\( 2 \times 2 \) Contingency Table The following shows the results of a survey of a sample of \( 200 \) randomly chosen adults classified according to gender and sport. These are called observed values or observed frequencies. $$ \begin{array}{|c|c|c|c|} \hline & \text{loves sport} & \text{hates sport} & \text{sum} \\ \hline \text{male} & 72 […]

# Drawing Venn Diagrams Effectively

Consider the following situation to illustrate through Venn diagrams. Two Circles There are \( 50 \) students in a certain high school. \( 16 \) study Physics, \( 13 \) study Chemistry, and \( 15 \) study both Physics and Chemistry. Illustrate this information on a Venn diagram. $$ \begin{align} a+b+c+d &= 50 \text{ total […]

# Mathematics Glossary

ABCDEFGHIJKLMNOPQRSTUVWXYZA Absolute error - The difference between the actual value and the measured value indicated by an instrument. Accessibility sampling - Selecting a sample from onlythose items that we most readily available. Acount servicing fee - Ongoing account keeping fees. Actual interest rate - Represents the actual oercentage return per year on an investment (the [...]

# Rewrite Expressions in Sine and Cosine involving Powers of Cosine or Sine by Complex Number

Let \( z = \cos \theta + i \sin \theta \). (a) By considering the real part of \( z^4 \), prove \( \cos 4\theta = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta \). \( \begin{align} \displaystyle \require{color} z &= \cos \theta + i \sin \theta \\ z^4 &= (\cos [...]

# Adding Multiples of Consecutive Odd Numbers by Mathematical Induction

(a) Factorise \( 4x^3 + 18x^2 + 23x + 9 \). \( \begin{align} \displaystyle 4x^3 + 18x^2 + 23x + 9 &= 4x^3 + 4x^2 + 14x^2 + 23x + 9 \\ &= 4x^2 (x+1) + 14x^2 + 14x + 9x + 9 \\ &= 4x^2 (x+1) + 14x(x+1) + 9(x+1) \\ &= (x+1)(4x^2 [...]

# Mathematical Induction Regarding Factorials

Prove by mathematical induction that for al lintegers \( n \ge 1 \) , $$ \dfrac{1}{2!} + \dfrac{2}{3!} + \dfrac{3}{4!} + \cdots + \dfrac{n}{(n+1)!} = 1 - \dfrac{1}{(n+1)!}$$ Step 1: Show it is true for \( n=1 \). \( \begin{align} \displaystyle \text{LHS } &= \dfrac{1}{2!} = \dfrac{1}{2} \\ \text{RHS } &= 1 - \dfrac{1}{2!} \\ [...]

# Applications of the Unit Circle

The identify $\cos^2 \theta + \sin^2 \theta = 1$ is required for finding trigonometric ratios. Example 1 Find exactly the possible values of $\cos \theta$ for $\sin \theta = \dfrac{5}{8}$. \( \begin{align} \displaystyle \cos^2 \theta + \sin^2 \theta &= 1 \\ \cos^2 \theta + \sin^2 \dfrac{5}{8} &= 1 \\ \cos^2 \theta + \dfrac{25}{64} &= 1 [...]

# Trigonometric Ratios

Circles with Cnetre $(0,0)$ Consider a circle with centre $(0,0)$ and radius $r$ units. Suppose $(x,y)$ is any point on this circle. Using ths distance formula; \( \begin{align} \displaystyle \sqrt{(x-0)^2+(y-0)^2} &= r \\ \therefore x^2+y^2 &= r^2 \end{align} \) $x^2+y^2 = r^2$ is the equation of a circle with centre $(0,0)$ and radius $r$. The [...]

# Arc Length and Sector Area

You should know with these terms relating to the parts of a circle. The centre of a circle is the point which is equidistant from all points on the circle. A radius of a circle a straight line joining the centre of a circle to any point on the circumference. A minor arc is an [...]

# Degree-Radian Conversions

Degree Measurement of Angles One full revolution makes an angle of $360^{\circ}$, and the angle on a straight line is $180^{\circ}$. Therefore, one degree, $1^{\circ}$, can be defined as $\dfrac{1}{360}$ of one full revolution. For greater accuracy we define one minute, $1'$, as $\dfrac{1}{60}$ of one degree and one second, $1^{\prime \prime}$, as $\dfrac{1}{60}$ of [...]