Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

$(2a - 4b)^8$

$=256a^8 - 4096a^7b \\+28672a^6b^2 ...$

Compound Interest

If £240 is invested with an interest rate of 2% compounded quarterly, find the value of the investment after 7 years. £275.97

Coordinates (Square)

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

$(4,5),(9,9),(0,10)$

(5,14)

Normal Distribution

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

Find

$ P(270\lt X \lt330) $

$0.997$

Factorise (Quadratic 1)

Factorise:

$x^2-2x-3$

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

Factorise (Quadratic 2)

Factorise:

$5x^2+7x-6$

$(x+2)(5x-3)$

Graph (Linear)

Draw a rough sketch of the graph of:

$2y=x+2$

Gradient 0.5
y intercept 1

Indices

What is the value of:

$9^{\frac{1}{2}}$

$= 3$

Trigonometry (Angle)

Find angle ABC if AC = 3.4m and BC = 4.6m. 47.7^o

Trigonometry (Side)

Find AC if angle ABC = 32^o and AB = 3m. 1.87m

Venn Diagrams

Describe the red region.

Differentiation (1)

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

Find $ \dfrac{dy}{dx}$

$6x^2 - 8x + 3$

Differentiation (2)

$y = \dfrac{8}{x^{8}} - 8\sqrt[9]{x}$

Find $ \frac{dy}{dx}$

$-\frac{64}{x^{9}} - \frac{8}{9}x^{-\frac{8}{9}}$

Differentiation (3)

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

Find $ \dfrac{dy}{dx}$

$72x(6x^2-9)^5$

Differentiation (4)

$y=(4x+9)(6x-3)$

Find $ \dfrac{dy}{dx}$

$48x+42$

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:
$y = x^2 - 2x + 1$
where $x = 0$
$y = 1 - 2x$

Differentiation (7)

Find the equation of the normal to the curve:
$y = -4x^2 + 8x - 5$
where $x = -1$
$y = -\frac{x}{16} - \frac{273}{16}$

Integration (1)

$y =27x^2 - 6x + 7$

Find $ \int y \quad dx$

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

Binomial Distribution

A game is played 10 times and the probability of winning is 0.4. Calculate the probability of winning exactly 5 times. 0.201

Formulas

Make up a maths question using this:

$ \triangle = b^2-4ac$

Quadratic equation discriminant

Greek Letters

What letter is this?

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
$u_{5} = 7$
$u_{14} = 25$
Find the sum of the first 48 terms.2208

Asymptotes (HV)

Find the equations of the asymptotes of:

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

$x=3,y=-6$

Trig Advanced

In the triangle ABC,
AB = 8.9cm.
BC = 9.9cm.
CÂB = 68.5°.
Find angle BĈA.

56.7°

Sigma

Evaluate:

$$\sum_{n=3}^{5} 2^n$$

56

Discriminant

$f(x)=8x^2+6x-8$

What is the value of the discriminant and what does it indicate?
292, Two distinct roots

Completing The Square

$f(x)=x^2-4x-8$

By completing the square find the coordinates of the vertex.
(2, -12)

Logarithms

What is the value of $\ln{e^3}$ ?

3

Integration (3)

Find the integral:

$\int \dfrac{x^2}{x^3-1} \;dx$

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

Graph (2 points)

Find the equation of the straight line that passes through:

(-3, -3) and (6, 15)

$y=2x+3$

Functions (Inverse)

Find the inverse of the function $f$:

$f(x)= \sqrt{x+18}$

$x²-18$

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

$y=3-\dfrac{10}{x}$

Graph (Fill)

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

Trig (Special Angles)

Without a calculator find the exact value of

$$\cos{\frac{\pi}{6}} \times \sin{45°}$$

$\dfrac{\sqrt{6}}{4}$

Trig (Large Angles)

Without a calculator find the exact value of

$$\tan{5\pi}$$

$0$

Simultaneous Eqns (3)*

Solve:

$ g-7h-7i=-97 \\ 2g-2h+i= 7\\ 5g+3h+i = 71$

g = 8, h = 8, i = 7

Radian Measures

Find the area of a sector with radius 6.7cm and angle $ \frac{\pi}{4}$

🍕

17.6cm²

Combinatorics*

How many ways can twenty four people be divided into two equal groups?

