Finding the lengths and angles within a right-angled triangle using: pythagoras theorem, SOHCAHTOA and principle of angles adding up to 180 degrees.

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Understanding order of operations on a calculator and using power and root keys.

Understanding order of operations on a calculator and using power and root keys.

Practice dividing polynomials using the long division method.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Finding the lengths and angles within a right-angled triangle using: pythagoras theorem, SOHCAHTOA and principle of angles adding up to 180 degrees.

Understanding order of operations on a calculator and using power and root keys.

Find a% of b using a calculator. Suggested method to use decimal equivalent.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

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. :)

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

Three graphs are given with areas underneath them shaded. The student is asked to calculate their areas, using integration.  Q1 has a polynomial. Q2 has exponentials and fractional functions. Q3 requires integration by parts.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Using simple substitution to find $\lim_{x \to a} bx+c$, $\lim_{x \to a} bx^2+cx+d$ and $\displaystyle \lim_{x \to a} \frac{bx+c}{dx+f}$ where $d\times a+f \neq 0$.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

This question shows how to ask for a number in scientific notation, by asking for the significand and exponent separately and using a custom marking algorithm in the gap-fill part to put the two pieces together.

Answers not in standard form, i.e. with a significand not in $[1,10)$, are accepted but given partial marks.

4 questions on using partial fractions to solve indefinite integrals.

4 questions on using partial fractions to solve indefinite integrals.

5 questions on using substitution to find indefinite integrals.

5 questions on using substitution to find indefinite integrals.

Practice dividing polynomials using the long division method.

An exam using an experimental theme to use KaTeX to render maths instead of MathJax

Using the unit circle definition of sin, cos and tan, to calculate the exact value of trig functions evaluated at angles that depend on 0, 30, 45, 60 or 90 degrees.

Find the sum of two 2-dimensional vectors, graphically and exactly using the parallelogram rule.