Find the gcd $d$ of two positive integers $a$ and $b$ also find integers $x,y$ such that $ax+by=d$, using the extended Euclidean algorithm.

Is this number divisible by 2? Half the time the number is, half the time it isn't. Steps give the divisibility test.

Is this number divisible by 9? Half the time the number is, half the time it isn't. Steps give the divisibility test.

Is this number divisible by 3? Half the time the number is, half the time it isn't. Steps give the divisibility test.

Is this number divisible by 10? Half the time the number is, half the time it isn't. Steps give the divisibility test and a way to determine the number of tens.

Is this number divisible by 5? Half the time the number is, half the time it isn't. Steps give the divisibility test and a way to determine the number of fives.

Tests Algebra, Graphing Straight Lines, Probability, Statistics

Display a fraction, given as a numerator and a denominator, as a mixed fraction when appropriate.

Open and answer

Straightforward question: student must find the general solution to a second order constant coefficient ODE. Uses custom marking algorithm to check that both roots appear and that the solution is in the correct form (e.g. two arbitrary constants are present). Arbitrary constants can be any non space-separated string of characters. The algorithm also allows for the use of $e^x$ rather than $\exp(x)$.

Unit tests are also included, to check whether the algorithm accurately marks when the solution is correct; when it's correct but deviates from the 'answer'; when one or more roots is incorrect; or when the roots are correct but constants of integration have been forgotten.

Simple addition 3

No description given

Practice of conversion between SI units of mass, volume & length.

Transposition of formulae. Changing the subject of an equation. 

rebel rebelmaths

Fourth part of the Core Foundation Maths pre-arrival self-assessment material:

Question 1:  Algebra IV: Properties of indices (1) - Multiplication/Division                                 

Question 2: Algebra IV: Properties of indices (2) - Fractions                                 

Question 3: Algebra IV - Properties of Indices (4) - Further

Question 4: Numbers V: standard index form (conversions and operations)                                 

6 questions which introduce the user to the Numbas system.

Practice with adding, subtracting and dividing basic algebraic fractions

Several problems involving dividing fractions, with increasingly difficult examples, including mixed numbers and complex fractions. 

This is the question for Lent Term week 10 of the MA100 course at the LSE. It looks at material from chapters 39.

This is the question for Lent Term week 9 of the MA100 course at the LSE. It looks at material from chapters 37 and 38.

This is the question for Lent Term week 8 of the MA100 course at the LSE. It looks at material from chapters 35 and 36.

This is the question for Lent Term week 7 of the MA100 course at the LSE. It looks at material from chapters 33 and 34.

The following is a description of parts a and b. In particular it describes the varaibles used for those parts.

This question (parts a and b) looks at optimisation problems using the langrangian method. parts a and b of the question we will ask the student to optimise the objective function f(x,y) = y + (a/b)x subject to the constraint function r^2 = (x-centre_x)^2 + (y-centre_y)^2.

The variables centre_x and centre_y take values randomly chosen from {6,7,...,10} and r takes values randomly chosen from {1,2,...,5}.

We have the ordered set of variables (a,b,c) defined to be randomly chosen from one of the following pythagorean triplets: (3,4,5) , (5,12,13) , (8,15,17) , (7,24,25) , (20,21,29). The a and b variables here are the same as those in the objective function. They are defined in this way because the minimum will occur at (centre_x - (a/c)*r , centre_y - (b/c)*r) with value centre_y - (b/c)r + (a/b) * centre_x - (a^2/bc)*r , and the maximum will occur at (centre_x + (a/c)*r , centre_y + (b/c)*r) with value centre_y + (b/c)r + (a/b) *centre_x + (a^2/bc)r. The minimisation problem has lambda = -c/(2br) and the maximation problem has lambda* = c/(2br).

We can see that all possible max/min points and values are nice rational numbers, yet we still have good randomisation in this question. :)

This is the question for Lent Term week 6 of the MA100 course at the LSE. It looks at material from chapters 31 and 32.

This is the question for Lent Term week 5 of the MA100 course at the LSE. It looks at material from chapters 29 and 30.

This is the question for Lent Term week 4 of the MA100 course at the LSE. It looks at material from chapters 27 and 28.

This is the question for Lent Term week 3 of the MA100 course at the LSE. It looks at material from chapters 25 and 26.

This is the question for Lent Term week 2 of the MA100 course at the LSE. It looks at material from chapters 23 and 24.

This is the question for Lent Term week 1 of the MA100 course at the LSE. It looks at material from chapters 21 and 22.

This is the question for week 10 of the MA100 course at the LSE. It looks at material from chapters 19 and 20.

This is the question for week 9 of the MA100 course at the LSE. It looks at material from chapters 17 and 18.

Description of variables for part b:
For part b we want to have four functions such that the derivative of one of them, evaluated at 0, gives 0; but for the rest we do not get 0. We also want two of the ones that do not give 0, to be such that the derivative of their sum, evaluated at 0, gives 0; but when we do this for any other sum of two of our functions, we do not get 0. Ultimately this part of the question will show that even if two functions are not in a vector space (the space of functions with derivate equal to 0 when evaluated at 0), then their sum could nonetheless be in that vector space. We want variables which statisfy:

a,b,c,d,f,g,h,j,k,l,m,n are variables satisfying

Function 1: x^2 + ax + b sin(cx)
Function 2: x^2 + dx + f sin(gx)
Function 3: x^2 + hx + j sin(kx)
Function 4: x^2 + lx + m sin(nx)

u,v,w,r are variables satifying
u=a+bc
v=d+fg
w=h+jk
r=l+mn

The derivatives of each function, evaluated at zero, are:
Function 1: u
Function 2: v
Function 3: w
Function 4: r

So we will define
u as random(-5..5 except(0))
v as -u
w as 0
r as random(-5..5 except(0) except(u) except(-u))

Then the derivative of function 3, evaluated at 0, gives 0. The other functions give non-zero.
Also, the derivative of function 1 + function 2 gives 0. The other combinations of two functions give nonzero.

We now take b,c,f,g,j,k,m,n to be defined as \random(-3..3 except(0)).
We then define a,d,h,l to satisfy
u=a+bc
v=d+fg
w=h+jk
r=l+mn

Description for variables of part e:

Please look at the description of each variable for part e in the variables section, first.
As described, the vectors V3_1 , V3_2 , V3_3 are linearly independent. We will simply write v1 , v2 , v3 here.
In part e we ask the student to determine which of the following sets span, are linearly independent, are both, are neither:

both: v1,v2,v3
span: v1,v1+v2,v1+v2+v3, v1+v2+v3,2*v1+v2+v3
lin ind: v1+v2+v3
neither: v2+v3 , 2*v2 + 2*v3
neither:v1+v3,v1-2*v3,2*v1-v3
neither: v1+v2,v1-v2,v1-2*v2,2*v1-v2