llege · 2020. 8. 30. · table of contents contents (unit-6) functions and graphs 1 6.1 domain,...

MILITARY TECHNOLOGICAL COLLEGE

GSD- Pure Mathematics MODULE CODE: MTCG1018

W

TERM-2 AY: 2019-2020 GFPMATH2

WORKBOOK-2

Table of Contents

Contents

(Unit-6) Functions and Graphs 1

6.1 Domain, range and function 1

6.2 Types of functions 4

6.3 Inverse function 8

6.4 Operations of Functions 10

6.5 Composite Function 12

Worksheet 6 14

(Unit-7) Exponential and logarithm functions 18

7.1 Exponential functions & Graphs 18

7.2 Logarithmic Function & Graphs 22

7.3 Applications in real life 24

Worksheet-7 28

(Unit-8) Inverse relation between exponential and logarithm functions 33

8.1 Inter-conversion of exponential and logarithm functions 33

8.2 Exponential and logarithmic equations 34

Worksheet-8 35

(Unit-9) Polar Coordinates 37

9.1 Polar Coordinate System-graphical representation 37

9.2 Conversion from Cartesian to Polar Coordinates 39

9.3 Conversion from Polar to Cartesian Coordinates 41

Worksheet 9 42

(Unit-10) Number Systems-Conversions 47

10.1 Converting different number systems to decimal (base 10) 47

10.2 Converting decimal (base 10) to other number systems 49

10.3 Converting binary to Octal or Hexadecimal system 52

10.4 Converting Octal or Hexadecimal to Binary system 54

10.5 Inter-conversion of Octal and Hexadecimal 57

10.6 Addition and subtraction of Binary numbers 60

Worksheet-10 61

References 64

Assessment Plan (Passing Mark: 50 %.)

Assessment Weightage

Quiz 10%

Midterm 40%

Final 50%

Total 100%

Note: Only Non-Programmable calculators are allowed.

Attendance Policy:

Warning Absence

First 10%

Second 15%

Third 20%

1

(UNIT-6) FUNCTIONS AND GRAPHS

6.1 DOMAIN, RANGE AND FUNCTION

Relation: A relation is simply a set of

ordered pairs (x, y).

The first elements in the ordered pairs

(the x-values), form the domain. The

second elements in the ordered pairs

(the y-values), form the range. Only

the elements "used" by the relation

constitute the range.

This mapping shows a relation from set A

into set B This relation consists of the ordered

pairs {(1,2), (3,2), (5,7), (9,8)}

The domain is the set {1, 3, 5, 9}.

The range is the set {2, 7, 8}.

The codomain is the set

B = {2, 3, 5, 6, 7, 8}.

3, 5 and 6 are not part of the range.

The range is the dependent variable

Example: Find the domain and range of the

relation. 𝑅 = {(1,1)(2,4)(3,9)}

Solution: Domain = {1,2,3}

Range = {1,4,9}

Class Activity 1

1) Find the domain and range of the relation

Solution:

2) Find the domain and range of the relation

x -2 0 2

y -8 0 8

Solution:

3) Find the domain and range of the relation.

𝑅 = {(𝑎, 1)(𝑏, 2)(𝑐, 3)}

Solution:

0

1

2

1

6

11

2

Definition of a Function

A function is a rule that produces a

correspondence between two sets of elements

such that to each element in the first set there

corresponds one and only one element in the

second set.

Note:

The set of all first elements is called

domain.

The set of all second elements is called

codomain.

The list of all elements which appears

as images is called range.

Example 1: Write domain, codomain and

range from the following figure. Is ‘f ‘a

function?

Solution: Domain = {1, 2. 3}

Codomain= {a, b, c, d}

Range = {a, b, c}

f (1)=a, f (2)=b, f (3)=c

Yes, ‘f’ is a function.

Example 2: Write domain, codomain and

range from the following figure. Is ‘g’ a

function?

Solution: Domain = {1, 2. 3}

Codomain= {a, b, c}; Range = {a, b}

g(1)=a, g(2)=b, g(3)=b

Yes, ‘g’ is a function.

Example 3: Is ‘ h’ a function?

Answer: No ‘ h’ is not a function since

element ‘3’ in Set-A have two images.

Example 4: Is ‘f ‘a function?

Answer: No ‘ f’ is not a function since

element ‘3’ in Set-A have no image.

Determining if a Graph Defines a Function

(Vertical Line Test):

A graph defines a function if each vertical line

passes through exactly one point on the graph

of the equation.

Example 5: Which of the following graphs are

function?

i)

-1 1

-2

-1

1

2

x

y

Answer: Yes it is function

g 1

3

2 b

a

c

B A

h 1

3

2 c

b

a

d

B A

f 1

3

2

c

b

a

d

B A

3

ii)

1

-1

1

x

y

Answer: Not a function

iii)

-1.5 -1 -0.5 0.5 1 1.5

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

Answer: Yes it is function

iv)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

x

y

Answer: Not a function

Class Activity 1

Circle the correct answer in the following

questions.

1) Which of the following ordered pairs are

function?

(a) )3,(),2,(),1,( bbb

(b) )2,(),2,(),2,( cba

(c) )3,2(),2,1(),1,1(

2) Which of the following is a function?

(a)

(b)

(c)

3) Which of the following is a function?

(a)

(b)

(c)

f 1

3

2

c

b

a

d

B A

h 1

3

2 c

b

a

d

B A

4

6.2 TYPES OF FUNCTIONS

One to one function: A function ‘f’ from a

set A to a set B is said to be one-to-one if no

two distinct elements in A have the same

image under ‘f’.

Note: If any horizontal line cuts the graph at a

single point then it is one-to-one function.

Example 1:

Example 2: Which of these graphs represent one

to one functions?

(i) Graph for 𝑓(𝑥) = √𝑥

Solution:

Since horizontal line cuts the graph only at

single point it is one to one function.

(ii) Graph for 𝑓(𝑥) = 𝑥3

Solution:

Since horizontal line cuts the graph only at

single point it is one-to-one function.

(iii) Graph for𝑓(𝑥) = 𝑥2.

Solution:

Since horizontal line cuts the graph at two

points it is not a one-to-one function.

(iv) Graph for 𝑓(𝑥) = 3𝑥 + 7.

Solution:

Since horizontal line cuts the graph only at a

single point, it is a one-to-one function.

Example 3: Which of the following are one to

one functions?

(i) 𝑓(𝑥) = 𝑥3

Answer: Yes, it is one to one function.

Reason: In 𝑓(𝑥) = 𝑥3

Put 𝑥 = 1we get 𝑓(1) = 13 = 1

Put 𝑥 = −1 we get 𝑓(−1) = (−1)3 = −1

Since 𝑓(−1) ≠ 𝑓(1) so it is one to one

function.

