Approximating integral of a linear function by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.

Express the equation of the given line in the form $y=mx+c$.

Express the equation of the given line in the form $y=mx+c$.

Find the equation of a straight line which has a given gradient $m$ and passes through the given point $(a,b)$.

Find the equation of the straight line which passes through the points $(a,b)$ and $(c,d)$.

Find the equation of the straight line perpendicular to the given line that passes through the given point $(a,b)$.

Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\displaystyle \int \frac{ax+b}{x^2+cx+d}\;dx,\;a \neq 0$ using partial fractions or otherwise.

Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\displaystyle \int \frac{a}{x^2+bx+c }\;dx$ using partial fractions or otherwise.

An object moves in a straight line, acceleration given by:

$\displaystyle f(t)=\frac{a}{(1+bt)^n}$. The object starts from rest. Find its maximum speed. 

The derivative of  $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.

Other method. Find $p,\;q$ such that $\displaystyle \frac{ax+b}{cx+d}= p+ \frac{q}{cx+d}$. Find the derivative of $\displaystyle \frac{ax+b}{cx+d}$.

Inverse and division of complex numbers.  Four parts.

Find the equation of the straight line parallel to the given line that passes through the given point $(a,b)$.

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

Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.

Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.

A quadratic is and a graph of it is given. A tangent is also sketch. The equation of the tangent line is asked for.

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

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This question tests the student's ability to solve simple linear equations by elimination. Part a) involves only having to manipulate one equation in order to solve, and part b) involves having to manipulate both equations in order to solve. 

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Straightforward solving linear equations question.

Adapted from 'Simultaneous equations by substitution 3 with parts' by Joshua Boddy.

Straightforward solving linear equations question.

Adapted from 'Simultaneous equations by elimination 1 with parts' by Joshua Boddy.

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