ADVANCED
Refreshing Revision

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 3% compounded quarterly, find the value of the investment after 6 years. £167.50

Coordinates (Square)

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

\((5,5),(8,10),(0,8)\)

(3,13)

Normal Distribution

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

Find

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

\(0.997\)

Factorise (Quadratic 1)

Factorise:

\(x^2-x-6\)

\((x+2)(x-3)\)

Factorise (Quadratic 2)

Factorise:

\(3x^2-7x-6\)

\((3x+2)(x-3)\)

Graph (Linear)

Draw a rough sketch of the graph of:


\(y=-x+1\)

Gradient -1
y intercept 1

Indices

What is the value of:

\(4^{0}\)

\(= 1\)

Trigonometry (Angle)

Find angle ABC if AC = 5.1m and BC = 6.2m. 55.3o

Trigonometry (Side)

Find AC if angle ABC = 32o and AB = 4.5m. 2.81m

Venn Diagrams

Describe the red region.

Circle part Circle part

Differentiation (1)

\(y = 3x^3 - 5x^2 + 6x\)

Find \( \dfrac{dy}{dx}\)

\(9x^2 - 10x + 6\)

Differentiation (2)

\(y = \dfrac{4}{x^6} - 2\sqrt[3]{x}\)

Find \( \frac{dy}{dx}\)

\(-\frac{24}{x^7} - \frac{2}{3}x^{-\frac{2}{3}}\)

Differentiation (3)

\(y=5\ln (8x^2+9)\)

Find \( \dfrac{dy}{dx}\)

\(80x(8x^2+9)^{-1}\)

Differentiation (4)

\(y=x \tan x\)

Find \( \dfrac{dy}{dx}\)

\(tanx+\frac{x}{cos^2x}\)

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:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = 1 - 2x\)

Differentiation (7)

Find the equation of the normal to the curve:
\(y = 5x^2 + 7x + 3\)
where \(x = 1\)
\(y = 15\frac{1}{17} - \frac{x}{17}\)

Integration (1)

\(y =24x^2 - 6x + 6\)

Find \( \int y \quad dx\)

\(8x^3 - 3x^2 + 6x+c\)

Binomial Distribution

A game is played 17 times and the probability of winning is 0.7. Calculate the probability of winning exactly 11 times.   0.178

Formulas

Make up a maths question using this:

\( A = 4\pi r^2 \)

Surface area of a sphere

Greek Letters

What letter is this?

Greek Letter Greek Letter

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
\(u_{5} = -39\)
\(u_{11} = -93\)
Find the sum of the first 42 terms.-7875

Asymptotes (HV)

Find the equations of the asymptotes of:

\(y=\dfrac{2x+5}{2x+3}\)

\(x=-\frac{3}{2},y=1\)

Trig Advanced

In the triangle ABC,
AB = 5.1cm.
BC = 9.7cm.
CA = 9.2cm.
Find angle CÂB.

79.8°

Sigma

Evaluate:

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

24

Discriminant

\(f(x)=-2x^2+3x+2\)

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

Completing The Square

\(f(x)=x^2-3x-6\)

By completing the square find the coordinates of the vertex.
(1.5, -8.25)

Logarithms

Solve for x:

\(\log_3x = 2\)


9

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:

(-3, 1) and (7, -19)

\(y=-2x-5\)

Functions (Inverse)

Find the inverse of the function \(f\):

\(f(x)=\sqrt{\frac{x-3}{7}}\)


\(7x²+3\)

Functions (Composite)

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

\(18x^2+24x+8\)

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

\(x=\pm \sqrt{y}\)

Sketch

Graph (Fill)

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

Jar Graph

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:

\(2d+3e-4f = -12 \\ d-e-f= -9\\ 9d+2e-2f=12\)

d = 2, e = 4, f = 7

Radian Measures

Find the area of a sector with radius 9.1cm and angle \( \frac{\pi}{4}\)

🍕

32.5cm2

Combinatronics*

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

4320

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:
$$ \sum_{k=1}^{15} 5 \times (-2)^k $$

-109230

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:

\(\int^{4}_{1} (x-8)^2 \; dx\)


\(93\)

Probability (Conditional)

32 Scouts went hiking. 18 got lost, 13 got blisters, and 8 got both lost and blisters. Find the probability that a randomly selected Scout got blisters, given that they were not lost.

\(\dfrac{5}{14}\)

Vectors*

Find the point of intersection of \(L_1\) and \(L_2\) if:

\(L_1: \quad \dfrac{x+4}{3} = y-2 = \dfrac{z+1}{2} \)

\(L_2: \quad x = \dfrac{y-5}{2} = \dfrac{-z-1}{2} \)

\( (-1,3,1) \)

Graph (Advanced)*

Sketch the graph of:

$$|x| + |y| = 1$$

Graph Plotter

Complex Numbers 1*

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

\(-4\)

Integration (4)*

Evaluate:

\(\int x\cos{x}\; dx\)


\(xsinx+cosx+c\)

Trig (Identities)*

Simplify:

$$\dfrac{\sin^2{x}-1}{\cos{x}}$$

\(-\cos{x}\)

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

Integration (Volume)*

Find the volume of revolution when \(y=x\) is rotated about the x-axis for \(0 \le x \le 1\)


\(\frac{\pi}{3}\) 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
\(f(x) = x^3\)

\(x^3 \text{ only 1 term}\)

Complex Numbers 2*


Find the four 4th roots of 1

\(1, i, -1, -i\)

Probability (Counting)*

5 alphabet blocks A, E, P, R and S are placed at random in a row. What is the likelihood that they spell out either SPEAR or PARSE?

1/60 or 1.67%

Proof by Induction*

Prove by mathematical induction that \( 11^n - 6 \) is divisible by 5 for all natural numbers \( n \)

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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