Alternate reason: Since the power of x is odd

so it is one to one.

5

Note: 1) If all the powers of x are odd then it

is one to one function.

2) Any constant has even power of x. For

example 2 = 2𝑥0

(ii) 𝑓(𝑥) = 𝑥2 𝑜𝑟 𝑦 = 𝑥2 Answer: No, it is not one to one function.

Reason: In 𝑓(𝑥) = 𝑥2

Put 𝑥 = 1 we get 𝑓(1) = 12 = 1

Put 𝑥 = −1 we get 𝑓(−1) = (−1)2 = 1

Since 𝑓(−1) = 𝑓(1) so it is not one to one

function.

Alternate reason: Since the power of x is even

so it is not one to one.

Class Activity-1

1) Which of the following graphs represent one to

one functions?

Answer:

Answer:

Answer:

Answer:

2) Which of the following are one to one

functions?

a) 𝑦 =7𝑥−2

3

Answer:

b) 𝑦 = 3𝑥2 − 1

Answer:

c) 𝑦 = 𝑥4 − 1

Answer:

d) 𝑦 = 2𝑥3 + 1

Answer:

3) Which of the following sets represent one to

one function?

a) ),3(),,2(),,1( bba

Answer:

b) )3,(),2,(),1,( cba

Answer:

6

Onto function: A function ‘f ’ from a set A

to a set B is said to be onto if for each element

y in B there exist an element x in A such

that 𝑓(𝑥) = 𝑦 In other words a function ‘f ’ from a set A

to a set B is said to be onto if the range of f

equals B , that is 𝑓(𝐴) = 𝐵

Example:

Constant function: A function ‘f ’ from a

set A to a set B is said to be Constant function

if all elements of A are mapped to single

element of B.

Example 1:

Example 2: 𝑓(𝑥) = 3 where x ∈ 𝑁 Solution: Its graph is

𝑓(1) = 3; 𝑓(2) = 3; 𝑓(3) = 3; 𝑓(100) = 3

so,

Applications of function:

Example: The number of computers infected

by a computer virus increases according to

𝑣(𝑡) = 𝑡2 + 2, where t is the time in hours.

Find

(a) The initial number of infected computers

(b) At 𝒗(𝟏) i.e. after one hour

(c) At 𝒗(𝟐) i.e. after two hours

Solution:

𝑣(0) = 02 + 2 = 2

𝑣(1) = 12 + 2 = 3

𝑣(2) = 22 + 2 = 6

Class Activity-2

I ) Circle the correct answer in the following

questions.

1) Which of the following relations is onto

function?

(a)

(b)

(c)

7

2) If 5)( xf where Nx then )(xf

is called ……………… function.

(a) Constant

(b) One-to-one

(c) Onto

II) The number of students in the ground is

given by 𝑓(𝑥) = 𝑥3 + 1, where ‘x’ is time in

hours.

Find number of students in the ground after 2

hours.

Solution:

8

6.3 INVERSE FUNCTION

Inverse Function: If f is a one-to-one and

onto function, then the inverse of f, denoted

by1f , is the function formed by reversing all

the ordered pairs in f.

Properties of Inverse Function

If 1f exists, then

1. 1f is a one-to-one and onto function.

2. Domain of 1f = Range of f

3. Range of 1f = Domain of f

Note: If f is not one-to-one, then f does not

have an inverse and 1f does not exist.

Example 1: Show that the function

43)( xxf is one-to-one. Find its

inverse.

Solution: Step 1: Verify that f is one-to-one.

ba

ba

babfaf

33

4343)()(

Hence, f is a one-to-one.

Step 2: Solve the equation )(xfy for x.

3

4

43

43

yx

yx

xy

Step 3: Interchange x and y

)(3

4 1 xfx

y

Example 2: Show that the function

43

32)(

x

xxf is one-to-one. Find its

inverse.

Solution:

Step 1:

ba

ba

abba

ababbaab

b

b

a

a

bfaf

1717

9898

1298612986

43

32

43

32

)()(

Hence, f is a one-to-one.

Step 2: Solve the equation )(xfy for x.

324343

32

xyxy

x

xy

3423 yxxy

34)23( yyx

23

34

y

yx

Step 3: Interchange x and y

)(23

34 1 xfx

xy

Class Activity

1) Find the inverse of the following functions:

i) )3,(),2,(),1,( cbaf

Ans:

ii) )8,4(),7,3(),6,2(),5,1(g

Ans:

9

2) Show that the function 32)( xxf is one

to one and find the inverse.

Solution:

3) Show that the function 34)( xxf is one

to one and find the inverse.

Solution:

4) Find the inverse of the following one to one

function 2

43)(

xxf

Solution:

10

6.4 OPERATIONS OF FUNCTIONS

Arithmetic of Functions

1). Sum function

( f + g )( x ) = f ( x ) + g ( x )

2). Difference function

( f – g )( x ) = f ( x ) – g ( x )

3) Product function

( f . g )( x ) = f ( x ) . g ( x )

4) Quotient function

(𝑓

𝑔) (𝑥) =

𝑓(𝑥)

𝑔(𝑥)

Example 1) Let f and g be the functions

defined by 𝑓(𝑥) = 𝑥2-1 and g(𝑥) = 𝑥 + 2.

Find the functions i) f + g ii) f – g

iii) f . g iv) 𝑓

g

Solution:

i) (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + g(𝑥)

= (x2 − 1) + (x + 2)

= 𝑥2 + 𝑥 + 1

ii) (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)

= (𝑥2 − 1) − (𝑥 + 2)

= 𝑥2 − 1 − 𝑥 − 2

= 𝑥2 − 𝑥 −3

iii) (𝑓. 𝑔)(𝑥) = 𝑓(𝑥). 𝑔(𝑥) = (𝑥2-1)(𝑥 + 2)

= 𝑥3 +2𝑥2-𝑥 − 2

iv) (𝑓

g) (𝑥) =

𝑓(𝑥)

g(𝑥)=

𝑥2−1

𝑥+2

Example 2) Let ‘f’ and ‘g’ be the functions

defined by 𝑓(𝑥) = √2𝑥 + 4 and g(𝑥) =

√𝑥 − 2. Find the functions

i) (𝑓 + g)(6)

ii) (𝑓 − 𝑔)(6)

iii) (𝑓. 𝑔)(2)

iv) (𝑓

𝑔) (4)

Solution:

i) (𝑓 + 𝑔)(6) = 𝑓(6) + 𝑔(6)

= √(12 + 4 ) + √(6 − 2)

= √16 + √4

= 4 + 2 = 6

ii) (𝑓 − 𝑔)(6) = 𝑓(6) − 𝑔(6)

