ADVANCED

Binomial Theorem (1)

Find the first three terms in the expansion of:

\((3a - 2b)^7\)

\(=2187a^7 - 10206a^6b \\+20412a^5b^2 ...\)

Compound Interest

If £240 is invested with an interest rate of 4% compounded monthly, find the value of the investment after 6 years. £304.98

Coordinates (Square)

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

\((3,3),(8,7),(-1,8)\)

(4,12)

Normal Distribution

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

Find

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

\(0.0478\)

Factorise (Quadratic 1)

Factorise:

\(x^2-3x-4\)

\((x+1)(x-4)\)

Factorise (Quadratic 2)

Factorise:

\(5x^2+13x-6\)

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

Graph (Linear)

Draw a rough sketch of the graph of:

\(y=x+2\)

Gradient 1
y intercept 2

Indices

What is the value of:

\(5^{-3}\)

\(= \frac{1}{125}\)

Trigonometry (Angle)

Find angle ABC if AC = 5.5m and AB = 6.9m. 38.6^o

Trigonometry (Side)

Find AB if angle ABC = 61^o and BC = 4.2m. 2.04m

Venn Diagrams

Describe the red region.

Differentiation (1)

\(y = 4x^3 - 6x^2 + 8x\)

Find \( \dfrac{dy}{dx}\)

\(12x^2 - 12x + 8\)

Differentiation (2)

\(y = \dfrac{3}{x^{9}} - 6\sqrt[7]{x}\)

Find \( \frac{dy}{dx}\)

\(-\frac{27}{x^{10}} - \frac{6}{7}x^{-\frac{6}{7}}\)

Differentiation (3)

\(y=\sqrt{4x^6-6x}\)

Find \( \dfrac{dy}{dx}\)

\((12x^5-3)(4x^6-6x)^{-\frac{1}{2}}\)

Differentiation (4)

\(y=\sin (x) \sqrt{ x^2 + 3}\)

Find \( \dfrac{dy}{dx}\)

\(cosx \sqrt{x^2+3}+\frac{xsinx}{\sqrt{x^2+3}}\)

Differentiation (5)

\(y=\frac{ \ln x}{x^2}\)

Find \( \dfrac{dy}{dx}\)

\(\frac{(1-2lnx)}{x^3}\)

Differentiation (6)

Find the equation of the tangent to the curve:
\(y = 3x^2 - 6x + 9\)
where \(x = 2\)
\(y = 6x - 3\)

Differentiation (7)

Find the equation of the normal to the curve:
\(y = 4x^2 + 2x - 1\)
where \(x = -2\)
\(y = \frac{x}{14} + 11\frac{1}{7}\)

Integration (1)

\(y =12x^2 - 6x + 6\)

Find \( \int y \quad dx\)

\(4x^3 - 3x^2 + 6x+c\)

Binomial Distribution

A game is played 10 times and the probability of winning is 0.2. Calculate the probability of winning exactly 5 times.   0.0264

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:
\(u_{6} = -51\)
\(u_{16} = -141\)
Find the sum of the first 21 terms.-2016

Asymptotes (HV)

Find the equations of the asymptotes of:

\(y=3\left(\dfrac{2x+3}{7-x}\right)\)

\(x=7,y=-6\)

Trig Advanced

In the triangle ABC,
BC = 7.8cm.
CA = 10.1cm.
BĈA = 49.7°
Find AB to 1 dp.

7.8cm

Sigma

Evaluate:

$$\sum_{n=2}^{8} n^2 - 9n$$

-112

Discriminant

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

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

Completing The Square

\(f(x)=x^2-8x-2\)

By completing the square find the coordinates of the vertex.
(4, -18)

Logarithms

Evaluate \(\log_2(32) \)

5

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:

(-1, 7) and (3, 11)

\(y=x+8\)

Functions (Inverse)

Find the inverse of the function \(f\):

\(f(x)= \sqrt{x-4}\)

\(x²+4\)

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:
\((a \times 10^p) \div (b\times 10^q)\)
where \(a \div b \) is a single digit number \((1 \le \frac{a}{b} \lt 10)\)

\(\frac{a}{b}\times10^{p-q}\)

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

\(1\)

Trig (Large Angles)

Without a calculator find the exact value of

\(\sqrt{3}\)

Simultaneous Eqns (3)*

Solve:

\( 5a+2b+c=44 \\ 3a+4b+2c= 46 \\ a+5b+c=38\)

a = 6, b = 6, c = 2

Radian Measures

Find the area of a sector with radius 4.1cm and angle \( \frac{\pi}{3}\)

🍕

8.80cm²

Combinatorics*

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

1716

Asymptotes (Ob)*

Find the equations of the asymptotes of:

x=1/2,y=15/2-4x

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:

\(\dfrac{1}{2-x}\)

\(\frac{1}{2}+\frac{x}{4}+\frac{x^2}{8}+\frac{x^3}{16}\)

Integration (2)

Evaluate:

\(\int^{6}_{2} (x-8)^2 \; dx\)

\(69.3333333333333\)

Probability (Conditional)

Given equal populations of Type X and Type Y bacteria, with mutation rates of 20% and 60% respectively, if a mutated bacterium is found, what's the probability it's Type Y?

\(0.750\)

Vectors Advanced*

Find the cartesian equation of this plane:

\( \mathbf{r} = \begin{pmatrix} -2 \\ 2 \\ 3 \end{pmatrix} \; + \; s \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix} \; + \; t \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} \)

2x-6y+5z=-1

Graph (Advanced)*

Sketch the graph of:

Graph Plotter

Complex Numbers 1*

Simplify
$$ (4-6i)(3-2i) $$

\(0-26i\)

Integration (4)*

Evaluate:

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

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

Trig (Identities)*

Simplify:

\(\cos{x}\)

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 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
\(f(x) = \sin(x)\)

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

Complex Numbers 2*

Solve for \(z\)
$$ z^3 = - 8i $$

\(\sqrt{3}-i,2i,-\sqrt{3}-i\)

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:
$$\sqrt{50}$$
\(5\sqrt{2}\)

Surds (2)

Simplify:
$$\dfrac{5}{\sqrt{8}}$$\(\frac{5\sqrt{8}}{8} = \frac{5\sqrt{2}}{4}\)

Surds (3)

Simplify

\(4\sqrt{7} - \sqrt{63}\)

\(\sqrt{7}\)

Surds (4)

Simplify:
$$\dfrac{4}{6 - \sqrt{5}}$$\(\frac{24 + 4\sqrt{5}}{31}\)

Standard Deviation

Calculate the standard deviation of the following numbers:

4, 2, 5, 8, 6

2

Inverse Function

Find the inverse function:

\(f(x)=(x-2)^2+1,\quad x\geq2\)
\(f^{-1}(x)=2+\sqrt{x-1},x\geq1\)

Changing the Subject

Make \(\theta\) the subject:

\(z=p+q\cos(r\theta) \\ \quad 0\leq r\theta\leq\pi\)
\(\theta=\dfrac{1}{r}\cos^{-1}\left(\dfrac{z-p}{q}\right)\)

Vector Products*

Find the scalar and vector products of:
\({\scriptsize\mathbf a=\left(\begin{smallmatrix}0\\4\\4\end{smallmatrix}\right)\text{ and }\mathbf b=\left(\begin{smallmatrix}8\\9\\5\end{smallmatrix}\right)}\)

\( {\scriptsize \mathbf a\cdot\mathbf b=56,\quad \mathbf a\times\mathbf b=\left(\begin{smallmatrix}-16\\32\\-32\end{smallmatrix}\right) } \)

Last Lesson

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

?