### 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:

$$x^2-9$$

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

Factorise:

$$2x^2-x-3$$

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

### Graph (Linear)

Draw a rough sketch of the graph of:

$$y=-x-2$$

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.

### 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?

### 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:

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

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

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

$$y=x^3-4x$$

### Graph (Fill)

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

### 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

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

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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