This is the question for week 8 of the MA100 course at the LSE. It looks at material from chapters 15 and 16.

This is the question for week 7 of the MA100 course at the LSE. It looks at material from chapters 13 and 14.

This is the question for week 6 of the MA100 course at the LSE. It looks at material from chapters 11 and 12.

This is the question for week 5 of the MA100 course at the LSE. It looks at material from chapters 9 and 10.

The following describes how we define our revenue and cost functions for part b of the question.

We have variables c, f, m, h.

The revenue function is R(q) = -c q^2 + 2mf q .
The cost function is C(q) = f q^2 - 2mc q + h .

The "revenue - cost" function is -(c+f) q^2 +2m(c+f) q - h

Differentiating, we see that there is a maximum point at m.

We pick each one of f, m, h randomly from the set {2, .. 6}, and we pick c randomly from {h+1 , ... , h+5}. This ensures that the discriminant of the "revenue - cost" function is positive, meaning there are two real roots, meaning the maximum point lies above the x-axis. I.e. we can actually make a profit.

This is the question for week 4 of the MA100 course at the LSE. It looks at material from chapters 7 and 8. The following describes how a polynomial was defined in the question. This may be helpful for anyone who needs to edit this question.

For parts a to c, we used a polynomial defined as m*(x^4 - 2a^2  x^2 + a^4 + b), where the variables "a" and "b" are randomly chosen from a set of reaosnable size, and the variable $m$ is randomly chosen from the set {+1, -1}. We can easily see that this polynomial has stationary points at -a, 0, and a. We introduced the variable "m" so that these stationary points would not always have the same classification. The variable "b" is always positive, and so this ensures that our polynomial does not cross the x-axis. The first and second derivatives; stationary points; the evaluation of the second derivative at the stationary points; the classification of the stationary points; and the axes intercepts can all be easily expressed in terms of the variables "a", "b", and "m". Indeed, this is what we did to mark the student's answers.

This is the question for week 3 of the MA100 course at the LSE. It looks at material from chapters 5 and 6. The following describes how two polynomials were defined in the question. This may be helpful for anyone who needs to edit this question.

In part a we have a polynomial. We wanted it to have two stationary points. To create the polynomial we first created the two stationary points as variables, called StationaryPoint1 and StationaryPoint2 which we will simply write as s1 ans s2 here. s2 was defined to be larger than s1. This means that the derivative of our polynomial must be of the form a(x-s1)(x-s2) for some constant a. The constant "a" is a variable called PolynomialScalarMult, and it is defined to be a multiple of 6 so that when we integrate the derivative a(x-s1)(x-s2) we only have integer coefficients. Its possible values include positive and negative values, so that the first stationary point is not always a max (and the second always a min). Finally, we have a variable called ConstantTerm which is the constant term that we take when we integrate the derivative derivative a(x-s1)(x-s2). Hence, we can now create a randomised polynomial with integers coefficients, for which the stationary points are s1 and s2; namely (the integral of a(x-s1)(x-s2)) plus ConstantTerm.

In part e we created a more complicated polynomial. It is defined as -2x^3 + 3(s1 + s2)x^2 -(6*s1*s2) x + YIntercept on the domain [0,35]. One can easily calculate that the stationary points of this polynomials are s1 and s2. Furthermore, they are chosen so that both are in the domain and so that s1 is smaller than s2. This means that s1 is a min and s2 is a max. Hence, the maximum point of the function will occur either at 0 or s2 (The function is descreasing after s2). Furthermore, one can see that when we evaluate the function at s2 we get (s2)^2 (s2 -3*s1) + YIntercept. In particular, this is larger than YIntercept if s2 > 3 *s1, and smaller otherwise. Possible values of s2 include values which are larger than 3*s1 and values which are smaller than 3*s1. Hence, the max of the function maybe be at 0 or at s2, dependent on s2. This gives the question a good amount of randomisation.

This is the question for week 2 of the MA100 course at the LSE. It looks at material from chapters 3 and 4.

This is the question for week 1 of the MA100 course at the LSE. It looks at material from chapter 2.

Slope of a curve at a point

Linear combinations of $2 \times 2$ matrices. Three examples.

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Use laws for addition and subtraction of logarithms to simplify a given logarithmic expression to an arbitrary base.

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Cofactors Determinant and inverse of a 3x3 matrix.

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

Find the determinant of a $3 \times 3$ matrix.

Elementary Exercises in multiplying matrices. 

Multiplication of $2 \times 2$ matrices.

Given random set of data (between 13 and 23 numbers all less than 100), find their stem-and-leaf plot.

rebelmaths

Just showing how to use the stdev function from the stats extension to calculate the standard deviation of a list of numbers.

rebelmaths

I feel this question has too many questions inside it, I have since made a question that just asks a single division problem called Decimals: Division (includes rounding the answer).

By powers of ten I mean a 1 followed by some 0s. The scientific notation questions will take care of the power of ten notation.

A set of MCQ designed to help Level 2 Engineering students prepare/practice for the on-line GOLA test that is used to assess the C&G 2850, Level 2 Engineering, Unit 202: Engineering Principles.

These practice questions cover:

• resolving axial forces in a truss structure
• stress & strain in a closed thin-walled pressure vessel (an example of 2D plane stress)
• use of Mohr's circle for determining principal stresses for a 2D plane stress case
• calculation of invariants for general 3D stress case, leading to von Mises stress and principal stresses

old question, way too many things in one question! I have made better questions out of each part now.

I think I prefer the other question I made called "Rounding to 0, 1, 2 and 3 decimal places"