= √(12 + 4 ) − √(6 − 2)

= 4 − 2 = 2

iii) (𝑓. 𝑔)(2) = 𝑓(2). 𝑔(2)

= √ (4 + 4) √ (2 − 2)

= √8. 0 = 0

iv) (𝑓

𝑔) (4) =

𝑓(4)

𝑔(4) =

√8+4

√4−2

=√12

√2 = √6

Example 3): 𝐼𝑓 𝑓(𝑥) = 𝑎𝑥 + 25 and

𝑓(3) = 7, find 𝑎. Solution:

𝑓(𝑥) = 𝑎𝑥 + 25

𝑓(3) = 𝑎(3) + 25 = 7

3𝑎 + 25 = 7

3𝑎 = 7 − 25

3𝑎 = −18

𝑎 =−18

3

𝑎 = −6

Class Activity

1) If f and g be the functions defined by

𝑓(𝑥) = 2𝑥 + 3, g(𝑥) =2x + 2, then find

i) (f + g) (x)

ii) (f – g) (2)

11

iii) (f. g) (2)

iv) (f

g) (2)

v) )1(3)1(2 gf

(2) If 15)( axxf and 10)5( f . Find

‘a’

Solution:

12

6.5 COMPOSITE FUNCTION

Let f and g be functions , then f o g is

called the composite of g and f and is defined

by the equation ( f o g ) ( x ) = f ( g ( x ) ).

Example 1: Let ‘f’ and ‘g’ be the functions

defined by 𝑓(𝑥) = 𝑥 − 4 and g(𝑥) = √𝑥.

Find (𝑓og)(𝑥) and (go𝑓)(𝑥).

Solution:

(𝑓og)(𝑥) = 𝑓(g(𝑥))

= 𝑓(√𝑥) = √𝑥 − 4

(𝑔𝑜𝑓)(𝑥) = 𝑔(𝑓(𝑥))

= 𝑔(𝑥 − 4) = √𝑥 − 4

Example 2: Let ‘f’ and ‘g’ be the functions

defined by 𝑓(𝑥) = √9 − 𝑥 and g(𝑥) = 10 − 𝑥

Find (𝑓𝑜𝑔)(𝑥) and (𝑓𝑜𝑔)(4).

Solution: (𝑓𝑜𝑔)(𝑥) = 𝑓(𝑔(𝑥))

= 𝑓(10 − 𝑥)

= √9 − (10 − 𝑥)

= √𝑥 − 1

(𝑓𝑜𝑔)(4) = √4 − 1=√3

Example 3: Let ‘f’ be the functions defined by

𝑓(𝑥) = 5𝑥 − 2. Find (𝑓𝑜𝑓)(𝑥) and (𝑓𝑜𝑓)(3)

Solution:

(𝑓𝑜𝑓)(𝑥) = 𝑓(𝑓(𝑥)) = 𝑓(5𝑥 − 2)

= 5(5𝑥 − 2) − 2

= 25𝑥 − 10-2

= 25𝑥 − 12

(𝑓𝑜𝑓)(3) = 25(3) − 12 = 75 −12

= 63

Class Activity

1) For the indicated functions ‘f ‘and ‘g’, find

the functions fog, gof and fof.

(i) 3

1

3 3)(;4)( xxgxxf

Solution:

(ii) 1)(;5)( 22 xxgxxxf

Solution:

13

2) Let ‘f’ and ‘g’ be the functions defined by

1)( xxf and 1)( xxg .

Find )2(fg

Solution:

3) Let ‘f ’ and ‘g ’ be the functions defined by

1)( xxf and 52)( xxg . Find

)1()( gf

Solution:

14

WORKSHEET 6

Section-A


questions.

(1) In the following figure the codomain is

…………..

(a) 9,5,3,1

(b) 8,7,6,5,3,2

(c) 8,7,2

(2) The range in the set of ordered pairs

)5,4(),2,3(),1,2(),1,1( is …………..

(a) 5,2,1

(b) 4,3,2,1

(c) 5,4,3,2,1

(3) Which of the following ordered pairs are

Functions?

(a) )3,(),2,(),1,( bbb

(b) )2,(),2,(),2,( cba

(c) )3,2(),2,1(),1,1(

(4) Which of the following is a function?

(a)

(b)

(c)

(5) Which of the following is a function?

(a)

(b)

(c)

f 1

3

2

c

b

a

d

B A

h 1

3

2 c

b

a

d

B A

15

(6) Which of the following sets represent one

to one function?

(a) ),3(),,2(),,1( bba

(b) )1,(),1,(),1,( cba

(c) )3,(),2,(),1,( cba

(7) Which of the following equations

represent one to one function?

(a) 23 xy

(b) 22 xy

(c) 2y

(8) Which of the following is constant

function?

(a)

(b)

(c)

(9) Let ‘f’ be the function defined by

1)( 2 xxf , then )2(f is …………..

(a) 9

(b) 5

(c) 1

(10) If 4)( xxf then )(1 xf

………………

(a) 4

(b) 4x

(c) 4x

16

Section-B

Show your solution step by step in the following questions.

1) 42)(,3)( 2 xxgxxxfIf then

find

(i) )2(3)1(2 gf

(ii) )2(

)1(

g

f

2) Let ‘f’ and ‘g’ be the functions defined by

𝑓(𝑥) = 3𝑥 + 1 and g(𝑥) = 2𝑥 − 3 Find

the functions

(i) (𝑓 + g)(𝑥)

(ii) (𝑓 − 𝑔)(3)

(iii) (𝑓. 𝑔)(−1)

(iv) (𝑓

𝑔) (4).

3) If 23)( xxf find

(i) h

xfhxf )()(

(ii) ax

afxf

)()(

17

4) Show that the function 3

5)(

xxf is

one to one and find the inverse.

5) Find the inverse of the following one to

one function 13

2)(

x

xxf

6) Using 𝑓(𝑥) = 𝑥 − 2 and 𝑔(𝑥) = 5𝑥 + 3

find

i) ( ( ))f g x

ii) ( ( 4))g f

iii) ( (1))f f

iv) ( (2))f g

18

(UNIT-7) EXPONENTIAL AND LOGARITHM FUNCTIONS

7.1 EXPONENTIAL FUNCTIONS & GRAPHS

Exponential function

The equation

1,0)( aawhereaxf x

is called an exponential function. The

constant a is called the base and x is called the

exponent or power.

Examples: x

x

xx 3.0,2

1,5,2

Properties of exponential function

1) yxyx aaa

2) xyyx aa

3) xxxbaab

4) x

xx

b

a

b

a

5) yx

y

x

aa

a

Where a, b are positive and yx, are real numbers.

