mean result
• calculate the standard error on the mean 
• write the mean±error to the correct precision as defined by the error written to 1 significant figure 

Advice is provided including on performing the calculations in Python or spreedsheets together with further reading. 

Tests students' ability to apply the recursive form of the Extended Euclidean Algorithm.  Random four-digit inputs are chosen subject to the condition that the Euclidean Algorithm terminates in seven steps.

Asks students to add and multiply two integers modulo another integer.  The modulus is a random number between 30 and 70, the summands are set to be large enough that modular reduction will be necessary.

Asks students to compute the multiplicative inverse of $a$ in $\mathbb{Z}_n$ where $n$ is an odd number between 31 and 61 and $a$ is an integer coprime to $n$ that lies between $n/4$ and $3n/4$.

A straightforward test of encrypting and decrypting an eight-letter message with the Vigenère Cipher using a four-letter key.  Message letters are generated uniformly at random from the English alphabet, as are the key letters.  Students are expected to be able to map English letters onto elements of $\mathbb{Z}_{26}$ in the usual order.

A straightforward test of encrypting and decrypting an eight-bit message with the Vernam Cipher.  The message and the key are uniform and independently generated eight-bit strings.

A straightforward test of encrypting and decrypting an eight-letter message with the Substitution Cipher.  Message letters are generated uniformly at random from the English alphabet and the key is a uniformly chosen random permutation of the alphabet.  Students are expected to be able to map English letters onto elements of $\mathbb{Z}_{26}$ in the usual order.

Manipulate surds and rationalise the denominator of a fraction when it is a surd.

The purpose of this question is to ask the students to go through one Rijndael round from the AES algorithm.

A straightforward test of encrypting and decrypting an eight-letter message with the Caesar Cipher.  Message letters are generated uniformly at random from the English alphabet, as is the key.  Students are expected to be able to map English letters onto elements of $\mathbb{Z}_{26}$ in the usual order.

Part of HELM Book 1.4.2

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The exercises for HELM Book 1.4.1

The 7th task in HELM book 1.4 section 1.

The 6th task in HELM book 1.4 section 1.