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 £160 is invested with an interest rate of 6% compounded quarterly, find the value of the investment after 5 years. £215.50

Coordinates (Square)

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

$(4,4),(7,8),(0,7)$

(3,11)

Normal Distribution

$ X \sim N(33, 6^2)$

Find

$ P(31\lt X \lt37) $

$0.378$

Factorise (Quadratic 1)

Factorise:

$x^2-x-12$

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

Factorise (Quadratic 2)

Factorise:

$6x^2-x-1$

$(3x+1)(2x-1)$

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:

$1^{0}$

$= 1$

Trigonometry (Angle)

Find angle ABC if AB = 4.5m and BC = 5.9m. 40.3^o

Trigonometry (Side)

Find AC if angle BCA = 46^o and AB = 3.7m. 3.57m

Venn Diagrams

Describe the red region.

Differentiation (1)

$y = 5x^3 - 7x^2 + 8x$

Find $ \dfrac{dy}{dx}$

$15x^2 - 14x + 8$

Differentiation (2)

$y = \dfrac{7}{x^{5}} - 7\sqrt[8]{x}$

Find $ \frac{dy}{dx}$

$-\frac{35}{x^{6}} - \frac{7}{8}x^{-\frac{7}{8}}$

Differentiation (3)

$y=\sin (5x^2+6)$

Find $ \dfrac{dy}{dx}$

$10xcos(5x^2+6)$

Differentiation (4)

$y=x^8 \sin x$

Find $ \dfrac{dy}{dx}$

$8x^7sinx+x^8cosx$

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

Differentiation (7)

Find the equation of the normal to the curve:
$y = 2x^2 - x + 3$
where $x = -1$
$y = \frac{x}{5} + 6\frac{1}{5}$

Integration (1)

$y =18x^2 - 8x + 2$

Find $ \int y \quad dx$

$6x^3 - 4x^2 + 2x+c$

Binomial Distribution

A game is played 17 times and the probability of winning is 0.6. Calculate the probability of winning exactly 6 times. 0.0242

Formulas

Make up a maths question using this:

$u_n=u_1+(n-1)d$

The n^th term of an arithmetic sequence

Greek Letters

What letter is this?

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
$u_{9} = 17$
$u_{16} = 24$
Find the sum of the first 41 terms.1189

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,
BĈA = 48.3°.
BC = 5.3cm.
AB̂C = 84.58°.
Find CA to 1 dp.

7.2cm

Sigma

Evaluate:

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

108

Discriminant

$f(x)=-7x^2+5x+1$

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

Completing The Square

$f(x)=x^2+7x+5$

By completing the square find the coordinates of the vertex.
(-3.5, -7.25)

Logarithms

Evaluate $\log_2(32) $

5

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:

(-8, -29) and (5, 10)

$y=3x-5$

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:
$(a \times 10^4) \div (b\times 10^{-2})$
where $a \div b $ is a decimal number $(0.1 \le \frac{a}{b} \lt 1)$

$\frac{10a}{b}\times10^5$

Graph (Mixed)

Draw a rough sketch of

$y=\dfrac{5}{x}$

Graph (Fill)

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

Trig (Special Angles)

Without a calculator find the exact value of

$$\tan{30°} \times \tan{\frac{\pi}{3}}$$

$1$

Trig (Large Angles)

Without a calculator find the exact value of

$$\cos{720°}$$

$1$

Simultaneous Eqns (3)*

Solve:

$ 5a+2b+c=41 \\ 3a+4b+2c= 47 \\ a+5b+c=39$

a = 5, b = 6, c = 4

Radian Measures

Find the perimeter of a sector with radius 6.1cm and angle $ \frac{5\pi}{6}$

🍕

28.2cm

Combinatorics*

In how many ways can 9 different books be arranged on a shelf if 3 of them must be together?

30240

Asymptotes (Ob)*

Find the equations of the asymptotes of:

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

x=0,y=3-x

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:

$(1+4x)^{\frac{3}{2}}$

$1+6x+6x^2-4x^3$

Integration (2)

Evaluate:

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

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

Probability (Conditional)

Each afternoon the probability my cat sleeps is 0.6 and the probability that my dog sleeps is 0.7. The probability that the dog sleeps given that the cat is sleeping is 0.9. Find the probability that both sleep in the afternoon.

$0.54$

Vectors*

Find the area of the triangle with sides:

$ \begin{pmatrix} 3 \\ 6 \\ 0 \end{pmatrix}, \; \begin{pmatrix} 8 \\ -4 \\ 7 \end{pmatrix} \; \text{and} \; \begin{pmatrix} 5 \\ -10 \\ 7 \end{pmatrix} $

38.1 square units

Graph (Advanced)*

Sketch the graph of:

$$y=1^{\sin{x}}$$

Graph Plotter

Complex Numbers 1*

Simplify
$$ (6+4i)(2+5i) $$

$-8+38i$

Integration (4)*

Evaluate:

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

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

Trig (Identities)*

Simplify:

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

$\sin{x}$

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

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

Describe the graph of an exponential function.

Clue: grow or decay rapidly, horizontal asymptote

Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
$f(x) = \sin(x)$

$x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040}$

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 committee of 4 is randomly selected from 8 men and 11 women. Determine the likelihood that it consists of all men.

35/1938 or 1.81%

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:
$$\sqrt{27}$$
$3\sqrt{3}$

Surds (2)

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

Surds (3)

Simplify

$7\sqrt{7} - 3\sqrt{7}$

$4\sqrt{7}$

Surds (4)

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

Standard Deviation

Calculate the standard deviation of the following numbers:

11, 17, 20, 23, 29

6

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