Find the points of intersection of a straight line and a circle.

Find the points of intersection of two circles.

No description given

Questions about percentage and ratio, applied to finance.

Based on section 3.2 of the Maths-Aid workbook on numerical reasoning.

Understanding of intersection and union symbols.

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

Understanding of intersection and union symbols.

Understanding of intersection and union symbols.

$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Solve for $x(t)$, $\displaystyle\frac{dx}{dt}=\frac{a}{(x+b)^n},\;x(0)=0$

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

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

Composite multiplication and division of complex numbers. Two parts.

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

Given ratio of ingredients in a preparation, and amounts of each ingredient, work out how much of the preparation you can make.

Based on question 5 from section 3 of the maths-aid workbook on numerical reasoning.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

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

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

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

The random variable $X$ has a PDF which involves a parameter $c$. Find the value of $c$. Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$.

Also find the expectation $\displaystyle \operatorname{E}[X]=\int_{-\infty}^{\infty}xf_X(x)\;dx$.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

Given percentages of males and females working on a project, and the percentage of the total staff who are male (or female), find the percentage of all staff working on the project.

Based on question 3 from section 3 of the maths-aid workbook on numerical reasoning.

Added to calculator test.

Added advice on using decimals.

Scale a page to some percentage of its original size, then increase/decrease by another percentage. Find the size of the final copy as a percentage of the original.

Based on question 2 from section 3 of the Maths-Aid workbook on numerical reasoning.

(Added a decimal version to advice - and changed increased to enlarged)

Used in non-calculator quiz.

Given the stakes of three people in a lottery syndicate, and the amount the syndicate won, work out each person's share of the winnings.

Based on question 4 from section 3.2 of the Maths-Aid workbook on numerical reasoning.

(Used in Non-Calculator test.)

Given the selling price of an item both as a cash amount and as a percentage of the cost of production, find the cost of production and the profit.

Based on question 1 from section 3 of the Maths-Aid workbook on numerical reasoning.

Added to Calculator test.

Given a PDF $f(x)$ on the real line with unknown parameter $t$ and three random observations, find log-likelihood and MLE $\hat{t}$ for $t$. 

Based on section 3 of the maths-aid/mathcentre workbook on numerical reasoning.

Questions about percentage and ratio, applied to finance.

Based on section 3.2 of the Maths-Aid workbook on numerical reasoning.

Scale a page to some percentage of its original size, then increase/decrease by another percentage. Find the size of the final copy as a percentage of the original.

Based on question 2 from section 3 of the Maths-Aid workbook on numerical reasoning.