Note:

1) yx aa if and only if x = y

2) xx ba if and only if a = b

Example 1: Solve 84 3 x for x

Solution: 33 284 x

332 2)2( x

3)3(2 x

362 x

2

9x

Class Activity

1) Solve 927 1 x for x

1. Solve 42 12 x for x

19

Basic exponential graphs

There are two cases in exponential functions.

Case 1: 1)( awhereaxf x

Case (1): 1a , e.g.here a = 5

Case 2: 10)( awhereaxf x

Case (2): 10 a e.g. here 𝑎 = 1

2

Basic properties of exponential graphs:

1) The domain of f is the set of all real

numbers ,

2) The range of f is the set of all positive real

numbers ,0 .

3) All graphs pass through the point (0, 1).

4) All graphs are continuous that is, there are

no holes or jumps.

5) The X-axis is a horizontal asymptote, that

is, there is no intercept on X-axis.

6) If 1a , then xa increases as x

increases.

7) If 10 a , then xa decreases as x

increases.

8) The function is one to one.

Exponential function with base e

The equation xexf )( ,

where x is a real number, is called an

exponential function with base e.

Note: e = 2.718 281 828 459 …

The constant e turns out to be an ideal base for

an exponential function because in calculus

and higher mathematics many operations take

on their simplest form using this base.

Graph of exponential function with base e

Graphing of exponential functions

Example 1: Use integer values of x from -3

to 3 to construct a table of values for

)4(2

1 xy

Method : Use a calculator to create the table

of values shown below

20

Then plot the points and join these points with

a smooth curve

Example 2: Use integer values of x from -4

to 4 to construct a table of values for

24

x

ey

Method: Use a calculator to create the table of

values shown below

Then plot the points and join these points with

a smooth curve

Class Activity

1) Use integer values of x from -3 to 3 to

construct a table of values for )4(2

1 xy , and

then graph this function.

21

2) Use integer values of x from -4 to 4 to

construct a table of values for 52 2

x

ey and

then graph this function.

22

7.2 LOGARITHMIC FUNCTION & GRAPHS Logarithmic Function

The inverse of exponential function is called

logarithmic function.

The equation

( ) log , 0, 1af x x where a a

is called a logarithmic function.

Note: There are two cases in logarithmic

functions.

Case (1): 1a

Case (2): 0 1a

Domain: ,0 , Range: ,

Note: Case(1) If 1a , the graph is an

increasing function.

Case(2) If 0 1a , the graph is a

decreasing function.

The logarithm of a number is the exponent to

which the base must be raised to give that

number.

In general, log a x = n implies that an = x.

and conversely, if x = an, then log a x = n

where, a > 0, a ≠ 1, 𝑎𝑛𝑑 𝑥 > 0.

an = x is the exponential form and log a x = n

is the logarithmic form.

23 = 8 Log28 =3

102 = 100 Log10100 =2

103 = 1000 Log101000=3

Class Activity 1

1) Write each of the following in

logarithmic form:

(i) 24 = 16

(ii) 33 = 27

(iii)53 = 125

(iv) 93

(v) 15

5

1

2) Write each of the following

logarithms in exponential form:

(i) log 2 16 = 4

(ii) log 4 64= 3

(iii) log 10 1000000 = 6

(iv) 2

15log25

(v) 24

1log2

23

Properties of Logarithms

If a, x and y are positive real numbers, 1a

and b is a real number then:

1) log 1 0a

Since a0 = 1, then, log 1 0a

Example : log2 (1) = 0 and log25 (1) = 0, etc.

2) log 1a a

Since 1a a , then, log 1a a

Example : log2 2 = 1 and log20 20 = 1

3) loga xy = logax + logay

Examples: a) 2 2 2log (8 4) log 8 log 4

b) 3 3 3 3log 12 log (3 4) log 3 log 4

4) 𝒍𝒐𝒈𝒂 𝒙

𝒚 = 𝒍𝒐𝒈𝒂x - 𝒍𝒐𝒈𝒂y

Examples: a) 𝑙𝑜𝑔2 100

3 = 𝑙𝑜𝑔2100 -

𝑙𝑜𝑔23

𝑏) 𝑙𝑜𝑔10 10000

10 = 𝑙𝑜𝑔1010000 - 𝑙𝑜𝑔1010

= 4 – 1 = 3

5) 𝒍𝒐𝒈𝒂𝒙𝒃 = 𝒃 𝒍𝒐𝒈𝒂𝒙

Example 1: 𝑙𝑜𝑔1010000 = 𝑙𝑜𝑔10104 =

4 𝑙𝑜𝑔1010 = 4

Example 2: 𝑙𝑜𝑔2 (√53

) = 𝑙𝑜𝑔2 (513)

= 1

3 𝑙𝑜𝑔2 (5)

Therefore, 𝒍𝒐𝒈𝟐 (√𝟓𝟑

) = 𝒍𝒐𝒈𝟐 𝟓

𝟑

The above rules are same for all positive bases.

The most common bases are the base 10 and

the base e. Logarithms with a base 10 are

called common logarithms, and logarithms

with a base e are natural logarithms. On your

calculator, the base 10 logarithm is noted by

log, and the base e logarithm is noted by ln.

Note: When the base is 10, we do not need to

state it.

Class Activity 2

1) Find the values of the following using

the definition of logarithm

(i) 𝑙𝑜𝑔416

(ii) 𝑙𝑜𝑔5125

(iii) 𝑙𝑜𝑔81

(iv) 𝑙𝑜𝑔88

(v) 𝑙𝑜𝑔 0.1

2) Assume that 3010.02log10 , find:

(i) 4log10

(ii) 5log10 [Hint: 2

10log5log 1010 ]

24

7.3 APPLICATIONS IN REAL LIFE

Table-Exponential growth and decay

Description Equation Graph Uses

Unlimited growth ktcey

0, kc

Short-Term population

growth (people,

bacteria, etc. ) growth

of money at continuous

compound interest

Exponential decay ktcey

0, kc

Radioactive decay,

light absorption in

water, glass, etc.

atmospheric pressure,

electric circuits

Limited growth ktecy 1

0, kc

Sales fads, company

growth, electric

circuits

Logistic growth ktce

My

1

0,, Mkc

Long-term population

growth, epidemics,

sales of new products,

company growth

25

More applications of exponential function

Population growth and compound interest are

examples of exponential growth, while

radioactive decay is an example of negative

exponential growth.

Example 1: Mexico has a population of around

100 million people, and it is estimated that the

population will double in 21 years. If population

growth continues at the same rate and model of

population growth is given by : d

t

PP 20

Where, P = population at time t

P0 = population at time t = 0

d = doubling time

. What will be the population?

i) 15 years from now?

ii) 30 years from now?

