Revision Question

Sets(Venn Diagram)

Combination and Permutation:


1. A club has 30 members. In how many ways can the president, treasurer and secretary be chosen to form a committee of three?


2. In class of 25 students, a first and second prize are to be awarded. In how many different ways can this be done?

3. In a class of 22, 12 are boys and the rest are girls. A student body of 2 prefects are to be elected. How many ways can they be chosen if:
a. Both prefects elected are boys
b. Both prefects elected are girls
c. 1 boy and 1 girl prefects

4. In how many ways can the letters in the word below be arranged:
a. OFFICE
b. PEN
c. HOSPITAL

5. A committee of 5 men and 3 women is to be chosen to form a committee if 7 people from a group of 10 men and 8 women. How many possible committees can be selected?
6. In how many ways can a committee of 7 be chose from a 9 married couples if:
a) The committee must have 4 men and 3 women
b) The committee are all chaired by women
c) No restrictions are made

7. Suppose there are 20 people and only 8 people are invited to a party. In how many ways can the party be selected?

8. A typical football team consists of two goalies, six defenders, six midfielders and four attackers. In how many ways can the coach select a starting lineup of one goalie, four defenders, four midfielders and two attackers?

9. In how many ways can a football team (11 players) be chosen from 15 people?

10. In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10      How many ways can you choose the top 3 goals?

Indices Question:

http://mathematics.laerd.com/maths/indices-1.php

Logarithm Question:

.http://www.regentsprep.org/regents/math/algtrig/ate9/logprac.htm

Statistical Data question:

From the following information, distinguish whether it is (i) quantitative or qualitative, (ii) discrete or continuous data.     

a)      The weight of babies

b)      The body shop perfume fragrances

c)       BDTVEC grading system (e.g. grade A, grade B, etc)

d)      The number of Brunei population in the year of 2011

e)      The brands of car sold in Brunei

               f)       The number of voters that vote for Hairi as a president for Student Committee for CCCT .

Measure of Central Tendency        

      1.      Determine the mean, mode and median for the following data:                                                               

a)      2, 3, 4, 6, 11

b)      21, 23, 15, 21, 27, 30

2.      The qualifying mathematics test marks for Cosmopolitan College of Commerce and Technology are shown in the table below.                                                                                                                      

Marks awarded (x)	Number of students
0 ≤ x < 20	4
20 ≤ x < 40	6
40 ≤ x < 60	15
60 ≤ x < 80	5
80     ≤ x < 100	3

For the above data, determine the mean and median of the data.    

For Representation of Statistical Data question, the URL below will help you:

http://www.mathgoodies.com/lessons/graphs/bar_graph.html

Probability 

1.     

Logarithm

Law of Logarithm:

Examples:

These are the examples on writing a logarithm:

Below is how to solve a logarithm equation:

For an explanation, watch the video below:

Indices

Law of Indices:

Examples:

Calculate the following: 

a) 2^0 + 3^1+2^0

 =1+3+1= 5

b) 2^3 *2^4

=2^3+4 = 2^7 = 128

c) 25^-2

= 1/ 5^2 = 1/625

Explanation regarding this topic:

Introduction to Statistical Data

In this topic it helps to differentiate data. These are the main topic for Statistical Data:

1. Quantitative and Qualitative
2. Discrete and Continuous data

Quantitative and Qualitative

Quantitative is a numerical data. For example,  the cost of a shirt.

Qualitative is a non-numerical data. For example, the colour of a shirt.

Discrete and Continuous Data

Discrete data can be counted. For example, number of people.

Continuous data is measured. For example, the height of a person.

Example:

1.      The number of students in Cosmopolitan college in the year of 2012
2.   The color of car passing by my house
3.   The size of shoes from National Diploma in Computer Studies for intake 2.
4.       The height of people in Brunei

   Answer:

1.   1. Discrete and Quantitative
2.      Qualitative
3.      Continuous data and Quantitative
4.      Continuous

Permutation and Combination

Permutation

It is the number of different ways that a certain number of objects can be arranged in order from a large number of objects. Permutation is about arrangements.

Combination

It is the number of different ways that a certain number of objects as a group can be selected from a larger number of objects. Combination is about choice, selection and election.

Formula for Combination and Permutation:

Example 1:

There are 5 boxes, how can the boxes be arranged?

From the question itself, it tells that it is Permutation because it mentions about arrangement .So, use the permutation formula for the answer.

 P   =    n! / (n-r)!

