ADVANCED
Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

\((2a - 3b)^4\)

\(=16a^4 - 96a^3b \\+216a^2b^2 ...\)

Compound Interest

If £120 is invested with an interest rate of 5% compounded monthly, find the value of the investment after 4 years. £146.51

Coordinates (Square)

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

\((3,3),(7,8),(-2,7)\)

(2,12)

Normal Distribution

\( X \sim N(4.5, 0.35^2)\)

Find

\( P(4.1\lt X \lt4.5) \)

\(0.373\)

Factorise (Quadratic 1)

Factorise:

\(x^2+x-2\)

\((x+2)(x-1)\)

Factorise (Quadratic 2)

Factorise:

\(x^2-4\)

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

Graph (Linear)

Draw a rough sketch of the graph of:


\(y=-x+1\)

Gradient -1
y intercept 1

Indices

What is the value of:

\(2^{-3}\)

\(= \frac{1}{8}\)

Trigonometry (Angle)

Find angle BCA if AB = 4m and AC = 5.5m. 36.0o

Trigonometry (Side)

Find AB if angle ABC = 53o and BC = 5.2m. 3.13m

Venn Diagrams

Describe the red region.

Circle part Circle part

Differentiation (1)

\(y = 8x^3 - 6x^2 + 2x\)

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

\(24x^2 - 12x + 2\)

Differentiation (2)

\(y = \dfrac{4}{x^5} - 2\sqrt[3]{x}\)

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

\(-\frac{20}{x^6} - \frac{2}{3}x^{-\frac{2}{3}}\)

Differentiation (3)

\(y=\sqrt{3x^4-8x}\)

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

\((6x^3-4)(3x^4-8x)^{-\frac{1}{2}}\)

Differentiation (4)

\(y=\sin x \sqrt{ x^2 + 4}\)

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

\(cosx \sqrt{x^2+4}+\frac{xsinx}{\sqrt{x^2+4}}\)

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:
\(y = -5x^2 + 7x - 3\)
where \(x = 2\)
\(y = \frac{x}{13} - \frac{119}{13}\)

Integration (1)

\(y =18x^2 - 16x + 3\)

Find \( \int y \quad dx\)

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

Binomial Distribution

A game is played 10 times and the probability of winning is 0.9. Calculate the probability of winning exactly 7 times.   0.0574

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:
\(u_{6} = 39\)
\(u_{12} = 81\)
Find the sum of the first 40 terms.5620

Asymptotes (HV)

Find the equations of the asymptotes of:

\(y=\dfrac{10-2x}{10x}\)

\(x=0,y=-{1}{5}\)

Trig Advanced

In the triangle ABC,
AB = 7.7cm.
BC = 6.6cm.
CÂB = 34.5°.
Find angle BĈA.

41.4° or 138.6°

Sigma

Evaluate:

$$\sum_{n=3}^{8} n^2 - 6n$$

1

Discriminant

\(f(x)=-3x^2-6x+9\)

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

Completing The Square

\(f(x)=x^2+8x+5\)

By completing the square find the coordinates of the vertex.
(-4, -11)

Logarithms

Solve for x:

\(\log_3x = 2\)


9

Integration (3)

Find the integral:

\(\int \sin(x)\cos^2(x) \;dx\)


\(-\frac{1}{3} \cos^3(x)+c\)

Graph (2 points)

Find the equation of the straight line that passes through:

(-7, -10) and (7, 18)

\(y=2x+4\)

Functions (Inverse)

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

\(f(x)=\sqrt{\frac{x-3}{4}}\)


\(4x²+3\)

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:
\(a \times 10^p \times b\times 10^q\)
where \(a \times b \) is a two digit number \((10 \le ab \lt 100)\)

\(\frac{ab}{10}\times10^{p+q+1}\)

Graph (Mixed)

Draw a rough sketch of

\(y=x^3-4x\)

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

$$\cos{\frac{\pi}{6}} \times \cos{60°}$$

\(\dfrac{\sqrt{3}}{4}\)

Trig (Large Angles)

Without a calculator find the exact value of

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

\(-\dfrac{1}{2}\)

Simultaneous Eqns (3)*

Solve:

\( j+k+l= 21 \\ 2j-3k+9l= 54\\ -j+k-3l=-21\)

j = 9, k = 6, l = 6

Radian Measures

Find the area of a sector with radius 4.7cm and angle \( \frac{\pi}{4}\)

🍕

8.67cm2

Combinatronics*

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

75600

Asymptotes (Ob)*

Find the equations of the asymptotes of:

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

x=0,y=3-x

Sequences (Geometric)

The 7th term of a geometric sequence is 128 and the sum of the first 7 terms is 254. Find the first term.

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:

\(\int^{\pi/3}_{\pi/6} \sin{x} \; dx\)


\(\dfrac{\sqrt{3}-1}{2}\)

Probability (Conditional)

The probability that it is cloudy on a particular day is 0.2. The probability that it is cloudy with a high level of pollution on a particular day is 0.1. Find the probability that there will be a high level of pollution on a day when it is cloudy.

\(0.500\)

Vectors*

Find the area of the triangle with sides:

\( \begin{pmatrix} 2 \\ 7 \\ 0 \end{pmatrix}, \; \begin{pmatrix} 6 \\ -5 \\ 7 \end{pmatrix} \; \text{and} \; \begin{pmatrix} 4 \\ -12 \\ 7 \end{pmatrix} \)

36.4 square units

Graph (Advanced)*

Sketch the graph of:

$$y=\cos^2x$$

Graph Plotter

Complex Numbers 1*

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

\(3-2i \; \text{ or } -3+2i\)

Integration (4)*

Evaluate:

\(\int xe^x\; dx\)


\(xe^x-e^x+c\)

Trig (Identities)*

Simplify:

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

\(\sec{x}\)

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

Integration (Volume)*

Find the volume of revolution when \(y=x^2\) is rotated about the x-axis for \(-2 \le x \le 2\)


\(\frac{64\pi}{5}\) cubic units

Miscellaneous

What is the formula for compound interest?

\( A = P(1 + \frac{r}{n})^{nt} \)

Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \ln(1 + x)\)

\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\)

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 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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