Calculate the answers up to 3 significant

digits.

Solution: We use the doubling time growth

model: d

t

PP 20

Substituting P0 = 100 and d = 21 , we get

21100 2t

P

i) When t = 15 years ,

peoplemillion164

2100 21

15

P

P

ii) When t = 30 years ,

peoplemillion269

2100 21

30

P

P

Example 2: The rate of decay of radioactive

isotope gallium 67 (67Ga), used in the

diagnosis of malignant tumors, is modelled as

h

t

AA

20

where A = amount at time t , A0 = amount at

time t = 0 and h = half-life.

If we start with 100 milligrams of the isotope

and it has a biological half- life of 46.5 hours,

how many milligrams will be left after

i) 24 hours?

ii) 1 week?

Calculate the answers up to 3 significant digits.

Solution: we use the half decay model:

h

th

t

AAA

2

2

100

Substituting A0 = 100 and h = 46.5, we get

46.5100 2t

A

i) When t = 24 hours, 24

46.5100 2 69.9 millgramsA

ii) When t = 1 week = 168 hours, 168

46.5100 2 8.17 millgramsA

Example 3: If a principal P is invested at an

annual rate r compounded n times a year, then

the amount A at the end of the t years is given

by

nt

n

rPA

1 .

Suppose 1000 RO is deposited in the account

paying 4% interest per year compounded

quarterly (four times per year).

i) Find the amount in the account

after 10 years with no

withdrawals.

ii) How much interest is earned over

the 10 year period?

Compute the answer to the nearest baiza.

26

Solution: i) Compound interest formula nt

n

rPA

1

Here P = 1000, r = 4% =0.04, n = 4 and

t = 10.

104

4

04.011000

A

4001.011000 A

864.1488A (rounded to nearest baiza)

Thus 1488.86 RO is in account after 10 years.

ii) The interest earned for that period is

1488.86 RO – 1000 RO = 488.864 RO

Class Activity

I) Circle the correct answer in the

following questions.

(1) The following graph describes …………

(a) Unlimited growth

(b) Limited growth

(c) Exponential decay

II) Show your solution step by step in the


1) Over short period of times the doubling time

growth model is often used to model population

growth:

d

t

PP 20

Where, P = population at time t

P0 = population at time t = 0

d = doubling time

Let in a particular laboratory, the doubling

time for bacterium Escherichia coli (E. Coli),

which is found naturally in the intestines of

many mammals, is found to be 25 minutes. If

the experiment starts with a population of

1,000 E. coli and there is no change in the

doubling time, how many bacteria will be

present after:

i) 10 minutes?

ii) 5 hours?


digits.

Solution:

27

2) The rate of decay of radioactive gold 198

(198Au), used in imaging the structure of the

liver, is modelled as h

t

AA

20

where, A = amount at time t , A0 = amount at

time t = 0 and h = half-life.

If we start with 50 milligrams of the isotope

and it has a biological half- life of 2.67 days,

how many milligrams will be left after:

i) Half day?

ii) 1 week?


digits.

Solution:

3) If a principal P is invested at an annual rate

r compounded n times a year, then the amount

A at the end of the t years is given by

nt

n

rPA

1 .


paying 6% interest per year compounded half

yearly. Find the amount in the account after 5

years with no withdrawals.

Solution:

28

WORKSHEET-7

Section-A


questions.

(1) The equivalent logarithm form of

3225 is …………..

(a) 232log5

(b) 532log2

(c) 325log2

(2) The equivalent exponential form of

364log4 is …………..

(a) 6434

(b) 6443

(c) 644 3

(3) 64log4 …………..

(a) 4

(b) 16

(c) 3

(4) 4log4 …………..

(a) 3

(b) 4

(c) 1

(5) 01.0log10 …………..

(a) 100

(b) 2

(c) 2

(6) If 69.02log b , then 3 2logb is equal to

…………..

(a) 2.07

(b) 0.23

(c) 1.2040

Section-B

Show your solution step by step in the


1) Simplify the following:

i) xx 2315 33

ii)

2

323

132

cba

cba

iii) )1()1( xxxx eeee

iv) 8

5354 45

x

exex xx

29

2) Solve the following for x

i) xx 2515 22

ii) 8)12( 3 x

iii) 927 1 x

iv) 222 1010

2 xx

3) Graph ]3,3[; xey

4) Simplify the following expression using the

laws of logarithm

i) 1000log10

ii) 5log1010

30

iii) )1(log3 xee

5) Write each of the following expression using

single logarithm

i) yxz bbb logloglog

ii) yxz bbb log5loglog3

6) If 10.13log,69.02log bb and

61.15log b , find the value of the following

i) 3 2logb

ii) 27logb

iii) 3

5logb

iv) 15logb

7) Cholera, an intestinal disease, is caused by

a cholera bacterium that multiplies

exponentially by cell division as modeled by teNN 386.1

0

Where N is the number of bacteria present

after t hours and 0N is the number of bacteria

present at t =0. If we start with 1 bacterium,

how many bacteria will be present in

i) 5 hours?

ii) 12 hours? Compute the answers to 3 significant digits.

31

8) If a principal P is invested at an annual rate

r compounded n times a year, then the amount

A at the end of the t years is given by

nt

n

rPA

1 .


paying 9% interest per year compounded

daily (365 days).

i) Find the amount in the account after 5

years with no withdrawals.

ii) How much interest is earned over the

5 year period.

Compute the answer to the nearest Baiza

33

(UNIT-8) INVERSE RELATION BETWEEN EXPONENTIAL AND LOGARITHM FUNCTIONS

8.1 INTER-CONVERSION OF EXPONENTIAL AND LOGARITHM FUNCTIONS

The exponential function xay is a One-to-

One function and hence its inverse exists. The

inverse of exponential function is logarithmic

function. The relation between these two

functions is given by

y

a axxy log

Example: xyx y2log2

or yxy x2log2

As xyandy x

2log2 are inverse to each

other, the graphs are symmetrical about the

line xy .

Example 1: Find x, a or y as indicated:

i) Find y : 8log4y

ii) Find x : 2log3 x

iii) Find a : 31000log a

Solution:

i) 8log4y

84 y

32 2)2( y

32 y

2

3y

ii) 2log3 x

9

1

3

13

2

2 x

iii) 31000log a

10003 a

3

1

)1000(a

10a

Class Activity

1) Find y : 8log2

1y

2) Find x : 2log5 x

3) Find a : 5.08log a

34

8.2 EXPONENTIAL AND LOGARITHMIC EQUATIONS

42 53 x is example of exponential equation

and 1log)3log( xx is example of

logarithmic equation.

