ADVANCED
Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

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

\(=81a^4 - 432a^3b \\+864a^2b^2 ...\)

Compound Interest

If £240 is invested with an interest rate of 6% compounded monthly, find the value of the investment after 8 years. £387.39

Coordinates (Square)

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

\((4,3),(7,9),(-2,6)\)

(1,12)

Normal Distribution

\( X \sim N(300, 10^2)\)

Find

\( P(270\lt X \lt330) \)

\(0.997\)

Factorise (Quadratic 1)

Factorise:

\(x^2-2x-3\)

\((x+1)(x-3)\)

Factorise (Quadratic 2)

Factorise:

\(5x^2+19x-4\)

\((x+4)(5x-1)\)

Graph (Linear)

Draw a rough sketch of the graph of:


\(y=2x\)

Gradient 2
y intercept 0

Indices

What is the value of:

\(5^{1}\)

\(= 5\)

Trigonometry (Angle)

Find angle ABC if AB = 5.1m and BC = 6.4m. 37.2o

Trigonometry (Side)

Find AB if angle ABC = 52o and BC = 5.1m. 3.14m

Venn Diagrams

Describe the red region.

Circle part Circle part

Differentiation (1)

\(y = 5x^3 - 4x^2 + 6x\)

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

\(15x^2 - 8x + 6\)

Differentiation (2)

\(y = \dfrac{3}{x^5} - 5\sqrt[6]{x}\)

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

\(-\frac{15}{x^6} - \frac{5}{6}x^{-\frac{5}{6}}\)

Differentiation (3)

\(y=\sqrt{3x^8+4}\)

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

\(12x^7(3x^8+4)^{-\frac{1}{2}}\)

Differentiation (4)

\(y=x^5 \sin x\)

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

\(5x^4sinx+x^5cosx\)

Differentiation (5)

\(y=\frac{2x^2}{4x-1}\)

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

\(\frac{(8x^2-4x)}{(4x-1)^2}\)

Differentiation (6)

Find the equation of the tangent to the curve:
\(y = 4x^2 + 2x - 1\)
where \(x = -2\)
\(y = -14x - 17\)

Differentiation (7)

Find the equation of the normal to the curve:
\(y = -5x^2 + 7x - 3\)
where \(x = 2\)
\(y = \frac{x}{13} - \frac{119}{13}\)

Integration (1)

\(y =24x^2 - 16x + 8\)

Find \( \int y \quad dx\)

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

Binomial Distribution

A game is played 16 times and the probability of winning is 0.9. Calculate the probability of winning exactly 11 times.   0.0137

Formulas

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:
\(u_{5} = -26\)
\(u_{17} = -110\)
Find the sum of the first 45 terms.-6840

Asymptotes (HV)

Find the equations of the asymptotes of:

\(y=\dfrac{2x+5}{2x+3}\)

\(x=-\frac{3}{2},y=1\)

Trig Advanced

In the triangle ABC,
AB = 5.4cm.
BC = 6.1cm.
CA = 7.2cm.
Find angle CÂB.

55.7°

Sigma

Evaluate:

$$\sum_{n=1}^{6} 2n+0$$

42

Discriminant

\(f(x)=-3x^2-6x-1\)

What is the value of the discriminent and what does it indicate?
24, Two distinct roots

Completing The Square

\(f(x)=x^2+2x+6\)

By completing the square find the coordinates of the vertex.
(-1, 5)

Logarithms

Evaluate \(\log_2(32) \)


5

Integration (3)

Find the integral:

\(\int \cos(x)e^{\sin(x)} \;dx\)


\(e^{\sin(x)}+c\)

Graph (2 points)

Find the equation of the straight line that passes through:

(-2, -1) and (4, -19)

\(y=-3x-7\)

Functions (Inverse)

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

\(f(x)=\frac{\sqrt{x}-15}{17}\)


\((17x+15)²\)

Functions (Composite)

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

\(16x^2+48x+39\)

Standard Form

Write in standard form:
\(a \times 10^p \times b\times 10^q\)
where \(a \times b \) is a single digit number \((1 \le ab \lt 10)\)

\(ab\times10^{p+q}\)

Graph (Mixed)

Draw a rough sketch of

\(y=2^x\)

Sketch

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

$$\cos{\frac{\pi}{4}} + \sin{45°}$$

\(\sqrt{2}\)

Trig (Large Angles)

Without a calculator find the exact value of

$$\sin{5\pi}$$

\(0\)

Simultaneous Eqns (3)*

Solve:

\( j+k+l= 17 \\ 2j-3k+9l= 24\\ -j+k-3l=-9\)

j = 3, k = 9, l = 5

Radian Measures

Find the perimeter of a sector with radius 4.2cm and angle \( \frac{2\pi}{3}\)

🍕

17.2cm

Combinatronics*

In how many ways can 12 different books be arranged on a shelf if 3 of them must be together?

21772800

Asymptotes (Ob)*

Find the equations of the asymptotes of:

$$y=\dfrac{2x^2-8x+8}{x-3}$$

x=3, y=2x-2

Sequences (Geometric)

Evaluate:
$$ \sum_{k=1}^{15} 5 \times (-2)^k $$

-109230

Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

\((1+x)^{-8}\)

\(1-8x-36x^2-120x^3\)

Integration (2)

Evaluate:

\(\int^{140}_{70} \dfrac{1}{x} dx\)


\(\ln{2} \approx 0.693\)

Probability (Conditional)

27 Scouts went hiking. 14 got lost, 11 got blisters, and 6 got both lost and blisters. Find the probability that a randomly selected Scout got blisters, given that they were not lost.

\(\dfrac{5}{13}\)

Vectors*

Find the area of the triangle with sides:

\( \begin{pmatrix} 9 \\ 4 \\ 0 \end{pmatrix}, \; \begin{pmatrix} 8 \\ -3 \\ 3 \end{pmatrix} \; \text{and} \; \begin{pmatrix} -1 \\ -7 \\ 3 \end{pmatrix} \)

33.0 square units

Graph (Advanced)*

Sketch the graph of:

$$y=\left|\cot\left(x\right)\right|$$

Graph Plotter

Complex Numbers 1*

Simplify
$$ \sqrt{5-12i} $$

\(3-2i \; \text{ or } -3+2i\)

Integration (4)*

Evaluate:

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


\(xe^x-e^x+c\)

Trig (Identities)*

Simplify:

$$\sec{x}-\tan{x}\sin{x}$$

\(\cos{x}\)

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

Integration (Volume)*

Find the volume of revolution when \(y=x^3\) is rotated about the y-axis for \(1 \le y \le 2\)


\(\approx 4.10\) cubic units

Miscellaneous

Describe the behavior of a function at its asymptote.

Clue: approaches but never reaches

Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = (1 + x)^n\)

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

Complex Numbers 2*


Find the four 4th roots of 1

\(1, i, -1, -i\)

Probability (Counting)*

A school committee of 8 is chosen at random from 12 senior students and 4 junior students. Find the probability that all four junior students are chosen.

1/26 or 3.85%

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

Answers appear here for Transum subscribers.



Try this Uniqueness Game with your class.

Transum.org/Maths/Game/Uniqueness/Game.asp?Level=8

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.

Teacher:
Scroll down the
page to see how
this Starter can be customised so that it
is just right for
your class.

Apple

©1997-2024 WWW.TRANSUM.ORG