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.

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-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.

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.

Dummy question to test random() function when first decimal place of upper limit of range is 4, and test of currency() function.

Given a random variable $X$  normally distributed as $\operatorname{N}(m,\sigma^2)$ find probabilities $P(X \gt a),\; a \gt m;\;\;P(X \lt b),\;b \lt m$.

Factorise the difference of two squares, t^2-number^2. Number is a random perfect square, or 1 over a perfect square.

DCS 2023 Maths Entrance Test to be used September 2023

1.5 hours with 3 randomised questions from each of these groups:

• Notation and Algebra
• Calculus - Differentiation
• Calculus - Integration
• Trigonometry

Expand (a+b)(c+d). a,b,c,d are random terms that can be +ve or -ve, and can consist of a number and/or a letter.

The answer must contain no brackets but will be accepted if it is not simplified from there.

Part of HELM Book 1.3

This uses an embedded Geogebra graph of a sine curve $y=a\sin (bx+c)+d$  with random coefficients set by NUMBAS.

This uses an embedded Geogebra graph of a cubic polynomial with random coefficients set by NUMBAS.  Student has to decide what kind of map it represents and whether an inverse function exists.

This uses an embedded Geogebra graph of a cubic polynomial with random coefficients set by NUMBAS.  Student has to decide what kind of map it represents and whether an inverse function exists.

This uses an embedded Geogebra graph of a cubic polynomial with random coefficients set by NUMBAS.  Student has to decide what kind of map it represents and whether an inverse function exists.

This uses an embedded Geogebra graph of amodulus function with random coefficients set by NUMBAS.

This uses an embedded Geogebra graph of a modulus function with random coefficients set by NUMBAS.

This uses an embedded Geogebra graph of a cubic polynomial with random coefficients set by NUMBAS.  Student has to decide what kind of map it represents and whether an inverse function exists.

This uses an embedded Geogebra graph of a polar function with random coefficients set by NUMBAS.

Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Students are given a proportion of students who DO speak a language other than English at home, and are asked to find the percentage who do NOT. The fraction denominator is either 20 or 25, and the numerator is randomised.

Given either (a) ax+bx+cy+dy, or (b) ax^2+bx+c, (c) ax^2+bx^2+cx^2, where a,b,c,d are randomised constants, and x and y are randomised letters, simplify by collecting like terms (if possible).

Part of HELM Book 1.3

Compare ax-bx, and ax(bx) and simplify, where a and b are +ve or -ve integers, and x is a randomised variable. Part of HELM Book 1.3

Write an expression (a^k1*a^k2)/a^k3 using a single positive index. Variable a is randomised and can be a number or a letter. k1,k2 and k3 are randomised positive numbers.

Part of HELM Book 1.2

Given an expression (either a^-k or 1/a^-k) with a negative index, rewrite it with a positive index.

The variable a and the index k are randomised.

Part of HELM Book 1.2

Calculate the product of two randomised scientific notation numbers and give the answer in scientific notation.

Part of HELM Book 1.2

Simplify three expressions: (a^b)^c, a^b * a^c, a^b/a^c where a, b and c are randomised. a is a letter, and b and c are rational numbers.

Part of HELM Book 1.2

Simplify (a^k1*a^k2)/(a^k3*a^k4) where a is a randomised variable and k1,k2,k3 and k4 are randomised fractions (k2 and/or k4 may be 0). They may be written in index form or in surd form, or even a combination of the two.

Part of HELM Book 1.2

Use a calculator to evaluate a number to the power of a fractional index. Both the number (a positive integer) and the index (a rational) are randomised.

Part of HELM Book 1.2

Given an expression 10^-k, rewrite it as a fraction with no index. k is a random positive integer from 1 to 6.

Part of HELM Book 1.2

Write an expression (a^k1*a^k2)/a^k3 using a single positive index. Variable a is randomised and can be a number or a letter. k1,k2 and k3 are randomised and can be positive or negative numbers.

Part of HELM Book 1.2

Given an expression (either a^-k or 1/a^-k) with a negative index, rewrite it with a positive index.

The variable a and the index k are randomised.

Part of HELM Book 1.2

Simplify n1.v^k1.(n2.v^k2), where n1, n2 are positive integers, v is a random letter variable, and k1 and k2 are nonzero integers.

The answer should be expressed as n.v^k

Part of HELM Book 1.2