# ADVANCED ### Binomial Theorem (1)

Find the first three terms in the expansion of:

$$(2a - 3b)^8$$

$$=256a^8 - 3072a^7b \\+16128a^6b^2 ...$$

### Compound Interest

If £140 is invested with an interest rate of 5% compounded monthly, find the value of the investment after 5 years. £179.67

### Coordinates (Square)

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

$$(1,2),(5,7),(-4,6)$$

(0,11)

### Normal Distribution

$$X \sim N(300, 10^2)$$

Find

$$P(270\lt X \lt330)$$

$$0.997$$

Factorise:

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

$$(x+4)(x-1)$$

Factorise:

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

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

### Graph (Linear)

Draw a rough sketch of the graph of:

$$y=x-1$$

y intercept -1

### Indices

What is the value of:

$$5^{-3}$$

$$= \frac{1}{125}$$

### Trigonometry (Angle)

Find angle ABC if AC = 4.7m and BC = 6.6m. 45.4o

### Trigonometry (Side)

Find AC if angle ABC = 22o and BC = 5.2m. 1.95m

### Venn Diagrams

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

$$y = 4x^3 - 5x^2 + 9x$$

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

$$12x^2 - 10x + 9$$

### Differentiation (2)

$$y = \dfrac{9}{x^6} - 3\sqrt{x}$$

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

$$-\frac{54}{x^7} - \frac{3}{4}x^{-\frac{3}{4}}$$

### Differentiation (3)

$$y=(9x^2-3)^6$$

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

$$108x(9x^2-3)^5$$

### Differentiation (4)

$$y=\sin x \cos x$$

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

$$cos^2x-sin^2x$$

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

### Integration (1)

$$y =27x^2 - 14x + 8$$

Find $$\int y \quad dx$$

$$9x^3 - 7x^2 + 8x+c$$

### Binomial Distribution

A game is played 17 times and the probability of winning is 0.2. Calculate the probability of winning exactly 6 times.   0.0680

### Formulas

What's 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:
$$u_{10} = -82$$
$$u_{13} = -106$$
Find the sum of the first 23 terms.-2254

### Asymptotes (HV)

Find the equations of the asymptotes of:

$$y=\dfrac{4-7x}{3-14x}$$

$$x=\frac{3}{14},y=\frac{1}{2}$$

In the triangle ABC,
AB = 5.3cm.
BC = 7.9cm.
CÂB = 95.2°.
Find angle BĈA.

41.9°

### Sigma

Evaluate:

$$\sum_{n=0}^{8} 116 - n^2$$

840

### Discriminant

$$f(x)=6x^2-5x-9$$

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

### Completing The Square

$$f(x)=x^2+8x-7$$

By completing the square find the coordinates of the vertex.
(-4, -23)

### Logarithms

Solve for x:

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

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

### Integration (3)

Find the integral:

$$\int (x+3)\sqrt{x^2+6x+8} \;dx$$

$$\frac{1}{3}(x^2+6x+8)^{\frac32}+c$$

### Graph (2 points)

Find the equation of the straight line that passes through:

(-7, 5) and (4, -17)

$$y=-2x-9$$

### Functions (Inverse)

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

$$f(x)= \sqrt{x}-7$$

$$(x+7)²$$

### Functions (Composite)

$$f(x)=5x+1 \\ g(x)=x^2 \\[1cm] \text{Find }gf(3x)$$

$$225x^2+30x+1$$

### 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

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

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

Without a calculator find the exact value of

$$\sin{\frac{\pi}{2}} \div \cos{45°}$$

$$\sqrt{2}$$

### Trig (Large Angles)

Without a calculator find the exact value of

$$\sin{5\pi}$$

$$0$$

### Simultaneous Eqns (3)*

Solve:

$$2d+3e-4f = 23 \\ d-e-f= -3\\ 9d+2e-2f=71$$

d = 7, e = 7, f = 3

Find the perimeter of a sector with radius 3.2cm and angle $$\frac{\pi}{3}$$

🍕

9.75cm

### Combinatronics*

How many ways can five children sit in a row without the youngest being in the middle?

96

### Asymptotes (Ob)*

Find the equations of the asymptotes of:

$$y=\dfrac{2x^2+3x-9}{x+2}$$

x=-2,y=2x-1

### Sequences (Geometric)

The 6th term of a geometric sequence is 160 and the sum of the first 6 terms is 315. Find the first term.

5

### Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

$$\dfrac{1}{(1+3x)^3}$$

$$1-9x+54x^2-270x^3$$

### Integration (2)

Evaluate:

$$\int^{5}_{2} x^2-2x+7 \; dx$$

$$6$$

### Probability (Conditional)

27 Scouts went hiking. 16 got lost, 15 got blisters, and 9 got both lost and blisters. Find the probability that a randomly selected Scout got blisters, given that they were not lost.

$$\dfrac{6}{11}$$

### 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:

$$|x|=|y|$$

Graph Plotter

### Complex Numbers 1*

Simplify
$$\sqrt{5-12i}$$

$$3-2i \; \text{ or } -3+2i$$

### Integration (4)

Evaluate:

$$\int x\cos{x}\; dx$$

$$xsinx+cosx+c$$

### Trig (Identities)*

Simplify:

$$\cosec{x}\tan{x}$$

$$\sec{x}$$

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

### 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

How do you solve a quadratic inequality?

Factorise the quadratic, then analyze the sign of each factor over its domain.

### Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
$$f(x) = x^3$$

$$x^3 \text{ only 1 term}$$

### Complex Numbers 2*

Expand and simplify:
$$(\sqrt{3}+i)^8$$

$$-128-128\sqrt{3}i \\ \text{or } 128(-1-\sqrt{3}i)$$

### Probability (Counting)*

A team of 11 is randomly chosen from a squad of 18 including the club captain and vice captain. Determine the probability that both the captain and vice-captain are chosen.

55/153 or 35.9%

### Proof by Induction*

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

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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