ADVANCED
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?

\((5,1),(9,4),(2,5)\)

(6,8)

Normal Distribution

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

Find

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

\(0.0288\)

Factorise (Quadratic 1)

Factorise:

\(x^2-9\)

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

Factorise (Quadratic 2)

Factorise:

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

\((x+1)(2x-3)\)

Graph (Linear)

Draw a rough sketch of the graph of:


\(y=-x-2\)

Gradient -1
y intercept -2

Indices

What is the value of:

\(125^{\frac{1}{3}}\)

\(= 5\)

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)

\(y=(9x^7+8)^7\)

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

\(441x^6(9x^7+8)^6\)

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)

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

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

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

Differentiation (6)

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

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

Asymptotes (HV)

Find the equations of the asymptotes of:

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

\(x=2,y=-2\)

Trig Advanced

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

7.7cm

Sigma

Evaluate:

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

-70

Discriminant

\(f(x)=2x^2-4x+3\)

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

Completing The Square

\(f(x)=x^2+7x-7\)

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

Logarithms

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)

\(y=x-3\)

Functions (Inverse)

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

\(f(x)=\sqrt{\frac{x-2}{7}}\)


\(7x²+2\)

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

\(\frac{ab}{10}\times10^6\)

Graph (Mixed)

Draw a rough sketch of

\(y=x^3-4x\)

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

$$\tan{\dfrac{19\pi}{3}}$$

\(\sqrt{3}\)

Simultaneous Eqns (3)*

Solve:

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

a = 3, b = 6, c = 6

Radian Measures

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

🍕

10.9cm

Combinatronics*

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

4320

Asymptotes (Ob)*

Find the equations of the asymptotes of:

$$y=\dfrac{-6x^2+5x}{2x+1}$$

x=-1/2,y=4-3x

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:

\((1-\dfrac{x}{2})^{\frac13}\)

\(1-\frac{x}{6}-\frac{x^2}{36}-\frac{5x^3}{648}\)

Integration (2)

Evaluate:

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


\(14\)

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.

\(\dfrac{9}{17}\)

Vectors*

Find the cartesian equation of this plane:

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

2x-6y+5z=-1

Graph (Advanced)*

Sketch the graph of:

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

Graph Plotter

Complex Numbers 1*

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

\(\frac{10}{17}+\frac{11}{17}i\)

Integration (4)*

Evaluate:

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


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

Trig (Identities)*

Simplify:

$$\sin{x}\cot{x}$$

\(\cos{x}\)

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

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

Miscellaneous

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:
$$ |(3+4i)z| $$

\(40\)

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.

?


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