### 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 £160 is invested with an interest rate of 6% compounded monthly, find the value of the investment after 9 years. £274.19

### Coordinates (Square)

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

$$(2,4),(6,9),(-3,8)$$

(1,13)

### Normal Distribution

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

Find

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

$$0.955$$

Factorise:

$$x^2-1$$

$$(x+1)(x-1)$$

Factorise:

$$10x^2+x-2$$

$$(2x+1)(5x-2)$$

### Graph (Linear)

Draw a rough sketch of the graph of:

$$y=-2x-1$$

y intercept -1

### Indices

What is the value of:

$$4^{\frac{1}{2}}$$

$$= 2$$

### Trigonometry (Angle)

Find angle ABC if AC = 4.9m and BC = 6.1m. 53.4o

### Trigonometry (Side)

Find AB if angle ABC = 43o and BC = 3.3m. 2.41m

### Venn Diagrams

Describe the red region.

### Differentiation (1)

$$y = 2x^3 - 8x^2 + 2x$$

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

$$6x^2 - 16x + 2$$

### Differentiation (2)

$$y = \dfrac{7}{x^4} - 7\sqrt[8]{x}$$

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

$$-\frac{28}{x^5} - \frac{7}{8}x^{-\frac{7}{8}}$$

### Differentiation (3)

$$y=(6x+5)^4$$

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

$$24(6x+5)^3$$

### Differentiation (4)

$$y=x(3x^2+4)^6$$

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

$$(3x^2+4)^6+36x^2(3x^2+4)^5$$

### Differentiation (5)

$$y=\frac{x^2+5}{3x-7}$$

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

$$\frac{(3x^2-14x-15)}{(3x-7)^2}$$

### Differentiation (6)

Find the equation of the tangent to the curve:
$$y = 5x^2 + 7x + 3$$
where $$x = 1$$
$$y = 17x - 2$$

### Differentiation (7)

Find the equation of the normal to the curve:
$$y = -x^2 + 4x + 2$$
where $$x = 1$$
$$y = 5\frac{1}{2} - \frac{x}{2}$$

### Integration (1)

$$y =24x^2 - 8x + 2$$

Find $$\int y \quad dx$$

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

### Binomial Distribution

A game is played 15 times and the probability of winning is 0.3. Calculate the probability of winning exactly 2 times.   0.0916

### Formulas

Make up a maths question using this:

$$\int \dfrac{1}{x} = \ln |x| + c$$

Reciprocal Integral formula

### Greek Letters

What letter is this?

### Sequences (Arithmetic)

Two terms of an arithmetic sequence:
$$u_{9} = 78$$
$$u_{18} = 168$$
Find the sum of the first 46 terms.10258

### Asymptotes (HV)

Find the equations of the asymptotes of:

$$y=\dfrac{2x-7}{6-2x}-5$$

$$x=3,y=-6$$

In the triangle ABC,
BĈA = 55.2°.
BC = 8.6cm.
AB̂C = 40.4°.
Find CA to 1 dp.

5.6cm

### Sigma

Evaluate:

$$\sum_{n=2}^{6} n^2 - 9n$$

-90

### Discriminant

$$f(x)=5x^2+8x-9$$

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

### Completing The Square

$$f(x)=x^2+5x+2$$

By completing the square find the coordinates of the vertex.
(-2.5, -4.25)

### Logarithms

Solve for x:

$$\log_3x = 2$$

9

### Integration (3)

Find the integral:

$$\int \dfrac{\ln(x)}{x} \;dx$$

$$\dfrac{\ln(x)^2}{2}+c$$

### Graph (2 points)

Find the equation of the straight line that passes through:

(-5, 4) and (1, -8)

$$y=-2x-6$$

### Functions (Inverse)

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

$$f(x)= \sqrt{x+15}$$

$$x²-15$$

### 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:
$$(a \times 10^4) \div (b\times 10^{-2})$$
where $$a \div b$$ is a decimal number $$(0.1 \le \frac{a}{b} \lt 1)$$

$$\frac{10a}{b}\times10^5$$

### Graph (Mixed)

Draw a rough sketch of

$$y=x(5-x)$$

### 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

$$\sin{5\pi}$$

$$0$$

### Simultaneous Eqns (3)*

Solve:

$$2x+y-3z= 9 \\ 3x+y+z= 33 \\ x-y+2z = 11$$

x = 8, y = 5, z = 4

Find the perimeter of a sector with radius 4.6cm and angle $$\frac{2\pi}{3}$$

🍕

18.8cm

### Combinatorics*

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)

The 6th term of a geometric sequence is 15625 and the sum of the first 6 terms is 19530. Find the first term.

5

### Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

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

$$\frac{1}{2}+\frac{x}{4}+\frac{x^2}{8}+\frac{x^3}{16}$$

### Integration (2)

Evaluate:

$$\int^{5}_{1} (x-8)^2 \; dx$$

$$105.333333333333$$

### Probability (Conditional)

The probability that I drop and brake my phone when I visit a coffee shop is 0.1. Today I visited two coffee shops and broke my phone in one of them. What is the probability that it was the first shop where the accident occurred?

$$0.526$$

### 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:

$$|x|=|y|$$

Graph Plotter

### Complex Numbers 1*

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

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

### Integration (4)*

Evaluate:

$$\int x\tan^{-1}x\; dx$$

$$(\frac{x^2+1}{2})tan^{-1}x-\frac{x}{2}+c$$

### Trig (Identities)*

Simplify:

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

$$\cos{x}$$

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

### 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 binomial theorem?

Clue: Expand $$(a + b)^n$$

### 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*

Find the four 4th roots of 1

$$1, i, -1, -i$$

### Probability (Counting)*

A committee of 5 is chosen from 15 people by random selection. Two Londoners were amongst the group from which the selection was made. Find the probability that both Londoners are chosen for the committee.

2/21 or 9.52%

### Proof by Induction*

Prove by mathematical induction that the sum of the first $$n$$ natural numbers is $$\frac{n(n + 1)}{2}$$

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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