Example 1: Solve 52 23 xto 2 decimal

places.

Solution: 52 23 x

Taking log on both the sides, we get

5log2log 23 x

5log2log)23( x

2log

5log)23( x

44.1x

Example 2: Solve 1log)3log( xx

Solution: 1log)3log( xx

1)]3(log[ xx

110)3( xx

01032 xx

0)2)(5( xx

2or5 xx

Since log of negative value is not defined so

2x

Example 3: Solve 3)]4)(73[(log2 xx

Solution: 3)]4)(73[(log2 xx32)4)(73( xx

828193 2 xx

020193 2 xx

0)5)(43( xx

5or3

4 xx

Example 4: Solve 2ln)3ln(ln ln xe x

Solution: 2ln)3ln(ln ln xe x

2ln)3ln(ln xx

2ln3

ln x

x

23

x

x

)3(2 xx

62 xx

6x

Example 5: Solve 22 ln)(ln xx

Solution: 22 ln)(ln xx

xx ln2)(ln 2

0)2(lnln xx

02lnor0ln xx

10 ex or 2ex

Class Activity

Solve the following up to 2 decimal places:

1) x4002.12

2) 735 21 x

3) )3log(2log5loglog xx

4) )2ln()12ln(ln xxx

log 53 2

log 2x

35

WORKSHEET-8

Section-A


questions.

(1) If 2100log a , then ‘a’ is equal to ……..

(a) 100

(b) 20

(c) 10

(2) If 3log5 x , then ‘x’ is equal to ……..

(a) 125

1

(b) 125

1

(c) 15

(3) If 16log4y , then ‘y’ is equal to ……..

(a) 4

(b) 2

(c) 12

Section-B



1) Find x, y or a as indicated in the following:

i) 2log5 x

ii) 31000log a

iii) 27log9y

2) Solve the following:

i) 2log3)5(log 1010 x

ii) )2(log2)22(log 2 xxx bb

iii) xx log2)10log(

36

iv) 14lnln x

v) 2ln8ln x

3) Solve the following:

i) 7.4310 52 x

ii) 62.931 xe

4) A certain amount of money P (principal) is

invested at an annual rate r compounded n

times a year. The amount of money A in the

account after t years, assuming no

withdrawals, is given by

nt

n

rPA

1

.

How many years to the nearest year will it take

money to double if it is invested at 6%

compounded annually (once in year).

Solution:

37

(UNIT-9) POLAR COORDINATES

9.1 POLAR COORDINATE SYSTEM-GRAPHICAL REPRESENTATION

Cartesian coordinate: (x, y) coordinates are

called Cartesian or Rectangular coordinates of

point P.

To reach the point P(x, y)

Start from Origin (0, 0)

Move ’x’-units horizontally

Move ‘y’-units vertically

The joining units of both x and y meet

at a point P(x, y).

Polar coordinates:

P ),( r is called polar coordinates, where r is

called radius vector and is called the polar

angle.

The point O is called the pole.

Move ‘r’ units along the line making an

angle '' with positive x-axis and we

can see the point ),( rP .

Terminology for Polar coordinates

Example: Plot the point

4,5

First find the angle 4

and then move 5 units

along the terminal side

P(x, y)

x

y

O

(0,0)

38

Some important facts to remember

Measuring angles

Positive: It is measured in the

counterclockwise direction from the polar axis.

Negative: It is measured in the clockwise

direction from the polar axis.

Example: Write down the polar coordinates

of the points A and B in the given Polar

coordinate plane.

Answer a) The Polar coordinates of ‘A’ are

(8, 85).

b) The Polar coordinates of ‘B’ are (4, -650).

Class Activity

1) Write down the polar coordinates of the

points C and D in the given Polar coordinate

plane.

2) Write down the Polar coordinates of the

points A, B and C in the given Polar

coordinate plane.

Ans:

(i)

Ans:

(ii)

Ans:

39

9.2 CONVERSION FROM CARTESIAN TO POLAR COORDINATES

To Convert Cartesian coordinates P(x, y) to

Polar coordinates ),( rP

For ‘r’ by Pythagoras theorem 222 yxr

This gives 22 yxr

For ‘ ’ using trigonometry we have

x

ytan

This gives

x

y1tan where can be in

radians or degrees.

Note: (1) As inverse tangents only return

values in the range

2,

2

, check the

quadrant the Cartesian coordinates belongs to.

If the point lies in second quadrant or third

quadrant then the polar angle can be taken as

)( .

(2) If the Cartesian coordinates lies on y-axis,

then 𝑥 = 0. In this case, if calculator is used

x

y1tan gives ERROR. Polar angle in

such cases can be taken as

2

if y coordinate is positive and

2

if y coordinate is negative.

Alternative method to calculate depending

on the Cartesian coordinate lies on the

quadrant:

Step1: Calculate : where

x

y1tan

Step 2:

(i) If (x, y) lies in the first (I) Quadrant

(ii) If (x, y) lies in the second (II) Quadrant

or 180

(iii) If (x, y) lies in the third (III) Quadrant

)( or )180(

(iv) If (x, y) lies in the fourth (IV) Quadrant

Example1: Convert the Rectangular

(Cartesian) coordinate system (3, 2) to the Polar

coordinate in terms of radians (up to 2

decimals).

Solution: Here x = 3 and y = 2

For ‘r’ by Pythagoras theorem 222 yxr or

22 yxr = 1323 22

and

x

y1tan =

3

2tan 1 = 0.59 radians

Polar coordinates of (3, 2) are 59.0,13

I II

III IV

40

Example 2: Convert the Cartesian coordinates

(1, –1) to Polar coordinates in terms of degrees.

Solution: Here x = 1 and y = –1

For ‘r’ by Pythagoras theorem 222 yxr or

22 yxr = 2)1(1 22

and

x

y1tan =

1

1tan 1 = 45

Polar coordinates of (1, –1) are 45,2

Class Activity

1) Convert the following Cartesian coordinates

P(x, y) to Polar coordinates ),( rP in terms

of degrees

(i) 3,1

(ii) (-5, -5)


P(x, y) to Polar coordinates ),( rP in terms

of radians.

(i) (4, 3)

(ii) (0, -5)

(iii) 3,1

41

9.3 CONVERSION FROM POLAR TO CARTESIAN COORDINATES

To Convert Polar coordinates ),( rP to

Cartesian coordinates P(x, y)

Using trigonometric ratios we get

r

ysin or sinry

r

xcos or cosrx

So, ),( rP = P(x, y) = )sin,cos( rrP

Example1: Convert the polar coordinates (2,

π/3) to Cartesian coordinates P(x, y)

Solution: Here r = 2 and 3

so cosrx = 3

cos2

=

2

12 =1

and sinry = 3

sin2

=

2

32 = 3

Thus (2, π/3) = )3,1( in Cartesian coordinates.

