MODULE : MATHEMATICS 2 (Diploma) GOALS: The goal of this course is to enhance students’ understanding and knowledge in Mathematics for technical applications. LEARNING OUTCOMES : After completing this course successfully, students should be able to : 1. Apply Pyhtagoras Theorem to solve problems in geometry for a right angled triangle. 2. Compute the distance, mid-point and slope between two coordinates points in Cartesian Coordinate System. 3. Derive the equation of lines based on the linear equation, y = mx + c. 4. Solve the linear equations problems with one unknown. Solve the system of linear equations by using the inverse matrix method for 2 by 2 matrices. LEARNING CONTENT: 2
Transcript
MODULE : MATHEMATICS 2
(Diploma)
GOALS:
The goal of this course is to enhance students’ understanding and knowledge in Mathematics for technical applications.
LEARNING OUTCOMES :
After completing this course successfully, students should be able to :
1. Apply Pyhtagoras Theorem to solve problems in geometry for a right
angled triangle.
2. Compute the distance, mid-point and slope between two coordinates
points in Cartesian Coordinate System.
3. Derive the equation of lines based on the linear equation, y = mx + c.
4. Solve the linear equations problems with one unknown.
Solve the system of linear equations by using the inverse matrix method for 2 by 2
matrices.
LEARNING CONTENT:
This module consists of :
Unit Title Hours
(45)
Unit 1 INDICES, LOGARITHM AND SURDS 6
Unit 2 MATRICES 10
Unit 3 TRIGONOMETRY 10
Unit 4 FUNDAMENTAL STATISTICS 9
Unit 5 STATISTICS 10
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ASSESSMENT :
Assignments : 10 %
PBL/Presentations : 30 %
Quizzes : 10 %
Tests : 20 %
Final Examinations : 30%
REFERENCES :
1. Abd. Wahid Md Raji & Rakan-rakan. (2000). Matematik Asas - Jilid I, Jabatan Matematik, Fakulti Sains, UTM.
2. Abd Wahid Md Raji & Rakan-rakan.(2000) Matematik Asas- Jilid II, Jabatan Matematik, Fakulti Sains, UTM.
3. Barnett, R. A., Ziegler, M.R. and Byleen, K.E. (2000). College Algebra with Trigonometry, 7th Ed. New York: Mc Graw Hill.
4. Kaufmann, J.E. and Schwitters,K.(2000). Algebra for College Students, 6th Ed. Pacific Grove: Brooks/Cole Thomson Learning.
5. Quek Suan Goen, Leng Ka Man & Yong Ping Kiang (2004). Mathematics STPM. Federal Publications, Selangor.
6. Woodbury, G.(2004). An introduction to Statistics. Thomson Learning.
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UNIT 1
INDICES, LOGARITHM AND SURDS
1.1 Introduction
In this topic students will be exposed to the rules of fundamental algebra in
solving problems dealing with indices, logarithms and surds. At the end of it,
students will see the connection between them.
Objectives
At the end of the topic, students should be able to:
use the laws of indices in solving problems
use the laws of logarithms in solving problems with general based logarithms,
common logarithms and natural logarithms.
rationalize the numerator or the denominator of surds and solve problems
concerning surds.
1.2 Indices and Laws of Indices
1.2.1 Indices
(A) Positive integer indices
If a is a non-zero numeral and n is a positive integer, then means a is
multiplied by itself n times and is called a to the power of n.
with
n times
In , a is the base and n is the index.
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Example 1.1
Find the value of
(a) . (b) .
(c) . (d) .
(B) Zero index
Any number with zero indexes equals to one. For instance,
.
Example 1.2
Find the value of each of the following without using a calculator.
(a) . (b) ( . (c) .
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(C) Negative indices
is the reciprocal of
Example 1.3
Find the value of each of the following and verify your answers using a
calculator.
(a) . (b) . (c) .
(D) Fractional Indices
1. is the nth root of a, that is
where
2. is the nth root of , that is
where
Example 1.4
Find the value of
(a) . (b) .
(c) . (d) .
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Example 1.5
Find the value of
(a) . (b) .
1.2.2 Laws of Indices
Example 1.6
Find the value of
(a) . (b) .
(c) . (d) .
Example 1.7
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Simplify and evaluate each of the following.
