ADVANCED
Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

\((3a - 4b)^8\)

\(=6561a^8 - 69984a^7b \\+326592a^6b^2 ...\)

Compound Interest

If £120 is invested with an interest rate of 3% compounded quarterly, find the value of the investment after 6 years. £143.57

Coordinates (Square)

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

\((1,3),(6,8),(-4,8)\)

(1,13)

Normal Distribution

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

Find

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

\(0.373\)

Factorise (Quadratic 1)

Factorise:

\(x^2+2x-8\)

\((x+4)(x-2)\)

Factorise (Quadratic 2)

Factorise:

\(8x^2-10x-3\)

\((4x+1)(2x-3)\)

Graph (Linear)

Draw a rough sketch of the graph of:


\(y=x\)

Gradient 1
y intercept 0

Indices

What is the value of:

\(1^{-3}\)

\(= 1\)

Trigonometry (Angle)

Find angle BCA if AC = 4.6m and BC = 5.6m. 34.8o

Trigonometry (Side)

Find AB if angle ABC = 65o and BC = 5.1m. 2.16m

Venn Diagrams

Describe the red region.

Circle part Circle part

Differentiation (1)

\(y = 3x^3 - 2x^2 + 7x\)

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

\(9x^2 - 4x + 7\)

Differentiation (2)

\(y = \dfrac{2}{x^3} - 5\sqrt[6]{x}\)

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

\(-\frac{6}{x^4} - \frac{5}{6}x^{-\frac{5}{6}}\)

Differentiation (3)

\(y=e^{5x+6}\)

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

\(5e^{5x+6}\)

Differentiation (4)

\(y=6x^2e^x\)

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

\(12xe^x+6x^2e^x\)

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:
\(y = x^2 + 6x + 9\)
where \(x = -3\)
\(y = 0\)

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)

\(y =6x^2 - 10x + 2\)

Find \( \int y \quad dx\)

\(2x^3 - 5x^2 + 2x+c\)

Binomial Distribution

A game is played 12 times and the probability of winning is 0.7. Calculate the probability of winning exactly 8 times.   0.231

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:
\(u_{9} = 46\)
\(u_{16} = 95\)
Find the sum of the first 29 terms.2552

Asymptotes (HV)

Find the equations of the asymptotes of:

\(y=3\left(\dfrac{2x+3}{7-x}\right)\)

\(x=7,y=-6\)

Trig Advanced

In the triangle ABC,
BĈA = 48.8°.
BC = 8.4cm.
AB̂C = 49.9°.
Find CA to 1 dp.

6.5cm

Sigma

Evaluate:

$$\sum_{n=2}^{7} n^2 - 3n$$

58

Discriminant

\(f(x)=-9x^2+5x-4\)

What is the value of the discriminant and what does it indicate?
-119, 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

Simplify:


\(\log_2(4\sqrt{16})\)


4

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:

(-2, -4) and (0, 2)

\(y=3x+2\)

Functions (Inverse)

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

\(f(x)= \sqrt{x-8}\)


\(x²+8\)

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:
\((a \times 10^3) \div (b\times 10^5)\)
where \(a \div b \) is a two digit number \((10 \le \frac{a}{b} \lt 100)\)

\(\frac{a}{10b}\times10^{-1}\)

Graph (Mixed)

Draw a rough sketch of

\(y=x^2+7x\)

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

$$\cos{0°} + \sin{\frac{\pi}{6}} + \cos{60°}$$

\(2\)

Trig (Large Angles)

Without a calculator find the exact value of

$$\tan{4\pi}$$

\(0\)

Simultaneous Eqns (3)*

Solve:

\(2x+y-3z= 14 \\ 3x+y+z= 25 \\ x-y+2z = 6\)

x = 7, y = 3, z = 1

Radian Measures

Find the perimeter of a sector with radius 4.2cm and angle \( \frac{\pi}{6}\)

🍕

10.6cm

Combinatorics*

In how many ways can 11 different books be arranged on a shelf if 2 of them must be together?

7257600

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 4th term of a geometric sequence is 54 and the sum of the first 4 terms is 80. Find the first term.

2

Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

\((1+2x)^{\frac12}\)

\(1+x-\frac{x^2}{2}+\frac{x^3}{2}\)

Integration (2)

Evaluate:

\(\int^{6}_{1} (x-8)^2 \; dx\)


\(111.666666666667\)

Probability (Conditional)

Box A contains 3 red and 4 blue cubes, and box B contains 5 red and 7 blue cubes. Finn selects a box at random and takes a cube from that box. Given that the cube is red, what is the probability that it came from box B?

\(\dfrac{35}{71}\)

Vectors*

Find the area of the triangle with sides:

\( \begin{pmatrix} 6 \\ 9 \\ 0 \end{pmatrix}, \; \begin{pmatrix} 7 \\ -9 \\ 2 \end{pmatrix} \; \text{and} \; \begin{pmatrix} 1 \\ -18 \\ 2 \end{pmatrix} \)

59.5 square units

Graph (Advanced)*

Sketch the graph of:

$$y=\sec\left(x\right)$$

Graph Plotter

Complex Numbers 1*

Simplify
$$ \dfrac{2-i}{1+3i}$$

\(-\frac{1}{10}-\frac{7}{10}i\)

Integration (4)*

Evaluate:

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


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

Trig (Identities)*

Simplify:

$$\tan{x}\cot{x}$$

\(1\)

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

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

Describe the behavior of a function at its asymptote.

Clue: approaches but never reaches

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*


Solve for \(z\)
$$ z^4 = \sqrt{3}+i $$

\(\sqrt[4]{2} cis \frac{\pi}{24},\sqrt[4]{2} cis \frac{13\pi}{24} \\ \sqrt[4]{2} cis \frac{-11\pi}{24}, \sqrt[4]{2} cis \frac{-23\pi}{24}\)

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 squares of the first \( n \) natural numbers is \( \frac{n(n + 1)(2n + 1)}{6} \)

Show true for n=1, assume true for n=k, prove for n=k+1

Surds (1)

Simplify:
$$\sqrt{75}$$
\(5\sqrt{3}\)

Surds (2)

Simplify:
$$\dfrac{6}{\sqrt{11}}$$\(\frac{6\sqrt{11}}{11}\)

Surds (3)

Simplify

\((1 + \sqrt{2})(2 + \sqrt{2})\)


\(4 + 3\sqrt{2}\)

Surds (4)

Simplify:
$$\dfrac{5}{3 - \sqrt{2}}$$\(\frac{15 + 5\sqrt{2}}{7}\)

Last Lesson

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?


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