Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

\((3a - 2b)^4\)

\(=81a^4 - 216a^3b \\+216a^2b^2 ...\)

Compound Interest

If £200 is invested with an interest rate of 3% compounded quarterly, find the value of the investment after 8 years. £254.02

Coordinates (Square)

Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?



Normal Distribution

\( X \sim N(65, 9^2)\)


\( P(36\lt X \lt48) \)


Factorise (Quadratic 1)




Factorise (Quadratic 2)




Graph (Linear)

Draw a rough sketch of the graph of:


Gradient -1
y intercept -2


What is the value of:


\(= 5\)

Trigonometry (Angle)

Find angle ABC if AC = 3.8m and BC = 5.3m. 45.8o

Trigonometry (Side)

Find AC if angle ABC = 61o and AB = 3.4m. 6.13m

Venn Diagrams

Describe the red region.

Circle part Circle part

Differentiation (1)

\(y = 6x^3 - 3x^2 + 7x\)

Find \( \dfrac{dy}{dx}\)

\(18x^2 - 6x + 7\)

Differentiation (2)

\(y = \dfrac{9}{x^3} - 8\sqrt[9]{x}\)

Find \( \frac{dy}{dx}\)

\(-\frac{27}{x^4} - \frac{8}{9}x^{-\frac{8}{9}}\)

Differentiation (3)


Find \( \dfrac{dy}{dx}\)


Differentiation (4)

\(y=\sin x \sqrt{ x^2 + 3}\)

Find \( \dfrac{dy}{dx}\)

\(cosx \sqrt{x^2+3}+\frac{xsinx}{\sqrt{x^2+3}}\)

Differentiation (5)


Find \( \dfrac{dy}{dx}\)


Differentiation (6)

Find the equation of the tangent to the curve:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = 1 - 2x\)

Differentiation (7)

Find the equation of the normal to the curve:
\(y = 4x^2 + 2x - 1\)
where \(x = -2\)
\(y = \frac{x}{14} + 11\frac{1}{7}\)

Integration (1)

\(y =9x^2 - 16x + 2\)

Find \( \int y \quad dx\)

\(3x^3 - 8x^2 + 2x+c\)

Binomial Distribution

A game is played 14 times and the probability of winning is 0.8. Calculate the probability of winning exactly 3 times.   0.00000382


Make up a maths question using this:

\( A = 4\pi r^2 \)

Surface area of a sphere

Greek Letters

What letter is this?

Greek Letter Greek Letter

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
\(u_{10} = 90\)
\(u_{19} = 180\)
Find the sum of the first 32 terms.4960

Asymptotes (HV)

Find the equations of the asymptotes of:



Trig Advanced

In the triangle ABC,
BC = 7.7cm.
CA = 9.2cm.
BĈA = 53.3°
Find AB to 1 dp.




$$\sum_{n=2}^{6} n^2 - 8n$$




What is the value of the discriminent and what does it indicate?
-8, No real roots

Completing The Square


By completing the square find the coordinates of the vertex.
(-3.5, -19.25)


Solve for x:

\( \log(x) + \log(29-x) = 2\)

\(x = 4 \text{ or } x = 25 \)

Integration (3)

Find the integral:

\(\int \dfrac{5x}{x^2-3} \;dx\)

\(\frac{5}{2} \ln(x^2-3)+c\)

Graph (2 points)

Find the equation of the straight line that passes through:

(-9, -12) and (6, 3)


Functions (Inverse)

Find the inverse of the function \(f\):



Functions (Composite)

\(f(x)=2x+3 \\ g(x)=2x^2 \\[1cm] \text{Find }fgf(x)\)


Standard Form

Write in standard form:
\(a \times 10^2 \times b\times 10^3\)
where \(a \times b \) is a two digit number \((10 \le ab \lt 100)\)


Graph (Mixed)

Draw a rough sketch of



Graph (Fill)

Sketch a height-time graph as this jar is filled.

Jar Graph

Trig (Special Angles)

Without a calculator find the exact value of

$$\cos{\frac{\pi}{4}} + \sin{45°}$$


Trig (Large Angles)

Without a calculator find the exact value of



Simultaneous Eqns (3)*


\( 5a+2b+c=33 \\ 3a+4b+2c= 45 \\ a+5b+c=39\)

a = 3, b = 6, c = 6

Radian Measures

Find the perimeter of a sector with radius 4.3cm and angle \( \frac{\pi}{6}\)




Ansh is with six people in a queue. How many ways can they line up without Ansh being at the back?


Asymptotes (Ob)*

Find the equations of the asymptotes of:



Sequences (Geometric)

The first term of a geometric sequence is 48 and the fourth term is twice this value. What is the common ratio?

\( \sqrt[3]{2} \)

Binomial Theorem (2)*

Find the first 4 terms in the expansion of:



Integration (2)


\(\int^{8}_{1} x^2-2x+7 \; dx\)


Probability (Conditional)

28 Scouts went hiking. 11 got lost, 14 got blisters, and 5 got both lost and blisters. Find the probability that a randomly selected Scout got blisters, given that they were not lost.



Find the cartesian equation of this plane:

\( \mathbf{r} = \begin{pmatrix} -2 \\ 2 \\ 3 \end{pmatrix} \; + \; s \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix} \; + \; t \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} \)


Graph (Advanced)*

Sketch the graph of:


Graph Plotter

Complex Numbers 1*

$$ \dfrac{3+2i}{4-i}$$


Integration (4)*


\(\int xe^x\; dx\)


Trig (Identities)*




$$ \DeclareMathOperator{cosec}{cosec} $$

Integration (Volume)*

Find the volume of revolution when \(y=\frac{1}{x}\) is rotated about the x-axis for \(1 \le x \le 2\)

\(\frac{\pi}{2}\) cubic units


What is the inverse of a function?

Clue: swaps the roles of x and y

Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = e^x\)

\(1 + x + \frac{x^2}{2} + \frac{x^3}{6}\)

Complex Numbers 2*

Given |z| = 8, find:
$$ |(3+4i)z| $$


Probability (Counting)*

6 girls sit at random on 6 seats in a row. Determine the probability that the two friends Karen and Julie sit at the ends of the row.

1/15 or 6.67%

Proof by Induction*

Prove by mathematical induction that the product of \( n \) consecutive integers is divisible by \( n! \) (n factorial)

Show true for n=1, assume true for n=k, prove for n=k+1

Last Lesson

Write down a summary of your last Maths lesson focussing on what you learnt.


An Advanced Mathematics Lesson Starter Of The Day


Concept Selection

Tick (or untick) the boxes above to select the concepts you want to be included in this Starter [untick all]. The display at the top of this page will change instantly to show your choices. You can also drag the panels above so that the questions are ordered to meet your needs.

* Topics shown with an asterix are on the IB Higher Level syllabus but not included in the Standard Level syllabus.

This Starter is called Refreshing Revision because every time you refresh the page you get different revision questions.

Regularly use this Starter to keep that important learning from being forgotten. Here is the web address (URL) for the version of this page with your currently selected concepts:

Copy and paste the URL above into your lesson plan or scheme of work.

For more ideas on revision there are plenty of tips, suggestions and links on the Mathematics Revision page.


Answers appear here for Transum subscribers.

Try this Uniqueness Game with your class.

Uniqueness Game



Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.


©1997-2024 WWW.TRANSUM.ORG