(a) . (b) .
Example 1.8
Simplify each of the following.
(a) . (b) .
(c) . (d) .
Example 1.9
Simplify each of the following.
(a) . (b) .
(c) . (d) .
Practice 1.1
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1. Find the value of
(a) . (b) .
Solution
(a) means _____ is multiplied by itself ________ times.
So, .
(b) means _____ is multiplied by itself ________ times.
Hence, .
2. Find the value of
(a) . (b) .
Solution
(a) .
(b) .
3. Find the value of
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(a) . (b) . (c) .
Solution
(a) .
(b) .
(c) .
4. Find the value of
(a) . (b) . (c) . (d) .
Solution
(a) Use the laws of indices. What should be done to the index if indices with the
same base are multiplied?
.
(b) By applying the laws of indices, what should be done to the index?
.
(c) Do you know which laws to apply? Check the base and the operation involve!
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(d) Refer to the laws of indices. What should be done to the index?
.
5. Simplify each of the following.
(a) . (b) . (c) . (d) .
Solution
(a) Same base multiplied, so the indices should be ……..…
.
(b) Same base divided, so the indices should be ………….
.
(c) What should be done to the index?
.
(d) Please refer to the laws of indices.
1.3 Logarithms and Laws of Logarithms
1.3.1 Definition of Logarithm
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If a is a positive number and then
.
is read as ‘logarithm of N to the base of a’.
Note : 1. .
2. .
Example 1.10
Express each of the following in logarithmic form.
(a) .
(b) .
(c)
Example 1.11
Express each of the following in index form.
(a) .
(b) .
1.3.2 Logarithm of a number
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1. Logarithms to the base of 10 are called common logarithms. The values
of common logarithms can be determined by using a scientific
calculator.
2. Logarithms to the base of e are called the natural logarithms. The natural
logarithms of x is usually written as ln x instead of .
Example 1.12
By using a scientific calculator, find the value of
(a) . (b) .
Note : 1. Logarithm of negative numbers is undefined.
2. Logarithm of zero is undefined.
Example 1.13
Find the value of
(a) . (b) .
Example 1.14
Find the value of x in each case.
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(a) . (b) .
(c) . (d) .
1.3.3 Laws of Logarithm
There are three basic laws of logarithms.
Example 1.15
Given that and , find the value of each of the
following without using a calculator.
(a) . (b) . (c) .
Example 1.16
Evaluate the following without using a calculator.
(a) . (b) .(c) .
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Example 1.17
Given that and find the value of .
Example 1.18
Given that and , express in terms of x and y.
Practice 1.2
1. Express each of the following in logarithmic form.
(a) . (b) . (c) .
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Solution
(a) If , then .
(b) If , then
(c) If , then .
2. Express each of the following in index form.
(a) . (b) .
Solution
(a) If , so .
(b) If , so .
3. Evaluate the following without using a calculator.
(a) . (b) . (c) .
Solution
(a) Write 81 in the index form. Can you determine the
base?
(b) Write 2 as the fractional indices of 8. What should be the index?
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(c) . Write 100 in index form with base ______.
4. Find the value of without using a calculator.
Solution
1.3.4 Change of Base of Logarithms
Let So .
By taking logarithm to base c on both sides, we have
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As , hence
.
Note : We can find the logarithm of a number by changing the base of the
logarithm to any appropriate base.
Example 1.19
Find the value of each of the following without using a scientific calculator.
(a) . (b) . (c) .
Example 1.20
Find the value of
1.3.5 Solving problems involving the change of base and laws of
logarithms
Example 1.21
If and , express each of the following in terms of m and n.
(a) . (b) .
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Example 1.22
Simplify .
1.3.6 Equation Involving Indices and Logarithms
(A) Solving equations involving indices
Steps in solving :
- simplify the algebraic expressions on both sides of the equation
- express the expressions in terms of the same base or the same index
- equate the base or the indices accordingly
For example,
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if , then m = n.
if , then a = b.
Example 1.23
Solve the following equations.
(a) . (b) . (c) .
(B) Solving equations involving logarithms
Example 1.24
Solve the following equations.
(a) .
(b) .
Practice 1.3
1. Solve for x if .
Solution
Apply log with base 10 to both sides of the equation.
