Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

$(3a - 2b)^8$

$=6561a^8 - 34992a^7b \\+81648a^6b^2 ...$

Compound Interest

If £240 is invested with an interest rate of 3% compounded quarterly, find the value of the investment after 6 years. £287.14

Coordinates (Square)

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

$(2,1),(7,6),(-3,6)$

(2,11)

Normal Distribution

$ X \sim N(-25, 3^2)$

Find

$ P(-20\lt X \lt-10) $

$0.0478$

Factorise (Quadratic 1)

Factorise:

$x^2-2x-3$

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

Factorise (Quadratic 2)

Factorise:

$3x^2+8x-16$

$(x+4)(3x-4)$

Graph (Linear)

Draw a rough sketch of the graph of:

$y=x+2$

Gradient 1
y intercept 2

Indices

What is the value of:

$2^{0}$

$= 1$

Trigonometry (Angle)

Find angle BCA if AB = 3.6m and BC = 5.3m. 42.8^o

Trigonometry (Side)

Find AC if angle BCA = 66^o and AB = 4.5m. 2.00m

Venn Diagrams

Describe the red region.

Differentiation (1)

$y = 9x^3 - 5x^2 + 6x$

Find $ \dfrac{dy}{dx}$

$27x^2 - 10x + 6$

Differentiation (2)

$y = \dfrac{9}{x^{3}} - 8\sqrt[9]{x}$

Find $ \frac{dy}{dx}$

$-\frac{27}{x^{4}} - \frac{8}{9}x^{-\frac{8}{9}}$

Differentiation (3)

$y=(6x+5)^3$

Find $ \dfrac{dy}{dx}$

$18(6x+5)^2$

Differentiation (4)

$y=(4x+7)(9x-4)$

Find $ \dfrac{dy}{dx}$

$72x+47$

Differentiation (5)

$y=\frac{x}{\sin x}$

Find $ \dfrac{dy}{dx}$

$\frac{(sinx-xcosx)}{sin^2x}$

Differentiation (6)

Find the equation of the tangent to the curve:
$y = -5x^2 + 7x - 3$
where $x = 2$
$y = 17 - 13x$

Differentiation (7)

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

Integration (1)

$y =15x^2 - 18x + 5$

Find $ \int y \quad dx$

$5x^3 - 9x^2 + 5x+c$

Binomial Distribution

A game is played 10 times and the probability of winning is 0.5. Calculate the probability of winning exactly 8 times. 0.0439

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:
$u_{5} = -4$
$u_{17} = 8$
Find the sum of the first 50 terms.825

Asymptotes (HV)

Find the equations of the asymptotes of:

$y=5\left(\dfrac{3x}{5+x}\right)$

$x=-5,y=15$

Trig Advanced

In the triangle ABC,
BC = 7.8cm.
CA = 10.3cm.
BĈA = 60.0°
Find AB to 1 dp.

9.3cm

Sigma

Evaluate:

$$\sum_{n=4}^{6} 4n+1$$

63

Discriminant

$f(x)=-4x^2+3x+1$

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

Completing The Square

$f(x)=x^2-4x+9$

By completing the square find the coordinates of the vertex.
(2, 5)

Logarithms

Solve for x:

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

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

Integration (3)

Find the integral:

$\int \cos(x)e^{\sin(x)} \;dx$

$e^{\sin(x)}+c$

Graph (2 points)

Find the equation of the straight line that passes through:

(-6, -3) and (6, 21)

$y=2x+9$

Functions (Inverse)

Find the inverse of the function $f$:

$f(x)=\sqrt{\frac{x-6}{5}}$

$5x²+6$

Functions (Composite)

$\text{Find }f(x) \text{ if} \\ f(a^3)=2a^6 \\$

$f(x)=2x^2$

Standard Form

Write in standard form:
$a \times 10^{-1} \times b\times 10^{-1}$
where $a \times b $ is a three digit number $(100 \le ab \lt 1000)$

$\frac{ab}{100}\times10^0$

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

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

$\sqrt{2}$

Trig (Large Angles)

Without a calculator find the exact value of

$$\sin{\dfrac{13\pi}{6}}$$

$\dfrac{1}{2}$

Simultaneous Eqns (3)*

Solve:

$ 5a+2b+c=22 \\ 3a+4b+2c= 30 \\ a+5b+c=29$

a = 2, b = 5, c = 2

Radian Measures

Find the perimeter of a sector with radius 5.9cm and angle $ \frac{2\pi}{3}$

🍕

24.2cm

Combinatorics*

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

92378

Asymptotes (Ob)*

Find the equations of the asymptotes of:

$$y=\dfrac{2x^2-8x+8}{x-3}$$

x=3, y=2x-2

Sequences (Geometric)

Evaluate:
$$ \sum_{k=1}^{10} 3 \times 2^{k-1} $$

3069

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:

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

$\dfrac{\sqrt{3}-1}{2}$

Probability (Conditional)

Tin A contains 5 red balls and 6 green balls. Tin B contains 9 red balls and 10 green balls. A dice is thrown and if the score is less than 3 a ball is selected from tin A; otherwise a ball is selected from tin B. Given that the ball selected was red, calculate the probability that it came from tin B.

$\frac{198}{293}$

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

Graph (Advanced)*

Sketch the graph of:

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

Graph Plotter

Complex Numbers 1*

Simplify
$$ (2-i)^{-2} $$

$\frac{3}{25}+\frac{4}{25}i$

Integration (4)*

Evaluate:

$\int x\sec^2x\; dx$

$xtanx+\ln|cosx|+c$

Trig (Identities)*

Simplify:

$$\sin{x}\cot{x}$$

$\cos{x}$

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

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 formula for compound interest?

$ FV = PV(1 + \frac{r}{100k})^{kn} $

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*

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

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

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 sum of the first $ n $ even numbers is $ n(n + 1) $

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

Surds (1)

Simplify:
$$\sqrt{32}$$
$4\sqrt{2}$

Surds (2)

Simplify:
$$\dfrac{4}{5\sqrt{3}}$$$\frac{4\sqrt{3}}{15}$

Surds (3)

Simplify

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

$4 + 3\sqrt{2}$

Surds (4)

Simplify:
$$\dfrac{5}{2 - \sqrt{3}}$$$\frac{10 + 5\sqrt{3}}{1} = 10 + 5\sqrt{3}$

Standard Deviation

Calculate the standard deviation of the following numbers:

15, 19, 19, 21, 21, 25

3

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