      =   5! / (5-5)! = 5*4*3*2*1 / 2 = 120 

Example 2:

In how many ways can three student council members be elected from five candidates?

It is a combination. and the combination formula will be use for the answer.

C= n! / r!(n-r)! 

  = 5! / 3!(5-3)!=5*4*3*2*1 / 3*2*1*2*1= 20/2 =10

For deep explanation, watch the video below:

Representation of Statistical Data

There are many ways to present data, either by using graph or table. These are some of graphs and tables that are mainly use to convey useful information.

1. Frequency Table
2. Bar Graph
3. Pictograph
4. Line Graph
5. Pie Chart
6. Histogram

Frequency Table

Frequency Table Example

A frequency table is created by making a table with 3 columns where the first column for Intervals, the second column for Tally and the third column for Frequency. Picture above is the example of frequency table.

Bar Graph

Bar Graph Example

Bar graph is a graph drawn using rectangular bars to show how large each value is. The arrangement of the bars can be horizontal or vertical.

Pictograph

Pictograph tells a value of data by using pictures or symbol. For example in the above picture, a symbol of cupcake is equivalent to 6 cupcakes.So, in Monday, there 30 cupcakes.

Line Graph

 Line graph is a type of chart that displays information by using a data points called 'markers' and it is connected by a line. It is a basic type of chart and commonly used in many fields.

Pie Chart

 Pie chart is a type of graph which is a circle that will be divided into sectors and each sector represent its own part. For example, the blue colour in above pie chart is representing 'drama' and 5% of people chose it as their favorite type of movie.

. Histogram

 Histogram is a graph that uses rectangular shape to show data and it usually in vertical position. It is quite similar with Bar Chart but histogram uses 'ranges' in the x-axis.

Measures of Central Tendency

There are 3 main things in Measures of Central Tendency:

1. Mean
2. Median
3. Mode

Mean

It is a sum of all the data values divided by the total number of data.

Formula:

Median

It is the middle value when the data are arranged in ascending order.

Mode

It is the value that occurs most often in a set of data.

Example 1:

Show the mean,median and mode in the set of data shown:

                           

                          {2,5,4,6,8,3,7,3,1}

Mean = 2+5+4+6+8+3+7+3+1 / 9

          = 4.3

Median= 1,2,3,3,4,5,6,7,8

            = 4 is the median.

Mode = 3

Example 2:

Show the mean,median and mode in the set of data shown:

                           

                          {32,35,34,36,31,33}

Mean = 32+35+34+36+31+33 / 6

          = 33.5

Median= 31,32,33,34,35,36

            = 33+34 / 2

            = 33.5

Mode = There is no mode.

Sets

Sets is the collection of distinct objects.

There are 3 common things for sets:

1. Unions(∪)-to collect the sets together and combine it.
2. Intersection(∩)-to see the common in a sets.
3. Complement(')- the sets that are complemented will be subtracted.

Examples 1:

x={3,8,5,14}

y={4,3,13,15}

x ∪ y={3,8,5,14,4,3,13,15}- all of the numbers are combined since it is Union.

x ∩ y={3}- it is because the number 3 are mentioned in both sets which makes it as common number.

Examples 2:

Universal Set:{7,8,9,10,11,12,13}

A={8,10,12}  B={10,12,13} C={7,8} 

A ∪ B= {8,10,12,13}

A∩ B={10,12}

A ∪ B'={8,10,12} {7,8,9,11}

           ={7,8,9,11,12}-since the B is compliment, all of the numbers except the number in B are the                                          answer.

Venn Diagram

It is a diagram representing mathematical or logical sets pictorially as circles or closed curves witin an enclosing rectangle(which known as universal set), common elements of the sets being represented by intersection of the circles.

Example:

In the picture above, it shows that the A is Union with B as both shape are shaded.

While the picture above is an intersection where A intersect B since only the joint between the two shape are shaded.

For more explanation, watch the video below:

Probabilty

Probability is the quality of being probable or the extent to which something likely to happen.

For example, when tossing a coin, there are two possibles outcomes which is the Head or the Tail. So, the probability of the coin landing either Head or Tail is 1/2 . 

 This is also the same with throwing dice, when it is thrown there are six possible outcomes which is 1,2,3,4,5 or 6. The probability of any one of them is 1/6.

Tree Diagram

Below is the picture of a Tree diagram:

We can extent it into two tosses of a coin and calculate by multiplying probabilities along the branches and add probabilities down the columns:

More explanation below: