Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

$(3a - 2b)^5$

$=243a^5 - 810a^4b \\+1080a^3b^2 ...$

Compound Interest

If £140 is invested with an interest rate of 5% compounded monthly, find the value of the investment after 6 years. £188.86

Coordinates (Square)

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

$(4,1),(9,7),(-2,6)$

(3,12)

Normal Distribution

$ X \sim N(65, 9^2)$

Find

$ P(36\lt X \lt48) $

$0.0288$

Factorise (Quadratic 1)

Factorise:

$x^2-x-12$

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

Factorise (Quadratic 2)

Factorise:

$2x^2+7x-4$

$(x+4)(2x-1)$

Graph (Linear)

Draw a rough sketch of the graph of:

$2y=x-4$

Gradient 0.5
y intercept -2

Indices

What is the value of:

$4^{0}$

$= 1$

Trigonometry (Angle)

Find angle BCA if AB = 3.8m and AC = 5.1m. 36.7^o

Trigonometry (Side)

Find AC if angle ABC = 21^o and BC = 3m. 1.08m

Venn Diagrams

Describe the red region.

Differentiation (1)

$y = 2x^3 - 8x^2 + 8x$

Find $ \dfrac{dy}{dx}$

$6x^2 - 16x + 8$

Differentiation (2)

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

Find $ \frac{dy}{dx}$

$-\frac{6}{x^{4}} - \frac{4}{5}x^{-\frac{4}{5}}$

Differentiation (3)

$y=(9x+2)^3$

Find $ \dfrac{dy}{dx}$

$27(9x+2)^2$

Differentiation (4)

$y=\sin x \cos x$

Find $ \dfrac{dy}{dx}$

$cos^2x-sin^2x$

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:
$y = 5x^2 + 7x + 3$
where $x = 1$
$y = 17x - 2$

Differentiation (7)

Find the equation of the normal to the curve:
$y = x^2 + 6x + 9$
where $x = -3$
$x = -3$

Integration (1)

$y =21x^2 - 14x + 5$

Find $ \int y \quad dx$

$7x^3 - 7x^2 + 5x+c$

Binomial Distribution

A game is played 10 times and the probability of winning is 0.8. Calculate the probability of winning exactly 8 times. 0.302

Formulas

Make up a maths question using this:

$^nC_r=\dfrac{n!}{r!(n-r)!}$

Combinations
(from n choose r)

Greek Letters

What letter is this?

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
$u_{10} = 62$
$u_{16} = 104$
Find the sum of the first 40 terms.5420

Asymptotes (HV)

Find the equations of the asymptotes of:

$y=\dfrac{10-2x}{10x}$

$x=0,y=-\frac{1}{5}$

Trig Advanced

In the triangle ABC,
AB = 9.7cm.
BC = 9.3cm.
CA = 12.7cm.
Find angle CÂB.

46.7°

Sigma

Evaluate:

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

120

Discriminant

$f(x)=-5x^2-6x-7$

What is the value of the discriminant and what does it indicate?
-104, No real roots

Completing The Square

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

By completing the square find the coordinates of the vertex.
(-3, -17)

Logarithms

Evaluate $\log_5(625) $

4

Integration (3)

Find the integral:

$\int 3xe^{x^2} \;dx$

$\frac{3}{2}e^{x^2}+c$

Graph (2 points)

Find the equation of the straight line that passes through:

(-5, -6) and (6, 16)

$y=2x+4$

Functions (Inverse)

Find the inverse of the function $f$:

$f(x)=\frac{\sqrt{x-9}}{9}$

$81x²+9$

Functions (Composite)

$f(x)=2x-3 \\[1cm] \text{Find }f \bullet f(1-\sqrt{m}) \\$

$-5-4\sqrt{m}$

Standard Form

Write in standard form:
$a \times 10^p \times b\times 10^q$
where $a \times b $ is a single digit number $(1 \le ab \lt 10)$

$ab\times10^{p+q}$

Graph (Mixed)

Draw a rough sketch of

$y=x(5-x)$

Graph (Fill)

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

Trig (Special Angles)

Without a calculator find the exact value of

$$\sin{\frac{\pi}{4}} \times \cos{45°}$$

$\dfrac{1}{2}$

Trig (Large Angles)

Without a calculator find the exact value of

$$\tan{5\pi}$$

$0$

Simultaneous Eqns (3)*

Solve:

$2d+3e-4f = 14 \\ d-e-f= -6\\ 9d+2e-2f=26$

d = 2, e = 6, f = 2

Radian Measures

Find the area of a sector with radius 8.6cm and angle $ \frac{2\pi}{3}$

🍕

77.5cm²

Combinatorics*

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

126

Asymptotes (Ob)*

Find the equations of the asymptotes of:

$$y=\dfrac{-6x^2-4x-1}{3x+2}$$

x=-2/3, y=-2x

Sequences (Geometric)

The fourth term of a geometric sequence is $-16$ and the sum to infinity is $32$. What is the common ratio?

-0.669

Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

$\dfrac{1}{\sqrt[3]{1+x}}$

$1 - \frac{x}{3} + \frac{2x^2}{9} - \frac{14x^3}{81}$

Integration (2)

Evaluate:

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

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

Probability (Conditional)

32 Scouts went hiking. 17 got lost, 16 got blisters, and 9 got both lost and blisters. Find the probability that a randomly selected Scout got blisters, given that they were not lost.

$\dfrac{7}{15}$

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:

$$y=\left|\cot\left(x\right)\right|$$

Graph Plotter

Complex Numbers 1*

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

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

Integration (4)*

Evaluate:

$\int x^2 \ln{x}\; dx$

$\frac{x^3}{9}(3\ln x-1)+c$

Trig (Identities)*

Simplify:

$$\dfrac{\cot{x}}{\cosec{x}}$$

$\cos{x}$

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

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
$f(x) = \frac{1}{x^2 + 1}$

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

Complex Numbers 2*

Solve for $z$
$$ z^4 = - 16 $$

$\sqrt{2}+i\sqrt{2},-\sqrt{2}+i\sqrt{2} \\ \sqrt{2}-i\sqrt{2},-\sqrt{2}-i\sqrt{2}$

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 $ n! > 2^n $ for all integers $ n $ greater than 4

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

Surds (1)

Simplify:
$$\sqrt{48}$$
$4\sqrt{3}$

Surds (2)

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

Surds (3)

Simplify

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

$3\sqrt{5}-12$

Surds (4)

Simplify:
$$\dfrac{8}{3 + \sqrt{6}}$$$\frac{24 - 8\sqrt{6}}{3}$

Standard Deviation

Calculate the standard deviation of the following numbers:

9, 13, 15, 17, 21

4

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