1352078

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 5^th term of a geometric sequence is 1024 and the sum of the first 5 terms is 1364. Find the first term.

4

Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

$(1-4x)^{-3}$

$1+12x+96x^2+640x^3$

Integration (2)

Evaluate:

$\int^{5}_{0} e^x dx$

$e^{5}- 1 \approx 147$

Probability (Conditional)

34 Scouts went hiking. 17 got lost, 18 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.

$\dfrac{10}{17}$

Vectors*

Find the vector product:

$ \begin{pmatrix} 3 \\ 8 \\ 0 \end{pmatrix} \; \times \; \begin{pmatrix} 5 \\ -3 \\ 7 \end{pmatrix} $

$ \begin{pmatrix} 56 \\ -21 \\ -49 \end{pmatrix} $

Graph (Advanced)*

Sketch the graph of:

$$y=\sin(x)\cos(x)$$

Graph Plotter

Complex Numbers 1*

Simplify
$$ \dfrac{1-4i}{1+5i}$$

$-\frac{19}{26}-\frac{9}{26}i$

Integration (4)*

Evaluate:

$\int e^x\sin{x}\; dx$

$\frac{e^x}{2}(sinx-cosx)+c$

Trig (Identities)*

Simplify:

$$\dfrac{\tan{x}}{\sec{x}}$$

$\sin{x}$

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

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

How do you determine if a geometric series converges?

Clue: common ratio test

Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
$f(x) = \frac{1}{x^2 + 1}$

$1 - x^2 + x^4 - x^6$

Complex Numbers 2*

Given |z| = 8, find:
$$ |(3+4i)z| $$

$40$

Probability (Counting)*

A school committee of 8 is chosen at random from 12 senior students and 4 junior students. Find the probability that all four junior students are chosen.

1/26 or 3.85%

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

Surds (1)

Simplify:
$$\sqrt{45}$$
$3\sqrt{5}$

Surds (2)

Simplify:
$$\dfrac{7}{2\sqrt{5}}$$$\frac{7\sqrt{5}}{10}$

Surds (3)

Simplify

$(3 + 2\sqrt{5})(6 - 3\sqrt{5})$

$3\sqrt{5}-12$

Surds (4)

Simplify:
$$\dfrac{6}{5 + \sqrt{2}}$$$\frac{30 - 6\sqrt{2}}{23}$

Standard Deviation

Calculate the standard deviation of the following numbers:

27, 29, 30, 31, 33

2

Last Lesson

Write down a summary of your last Maths lesson focussing on what you learnt.

?

ADVANCED

Binomial Theorem (1)

Compound Interest

Coordinates (Square)

Normal Distribution

Factorise (Quadratic 1)

Factorise (Quadratic 2)

Graph (Linear)

Indices

Trigonometry (Angle)

Trigonometry (Side)

Venn Diagrams

Differentiation (1)

Differentiation (2)

Differentiation (3)

Differentiation (4)

Differentiation (5)

Differentiation (6)

Differentiation (7)

Integration (1)

Binomial Distribution

Formulas

Greek Letters

Sequences (Arithmetic)

Asymptotes (HV)

Trig Advanced

Sigma

Discriminant

Completing The Square

Logarithms

Integration (3)

Graph (2 points)

Functions (Inverse)

Functions (Composite)

Standard Form

Graph (Mixed)

Graph (Fill)

Trig (Special Angles)

Trig (Large Angles)

Simultaneous Eqns (3)*

Radian Measures

Combinatorics*

Asymptotes (Ob)*

Sequences (Geometric)

Binomial Theorem (2)*

Integration (2)

Probability (Conditional)

Vectors*

Graph (Advanced)*

Complex Numbers 1*

Integration (4)*

Trig (Identities)*

Integration (Volume)*

Miscellaneous

Maclaurin Series*

Complex Numbers 2*

Probability (Counting)*

Proof by Induction*

Surds (1)

Surds (2)

Surds (3)

Surds (4)

Standard Deviation

Last Lesson

An Advanced Mathematics Lesson Starter Of The Day

Concept Selection

Answers