# ADVANCED ### Binomial Theorem (1)

Find the first three terms in the expansion of:

$$(4a - 3b)^8$$

$$=65536a^8 - 393216a^7b \\+1032192a^6b^2 ...$$

### Compound Interest

If £240 is invested with an interest rate of 4% compounded quarterly, find the value of the investment after 9 years. £343.38

### Coordinates (Square)

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

$$(3,2),(9,5),(0,8)$$

(6,11)

### Normal Distribution

$$X \sim N(-25, 3^2)$$

Find

$$P(-20\lt X \lt-10)$$

$$0.0478$$

Factorise:

$$x^2-x-6$$

$$(x+2)(x-3)$$

Factorise:

$$5x^2+18x-8$$

$$(x+4)(5x-2)$$

### Graph (Linear)

Draw a rough sketch of the graph of:

$$y=2x+2$$

y intercept 2

### Indices

What is the value of:

$$1^{-2}$$

$$= 1$$

### Trigonometry (Angle)

Find angle BCA if AC = 4.1m and BC = 5.5m. 41.8o

### Trigonometry (Side)

Find AB if angle ABC = 50o and BC = 3m. 1.93m

### Venn Diagrams

Describe the red region.  ### Differentiation (1)

$$y = 7x^3 - 5x^2 + 9x$$

Find $$\dfrac{dy}{dx}$$

$$21x^2 - 10x + 9$$

### Differentiation (2)

$$y = \dfrac{9}{x^5} - 4\sqrt{x}$$

Find $$\frac{dy}{dx}$$

$$-\frac{45}{x^6} - \frac{4}{5}x^{-\frac{4}{5}}$$

### Differentiation (3)

$$y=e^{\cos x}$$

Find $$\dfrac{dy}{dx}$$

$$-sinxe^{cosx}$$

### Differentiation (4)

$$y=\sin x \sqrt{ x^2 + 5}$$

Find $$\dfrac{dy}{dx}$$

$$cosx \sqrt{x^2+5}+\frac{xsinx}{\sqrt{x^2+5}}$$

### 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:
$$y = 3x^2 - 6x + 9$$
where $$x = 2$$
$$y = 9\frac{1}{3} - \frac{x}{6}$$

### Integration (1)

$$y =21x^2 - 10x + 2$$

Find $$\int y \quad dx$$

$$7x^3 - 5x^2 + 2x+c$$

### Binomial Distribution

A game is played 14 times and the probability of winning is 0.5. Calculate the probability of winning exactly 11 times.   0.0222

### Formulas

What's this?

$$z=\dfrac{x-\mu}{\sigma}$$

Standardised Normal Variable

### Greek Letters

What letter is this?  ### Sequences (Arithmetic)

Two terms of an arithmetic sequence:
$$u_{9} = -54$$
$$u_{15} = -102$$
Find the sum of the first 49 terms.-8918

### Asymptotes (HV)

Find the equations of the asymptotes of:

$$y=10+\dfrac{9x}{5-3x}$$

$$x=\frac{5}{3},y=7$$

In the triangle ABC,
AB = 9.3cm.
BC = 9.4cm.
CÂB = 67.1°.
Find angle BĈA.

65.7°

### Sigma

Evaluate:

$$\sum_{n=0}^{6} 101 - n^2$$

616

### Discriminant

$$f(x)=3x^2-3x-3$$

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

### Completing The Square

$$f(x)=x^2+3x+8$$

By completing the square find the coordinates of the vertex.
(-1.5, 5.75)

### Logarithms

Simplify $$\log_{10}10^5$$

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:

(-7, 3) and (2, -6)

$$y=-x-4$$

### Functions (Inverse)

Find the inverse of the function $$f$$:

$$f(x)=\sqrt{\frac{x-6}{4}}$$

$$4x²+6$$

### Functions (Composite)

$$\text{Find }f(x) \text{ if} \\ f(1-b)=3-b \\$$

$$f(x)=x+2$$

### Standard Form

Write in standard form:
$$(a \times 10^3) \div (b\times 10^5)$$
where $$a \div b$$ is a two digit number $$(10 \le \frac{a}{b} \lt 100)$$

$$\frac{a}{10b}\times10^{-1}$$

### Graph (Mixed)

Draw a rough sketch of

$$y=3-\dfrac{10}{x}$$ ### Graph (Fill)

Sketch a height-time graph as this jar is filled.  ### Trig (Special Angles)

Without a calculator find the exact value of

$$\tan{30°} \times \tan{\frac{\pi}{3}}$$

$$1$$

### Trig (Large Angles)

Without a calculator find the exact value of

$$\tan{4\pi}$$

$$0$$

### Simultaneous Eqns (3)*

Solve:

$$2d+3e-4f = -2 \\ d-e-f= -2\\ 9d+2e-2f=32$$

d = 4, e = 2, f = 4

Find the area of a sector with radius 6.9cm and angle $$\frac{2\pi}{3}$$

🍕

49.9cm2

### Combinatronics*

In how many ways can 7 different books be arranged on a shelf if 4 of them must be together?

576

### Asymptotes (Ob)*

Find the equations of the asymptotes of:

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

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

### Sequences (Geometric)

Evaluate:
$$\sum_{k=1}^{10} 3 \times 2^{k-1}$$

3069

### Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

$$(1+2x)^{\frac12}$$

$$1+x-\frac{x^2}{2}+\frac{x^3}{2}$$

### Integration (2)

Evaluate:

$$\int^{160}_{80} \dfrac{1}{x} dx$$

$$\ln{2} \approx 0.693$$

### Probability (Conditional)

What is the probability that it was machine B that broke down if at least one of the independent machines broke down today, given machine A has a 5% chance and machine B has a 8% chance of breaking down on any given day?

$$0.635$$

### 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

Sketch the graph of:

$$y=1^{\sin{x}}$$

Graph Plotter

### Complex Numbers 1*

Simplify
$$(1+i)^{4}$$

$$-4$$

### Integration (4)

Evaluate:

$$\int e^x\sin{x}\; dx$$

$$\frac{e^x}{2}(sinx-cosx)+c$$

### Trig (Identities)*

Simplify:

$$\dfrac{\tan{x}}{\sec{x}}$$

$$\sin{x}$$

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

### Integration (Volume)*

Find the volume of revolution when $$y=\sqrt{x^2}$$ is rotated about the y-axis for $$2 \le y \le 3$$

$$\frac{65\pi}{4}$$ cubic units

### Miscellaneous

What is the difference between a permutation and a combination?

Permutations consider order; combinations do not.

### Maclaurin Series*

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

$$1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}$$

### Complex Numbers 2*

Find the five 5th roots of 1

$$1, cis\frac{2\pi}{5}, cis\frac{4\pi}{5},\\ cis\frac{-2\pi}{5}, cis\frac{-4\pi}{5}$$

### 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 $$2^n > n^2$$ for all integers $$n$$ greater
than four

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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