# ADVANCED ### 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 £200 is invested with an interest rate of 6% compounded quarterly, find the value of the investment after 9 years. £341.83

### Coordinates (Square)

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

$$(2,4),(5,10),(-4,7)$$

(-1,13)

### Normal Distribution

$$X \sim N(50, 5^2)$$

Find

$$P(40\lt X \lt60)$$

$$0.955$$

Factorise:

$$x^2-2x-8$$

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

Factorise:

$$3x^2-10x-8$$

$$(3x+2)(x-4)$$

### Graph (Linear)

Draw a rough sketch of the graph of:

$$y=x-1$$

y intercept -1

### Indices

What is the value of:

$$27^{\frac{1}{3}}$$

$$= 3$$

### Trigonometry (Angle)

Find angle ABC if AC = 4.9m and AB = 6.5m. 37.0o

### Trigonometry (Side)

Find BC if angle BCA = 25o and AC = 5.5m. 6.07m

### Venn Diagrams

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

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

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

$$6x^2 - 18x + 3$$

### Differentiation (2)

$$y = \dfrac{7}{x^8} - 5\sqrt{x}$$

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

$$-\frac{56}{x^9} - \frac{5}{6}x^{-\frac{5}{6}}$$

### Differentiation (3)

$$y=e^{2x+3}$$

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

$$2e^{2x+3}$$

### Differentiation (4)

$$y=x^2 \ln x$$

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

$$2x^1lnx+x^1$$

### Differentiation (5)

$$y=\frac{e^{3x}}{ \cos 4x}$$

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

$$\frac{(3e^{3x}cos4x+4e^{3x}sin4x)}{cos^24x}$$

### Differentiation (6)

Find the equation of the tangent to the curve:
$$y = 2x^2 - x + 3$$
where $$x = -1$$
$$y = 1 - 5x$$

### 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)

$$y =21x^2 - 8x + 8$$

Find $$\int y \quad dx$$

$$7x^3 - 4x^2 + 8x+c$$

### Binomial Distribution

A game is played 20 times and the probability of winning is 0.6. Calculate the probability of winning exactly 5 times.   0.00129

### Formulas

What's this?

$$S_n = \dfrac{u_1(r^n-1)}{r-1}$$

The sum of a geometric sequence

### Greek Letters

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

Two terms of an arithmetic sequence:
$$u_{5} = 40$$
$$u_{12} = 117$$
Find the sum of the first 29 terms.4350

### Asymptotes (HV)

Find the equations of the asymptotes of:

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

$$x=2,y=-2$$

In the triangle ABC,
AB = 5.4cm.
BC = 7.8cm.
CÂB = 89.3°.
Find angle BĈA.

43.8°

### Sigma

Evaluate:

$$\sum_{n=4}^{5} n^2 - 3n$$

14

### Discriminant

$$f(x)=-4x^2+7x-5$$

What is the value of the discriminent and what does it indicate?
-31, No real roots

### Completing The Square

$$f(x)=x^2+7x+2$$

By completing the square find the coordinates of the vertex.
(-3.5, -10.25)

### Logarithms

Solve for x:

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

$$x = 4 \text{ or } x = 25$$

### 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:

(-1, -7) and (5, -19)

$$y=-2x-9$$

### Functions (Inverse)

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

$$f(x)=\frac{8+ x}{6}$$

$$6x-8$$

### Functions (Composite)

$$f(x)=7x-3 \\ g(x)=3x^2 \\[1cm] \text{Find }g \circ f(x)$$

$$147x^2-126x+27$$

### Standard Form

Write in standard form:
$$(a \times 10^p) \div (b\times 10^q)$$
where $$a \div b$$ is a single digit number $$(1 \le \frac{a}{b} \lt 10)$$

$$\frac{a}{b}\times10^{p-q}$$

### Graph (Mixed)

Draw a rough sketch of

$$x=\pm \sqrt{y}$$ ### Graph (Fill)

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

Without a calculator find the exact value of

$$\cos{30°} \div \sin{\frac{\pi}{3}}$$

$$1$$

### Trig (Large Angles)

Without a calculator find the exact value of

$$\cos{\dfrac{16\pi}{3}}$$

$$-\dfrac{1}{2}$$

### Simultaneous Eqns (3)*

Solve:

$$5a+2b+c=45 \\ 3a+4b+2c= 48 \\ a+5b+c=36$$

x=6, y=5, z=5

Find the perimeter of a sector with radius 4.8cm and angle $$\frac{\pi}{6}$$

🍕

12.1cm

### Combinatronics*

A safe has a four-digit code. How many possibilities are there if no digit can be repeated and the code must be odd?

2520

### Asymptotes (Ob)*

Find the equations of the asymptotes of:

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

x=1,y=x+2

### Sequences (Geometric)

Evaluate:
$$\sum_{k=1}^{8} \left( \dfrac12 \right)^{k-3}$$

7.97

### Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

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

$$1-8x-36x^2-120x^3$$

### Integration (2)

Evaluate:

$$\int^{100}_{50} \dfrac{1}{x} dx$$

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

### Probability (Conditional)

Box A contains 5 red and 7 blue cubes, and box B contains 9 red and 12 blue cubes. Finn selects a box at random and takes a cube from that box. Given that the cube is red, what is the probability that it came from box B?

$$\dfrac{36}{71}$$

### Vectors*

There are eight different ways that three planes can relate in three dimensions. One is where they all intersect at a point. How many of the other seven ways can you sketch or describe?

Solution

Sketch the graph of:

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

Graph Plotter

### Complex Numbers*

Simplify
$$(2-i)^{-2}$$

$$\frac{3}{25}+\frac{4}{25}i$$

### Integration (4)

Evaluate:

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

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

### Trig (Identities)*

Simplify:

$$\tan{x}\cot{x}$$

$$1$$

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

### 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

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
$$f(x) = \ln(1 + x)$$

$$x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$$

### Last Lesson

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