Example 2: Convert the Polar coordinate

system point P )45,2( to the Cartesian

coordinate system point P(x, y)

Solution: Here r = 2 and 45

so cosrx = 45cos2 = 22

12

and sinry = 45sin2 = 22

12

Class Activity

1) Convert the following Polar

coordinates ),( rP to Cartesian

coordinates P(x, y)

(i) 90,5

(ii) 270,7

(iii)

8

7,2

(iv)

7

3,3

42

WORKSHEET 9

Section-A

Circle the correct answer in the following questions.

1) In the following figure, '' represents

(a) Pole

(b) Polar Angle

(c) Polar Axis

2) The polar coordinates of A in the following figure

is

(a) 045,4

(b) 045,4

(c) 045,4

3) If OX is the polar axis then the polar coordinates

of ‘A’ in the following figure are …..............

(a) 7,65

(b) 65,7

(c) 65,9

4) The Cartesian coordinates (-5, -5) lies in

quadrant….

(a) I

(b) II

(c) IV

5) The polar coordinates for the point (0, -2) is …..

(a) ),2(

(b)

2,2

(c)

2,2

4

- 450

X

A

O

A

O 2

7

65o

X

43

6) The angle 2

radians in degree is ……………

(a) 180

(b) 90

(c) 1.57

7) In the following figure the polar coordinates of

the point ‘A’ are …...

(a)

3,3

(b)

3,4

(c)

3,

3

8) In the following figure the polar coordinates

of the point ‘A’ are …...

(a) 0,90

(b)

2,90

(c)

2,90

44

Section-B

Show your solution step by step in the following questions.

Give the answer up to 2 decimals in the following questions.

1) Convert the following Cartesian

coordinates P(x, y) to Polar coordinates

),( rP in terms of degrees.

i) 1,2

Solution:

ii) 4,3

Solution:

iii) 5,0

Solution:

45


P(x, y) to Polar coordinates P ),( r in terms of

radians.

i) 3,2

Solution:

ii) 4,1

Solution:

iii) 7,0

Solution:

46

3) Convert the following Polar coordinates P

),( r in terms of Cartesian coordinates P(x,

y)

i) 135,4

Solution:

ii) 90,1

Solution:

iii)

3,4

Solution:

iv)

3

2,2

Solution:

47

(UNIT-10) NUMBER SYSTEMS-CONVERSIONS

10.1 CONVERTING DIFFERENT NUMBER SYSTEMS TO DECIMAL (BASE 10)

Introduction: The number 25 can be

represented in different number system as

following: 2510 = 110012 = 318 =1916

Table for number system

Number

System

Base Symbols used Examples

Decimal 10 0, 1, 2, 3, 4, 5,

6, 7, 8, 9 10)2548(

Binary 2 0,1 2)10101(

Octal 8 0, 1, 2, 3, 4, 5,

6, 7 8)5421(

Hexadecimal 16 0, 1, 2, 3, 4, 5,

6, 7, 8, 9,

A for 10,

B for 11,

C for 12 ,

D for 13,

E for 14 and

F for 15

16)1CA2B(

Or

H)1CA2B(

Example: The place values for the integer

559 in the decimal system is

012 109105105559

Note: The digit 5 in position 1 has the value

50 but the same digit in position 2 has the

value 500.

Converting Binary to Decimal

Method:

1.Multiply each bit by n2 , where ‘n’ is the

weight of the bit.

2.Where 0n , starts from right of the

binary digit.

3. Add the results.

Examples: Convert the following, binary to

decimal system.

1) 2)11001(

Solution: 01234

2 2120202121)11001(

10)25(

2) 2)1001101(

Solution: 3456

2 21202021)1001101( 012 212021

10)77(

Class Activity 1

1) Convert the following binary to decimal

(i) 2)1101(

(ii) 2)100001(

48

Converting Octal to Decimal

Method:

1. Multiply each bit by n8 , where ‘n’ is the

weight of the bit.

2. Where 0n , starts from right of the octal

digit.

3. Add the results.

Example: Convert octal 8)352( to decimal

number.

Solution:

10

012

8 )234(828583)352(

Class Activity 2

1) Convert the following octal to decimal:

(i) 8(723)

(ii) 8(514)

Converting Hexadecimal to Decimal

Method:

1. Multiply each hexadecimal digit by n16 ,

where ‘n’ is the weight of the bit.

2. Where 0n starts from, right of the

hexadecimal digit.

3. Add the results.

Example: Convert hexadecimal 16)( BCA to

decimal number.

Solution:

10

012

16

)2763(

161116121610)(

BCA

Class Activity 3

1) Convert the following hexadecimal to

decimal:

(i) 16( )ABC

(ii) 16( 12 )A B

49

10.2 CONVERTING DECIMAL (BASE 10) TO OTHER NUMBER SYSTEMS

Converting Decimal to Binary

Method:

1.Divide progressively the given decimal

number by 2.

2.Write down the remainder after each

division.

3.The remainders taken in the reverse order

to form a Binary number.

Example: Convert decimal number 10)13( to

binary number.

Solution:

6213 with remainder 1



021 with remainder 1 Read

Upward

So, 210 )1011()13(

Class Activity 1

1) Convert the following decimal to binary:

10)23(

2) Convert the following decimal to binary:

10)501(

50

Converting Decimal to Octal Conversion

Method:

1. Divide progressively the given decimal

number by 8.

2. Write down the remainder after each

division.

3. The remainders taken in the reverse order

to form an octal number.

Example: Convert decimal 10)3485( to octal

number.

Solution:




086 with remainder 6 Read

Upward

810 )6635()3485(

Class Activity 2

1) Convert 10)23( to the octal number system.

2) Convert the following decimal to octal:

10)932(

51

Converting Decimal to Hexadecimal

Method:

1. Divide progressively the given decimal

number by 16.

2. Write down the remainder after each

division.

3. The remainders taken in the reverse order

to form a hexadecimal number.

Example: Convert decimal 10)3485( to

hexadecimal number.

Solution:

217163485 with remainder 13=D


01613 with remainder 13=D

Read Upward

1610 )9()3485( DD

Class Activity 3

1) Convert 10)43( to the hexadecimal number

system.

2) Convert the following decimal to

hexadecimal 10)935(

52

10.3 CONVERTING BINARY TO OCTAL OR HEXADECIMAL SYSTEM

Converting Binary to Octal

Method

1. As 823 , group bits in threes, starting on

right

2. Convert to octal digits

Example : Convert binary 21)(101101011 to

octal number.

