Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

$(2a - 3b)^6$

$=64a^6 - 576a^5b \\+2160a^4b^2 ...$

Compound Interest

If £200 is invested with an interest rate of 2% compounded quarterly, find the value of the investment after 9 years. £239.34

Coordinates (Square)

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

$(4,4),(8,10),(-2,8)$

(2,14)

Normal Distribution

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

Find

$ P(36\lt X \lt48) $

$0.0288$

Factorise (Quadratic 1)

Factorise:

$x^2-4$

$(x+2)(x-2)$

Factorise (Quadratic 2)

Factorise:

$2x^2+7x-4$

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

Graph (Linear)

Draw a rough sketch of the graph of:

$y=-x-2$

Gradient -1
y intercept -2

Indices

What is the value of:

$3^{-3}$

$= \frac{1}{27}$

Trigonometry (Angle)

Find angle ABC if AB = 4.6m and BC = 6.4m. 44.0^o

Trigonometry (Side)

Find BC if angle BCA = 50^o and AB = 4.6m. 6.00m

Venn Diagrams

Describe the red region.

Differentiation (1)

$y = 6x^3 - 4x^2 + 5x$

Find $ \dfrac{dy}{dx}$

$18x^2 - 8x + 5$

Differentiation (2)

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

Find $ \frac{dy}{dx}$

$-\frac{25}{x^{6}} - \frac{3}{4}x^{-\frac{3}{4}}$

Differentiation (3)

$y=\frac{1}{(8x+9)^8}$

Find $ \dfrac{dy}{dx}$

$-\frac{64}{(8x+9)^9}$

Differentiation (4)

$y=x \tan x$

Find $ \dfrac{dy}{dx}$

$tanx+\frac{x}{cos^2x}$

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:
$y = 4x^2 + 2x - 1$
where $x = -2$
$y = -14x - 17$

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)

$y =9x^2 - 8x + 2$

Find $ \int y \quad dx$

$3x^3 - 4x^2 + 2x+c$

Binomial Distribution

A game is played 13 times and the probability of winning is 0.5. Calculate the probability of winning exactly 12 times. 0.00159

Formulas

Make up a maths question using this:

$ \tan \theta \equiv \dfrac{ \sin \theta}{ \cos \theta}$

Trigonometric identity

Greek Letters

What letter is this?

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
$u_{7} = 62$
$u_{17} = 182$
Find the sum of the first 31 terms.5270

Asymptotes (HV)

Find the equations of the asymptotes of:

$y=\dfrac{5x}{2-x}+3$

$x=2,y=-2$

Trig Advanced

In the triangle ABC,
AB = 6.8cm.
BC = 5.6cm.
CA = 8.8cm.
Find angle CÂB.

39.5°

Sigma

Evaluate:

$$\sum_{n=4}^{9} n^2 - 4n$$

115

Discriminant

$f(x)=-8x^2+2x+5$

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

Completing The Square

$f(x)=x^2+5x+4$

By completing the square find the coordinates of the vertex.
(-2.5, -2.25)

Logarithms

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

3

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, -2) and (3, 8)

$y=2x+2$

Functions (Inverse)

Find the inverse of the function $f$:

$f(x)= \sqrt{x+13}$

$x²-13$

Functions (Composite)

$f(x)=3x+2 \\ g(x)=2x^2 \\[1cm] \text{Find }gf(x)$

$18x^2+24x+8$

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

$y=2^x$

Graph (Fill)

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

Trig (Special Angles)

Without a calculator find the exact value of

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

$\dfrac{\sqrt{3}}{4}$

Trig (Large Angles)

Without a calculator find the exact value of

$$\cos{7\pi}$$

$-1$

Simultaneous Eqns (3)*

Solve:

$ j+k+l= 11 \\ 2j-3k+9l= 28\\ -j+k-3l=-11$

j = 5, k = 3, l = 3

Radian Measures

Find the area of a sector with radius 9.9cm and angle $ \frac{\pi}{6}$

🍕

25.7cm²

Combinatorics*

How many ways can nine children sit in a row without the youngest being in the middle?

322560

Asymptotes (Ob)*

Find the equations of the asymptotes of:

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

x=3, y=2x-2

Sequences (Geometric)

The 6^th term of a geometric sequence is 64 and the sum of the first 6 terms is 126. 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:

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

$114.333333333333$

Probability (Conditional)

In a bookstore with equally sized fiction and non-fiction sections, if a hardcover book is selected (20% of fiction, 70% of non-fiction are hardcovers), what's the probability it's non-fiction?

$0.778$

Vectors*

Find the angle between the plane and the line:

$\Pi: \quad 4x+4y-2z=7$

$L: \quad x+1= \dfrac{4-y}{2} = 3-z $

$ \approx 7.82^o $

Graph (Advanced)*

Sketch the graph of:

$$y=\cos^2x$$

Graph Plotter

Complex Numbers 1*

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

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

Integration (4)*

Evaluate:

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

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

Trig (Identities)*

Simplify:

$$\sec{x}-\tan{x}\sin{x}$$

$\cos{x}$

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

Integration (Volume)*

Find the volume of revolution when $y=x^3$ is rotated about the y-axis for $1 \le y \le 2$

$\approx 4.10$ cubic units

Miscellaneous

How do you solve a quadratic inequality?

Factorise the quadratic, then analyze the sign of each factor over its domain.

Maclaurin Series*

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

$1 + x + x^2 + x^3$

Complex Numbers 2*

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

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

Probability (Counting)*

6 children, three boys and three girls, are randomly seated on a row of 6 chairs. Determine the likelihood that the three boys are seated together.

1/5 or 20%

Proof by Induction*

Prove by mathematical induction that the sum of the first $ n $ odd numbers is $ n^2 $

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:
$$\dfrac{7}{\sqrt{5}}$$$\frac{7\sqrt{5}}{5}$

Surds (3)

Simplify

$8\sqrt{3}(1 - \sqrt{3})$

$8\sqrt{3} - 24$

Surds (4)

Simplify:
$$\dfrac{3}{4 + \sqrt{2}}$$$\frac{12 - 3\sqrt{2}}{14} = \frac{6 - \sqrt{2}}{7}$

Standard Deviation

Calculate the standard deviation of the following numbers:

7, 9, 10, 11, 13

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