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What happen if you use log with base e, the natural log?
Does it give the same answer? Please try….
2. Solve the following equations.
(a) .
Solution
So, the answers for x are : ……………………..
(b)
Solution
Arrange the terms with x,
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Calculator!
Eliminate the denominator,
.
Thus, the answers for x are : ……………………..
3. Determine the value of x if
Solution
You have to solve it by substituting to the equation.
Please continue on the steps of solving. You are correct if the answers are :
x = 2 and x = 2.322.
1.4 Surds and Laws of Surds
The expression is a radical, the number a is the radicand, and n is the index of
the radical. The symbol is called a radical sign.
1.4.1 Laws of Surds
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The three listed laws in the next chart are true for positive integer’s m and n,
provided the indicated roots exist.
Example 1.25
Simplify the following expressions by using the laws of surds.
(a) . (b) . (c) .
1.4.2 Rationalizing the Numerator and Denominator of Surds
If the numerator and denominator of a quotient contains of the form , with k <
n and a > 0, then multiplying numerator and denominator by will eliminate
the radical.
Example 1.26
Rationalize the denominator or the numerator.
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(a) .
(b) .
(c) .
(d) .
(e) .
Practice 1.4
1. Rationalize the numerator.
(a) . (b) .
Solution
(a) The numerator and the denominator should be multiplied by ………..
So,
= ………..
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(b) The numerator and the denominator should be multiplied by ……………….
Hence,
= …………………….
2. Rationalize the denominator.
(a) . (b) . (c)
Solution
(a) The …………………….. and the ……………………… should be multiplied
by ……………..
So, the answer is .
(b) The …………………….. and the ……………………… should be multiplied
by ……………..
Hence,
= .
(c) Try this on your own!
The …………………….. and the ……………………… should be multiplied
by ……………..
You got it right if the answer is .
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EXERCISE
1. Find the value of the following.
(a) . (b) .
(c) . (d) .
(e) . (f) .
(g) . (h) .
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(i) . (j) .
2. Find the value of
(a) . (b) .
(c) . (d) .
(e) . (f) .
(g) . (h) .
(i) . (j) .
3. Simplify and evaluate each of the following.
(a) . (b) .
(c) . (d) .
(e) . (f) .
4. Simplify.
(a) . (b) .
(c) . (d) .
(e) . (f) .
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5. Express each of the following in logarithmic form.
(a) . (b) .
(c) . (d) .
6. Convert each of the following to index form.
(a) . (b) .
(c) . (d) .
7. Find the value of x in each case.
(a) . (b) .
(c) . (d) .
(e) . (f) .
8. Given that and find the value of the following
without using a calculator.
(a) . (b) .
(c) . (d) .
(e) . (f) .
9. Simplify the following logarithmic expressions to the simplest form.
(a) .
(b) .
(c) .
(d) .
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(e) .
(f) .
10. Find the value of
(a) . (b) .
(c) . (d) .
(e) . (f) .
11. Given that and , find the value of each of the
following without using a scientific calculator.
(a) . (b) .
(c) .
12. If , express each of the following in term of t.
(a) . (b) .
(c) . (d) .
(e) . (f) .
13. If and , express each of the following in terms of p and q.
(a) . (b) .
(c) . (d) .
14. Find the value of
(a) .
(b) .
(c) .
15. Rationalize the denominator.
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(a) . (b) .
(c) . (d) .
(e) . (f) .
16. Rationalize the numerator.
(a) . (b) .
Activity
1. Savings account
One of the oldest bank in Malaysia is the Standard Chartered Bank. If RM 200
had been deposited at that time into an account that paid 4% annual interest,
then 180 years later the amount would have grown to dollars.
Approximate this amount to the nearest cent.
2. Length of a halibut
The length-weight relationship for Pacific halibut can be approximated by
, where W is in kilograms and L is in meters. The largest
documented halibut weighed 230 kg. Estimate its length.
3. Weight lifter’s handicaps
O’Carroll’s formula is used to handicap weight lifters. If a lifter who weighs b
kg lifts w kg of weight, then the handicapped weight W is given by
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.
Suppose two lifters weighing 75 kg and 120 kg lift weights of 180 kg and 120
kg lift weights of 180 kg and 250 kg, respectively. Use O’Carroll’s formula to