82 )1327(1)(101101011

Note: -The inter-conversions can also be

done as following:

Step1: Convert binary base to decimal.

Step2: Convert decimal to the octal base.

Class Activity 1

1) Convert the following binary to octal:

a) 2)1010101(

b) (10111100)2

c) (1011110100)2

53

Converting Binary to Hexadecimal

Method

1. As 1624 , group bits in fours, starting

on right

2. Convert to hexadecimal digits

Example : Convert binary 21)(101101011

to hexadecimal number.

162 )72()1011010111( D


done as following:

Step1: Convert binary base to decimal.

Step2: Convert decimal to the hexadecimal

base.

Class Activity 2

1) Convert the following binary to

hexadecimal:

a) 2)1010101(

b) (11001101)2

c) (10000101010100)2

54

10.4 CONVERTING OCTAL OR HEXADECIMAL TO BINARY SYSTEM

Converting Octal to Binary

Method

1. Convert each octal digit to a 3-digit

Example : Convert 8)507( to binary number

28 )111000101()507(


done as following:

Step1: Convert octal base to decimal.

Step2: Convert decimal to the binary base.

Class Activity 1

1) Convert the following octal to binary:

a) 8)73(

b) 8)723(

c) (1416)8

55

Converting Hexadecimal to Binary

Method:

1. Convert each given hexadecimal digit to a

4-bit equivalent binary representation.

Example: Convert 16)10( BE to binary

number.

216 )1111100001000010()10( BE


done as following:

Step1: Convert hexadecimal base to decimal.

Step2: Convert decimal to the binary base.

Class Activity 2

1) Convert the following hexadecimal to

binary:

a) 16)45(

b) (3𝐹𝐷)16

c) (𝐴𝐵𝐶)16

57

10.5 INTER-CONVERSION OF OCTAL AND HEXADECIMAL

Converting Octal to Hexadecimal

Method

1. Use binary as an intermediary.

Example: Convert octal 8)6071( to

hexadecimal

168 )63()6071( C


done as following:

Step1: Convert octal base to decimal.

Step2: Convert decimal to the hexadecimal

base.

Class Activity 1

1) Convert the following octal to

hexadecimal:

a) 8)54(

b) (736)8

c) (1077)8

58

Converting Hexadecimal to Octal

Method

1. Use binary as an intermediary.

Example : Convert hexadecimal 16)01( FC

to octal.


done as following:

Step1: Convert hexadecimal base to decimal

Step2: Convert decimal to the octal base

Example: Convert hexadecimal 16)01( FC

to octal.

Solution: Step 1: Convert given base to

decimal

10

0123

16

)7183(

16151601612161)01(

FC

Step2: Convert decimal to the required base






816 )16017()01( FC

Class Activity 2

1) Convert the following hexadecimal to octal

16)(ABC

59

2) Convert the following hexadecimal to octal

a) 16)5( AB

b) 16(9 7)B

c) (𝐴𝐵𝐶)16

60

10.6 ADDITION AND SUBTRACTION OF BINARY NUMBERS

Binary Addition

Addition table

A B A+B Carry

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

Else, Remember in Binary:

0+0 = 0

0+1 = 1

1+0 = 1

1+1= 10

1+1+1=11

Example: Add 2)1110( and 2)1111(

1 1 1 0

+ 1 1 1 1

-------------

1 1 1 0 1

-------------

Class Activity 1

Simplify the following Binary numbers:

a. 1 1 0 1 1 b. 1 1 1

+ 0 0 0 1 0 + 1 0 1

------------------ --------------

------------------ -------------

c. 1 0 1 0 0 0 1 d. 1 0 1 0 1 1 1 1

+ 1 1 1 1 + 1 1 1

---------------------- ----------------------

--------------------- ---------------------

Binary Subtraction

Subtraction table

Example: Simplify the following:

a. (1100)2 ̶ (110)2

b. 22 0111100110011100

a. 1 1 0 0 b. 1 0 0 1 1 1 0 0

– 0 1 1 0 – 0 1 1 1 1 0 0 1

------------------ -------------------------

0 1 1 0 0 0 1 0 0 0 1 1

------------------ ------------------------

Class Activity 2

Simplify the following:

1) 22 )10()11011(

2) 22 )101()10111(

3) 22 )1111()1010001(

4) 22 )1000111()10101111(

61

WORKSHEET-10

Section-A


questions.

(1) The number system with base 16 is

called……

(a) Hexadecimal

(b) Octal

(c) Binary

(2) The octal number system has base……

(a) 2

(b) 8

(c) 16

(3) The total symbols used in the binary

number system are ……

(a) 0

(b) 1

(c) 2

(4) The binary code for (15)10 is ……

(a) (1111)2

(b) (70)2

(c) (F)2

(5) The value of (1101)2 + (111)2 is ……

(a) (1212)2

(b) (10101)2

(c) (10100)2

Section-B



1) Simplify the following Binary numbers

i) Add 2)101011( and 2)11011(

Solution:

ii) Subtract 2)10111( from 2)101001(

Solution:

2) Convert (33)10 to the following number

system:

i) Binary

ii) Octal

iii) Hexadecimal

62

3) Convert (1110101)2 to the following

numbers:

i) Decimal

ii) Octal

iii) Hexadecimal

4) Convert (703)8 to the following number

system:

i) Decimal

ii) Binary

iii) Hexadecimal

63

5) Convert 16)1( AF to the following number

system:

i) Decimal

ii) Binary

iii) Octal

64

REFERENCES

[1] Raymond A. Barnett, Michael R. Zigler and Karl E. Byleen, 7th edition, College Algebra

with Trigonometry, McGraw Hill.

[2] Margaret Lial John Hornsby, David I. Schneider Callie Daniels, College Algebra and

Trigonometry, Pearson, 5th Edition.

[3] John Bird, 6th Edition, Basic Engineering Mathematics, Routledge (Taylor and Francis Group).

[4] Ronald E Walpole, 3rd Edition, Introduction to Statistics, Pearson Prentice Hall.

[5] Websites/ HTML:

http://www.statisticshowto.com

http://math.tutorvista.com

www.mathsisfun.com

www.statcan.gc.ca

people.umass.edu/biep540w/pdf/Grouped%20Data%20Calculation.pdf

www.lboro.ac.uk/media/wwwlboroacuk/content/mlsc/.../var_stand_deviat_group.pdf

web.thu.edu.tw/wenwei/www/Courses/statistics/ch2.2.pdf

http://www.statisticshowto.com/

http://math.tutorvista.com/

http://www.mathsisfun.com/

http://www.statcan.gc.ca/

http://www.lboro.ac.uk/media/wwwlboroacuk/content/mlsc/.../var_stand_deviat_group.pdf

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