under standing with meths

7/30/2019 UNDER STANDING WITH METHS

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Essential MathematicsFor Decision Making

By,

Yaseen Ahmed MeenaiFaculty, DMS-FCS-IBA


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Decision Making Skills needs Maths


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Where is The Decision System?

It is a built-in system present inside the frontal Lobe of

our Brain

We mammalians are social creatures and we owe this

system mostly to our frontal lobes

We would not want to act socially inappropriate; thanksto our frontal lobes. Such as loosing our empathy

Frontal Lobe of the brain completes in 20s of an average

human.

So what can we see in a patient when the frontal lobe, apart of it per say, is injured

1- Attention deficits 2-Disinhibition;social inappropriateness

http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-

another-person-in-the-same-body/
http://www.ect.org/effects/lobe.htmlhttp://en.wikipedia.org/wiki/Disinhibitionhttp://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromes-another-person-in-the-same-body/http://en.wikipedia.org/wiki/Disinhibitionhttp://www.ect.org/effects/lobe.htmlhttp://www.ect.org/effects/lobe.htmlhttp://www.ect.org/effects/lobe.html


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How we can accelerate the frontal

Lobe Usage??

We can enhance our Decision System by

means of exercising the Logical Thinking

practices.

There is a difference b/w Thinking and

LogicalThinking

By Thinking Logically; we can minimize the

TIME and EFFORT for completion of any

assigned task.


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Logical Thinking motivation

Drawing a FISH can help us understanding thelogical thinking:

Now, try to re-draw the same fish, but withoutlifting your pen once it touches the paper andwithout striking out any of your drew line.


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Logical Thinking Motivation from

WEB

There are so many websites containing similar

stuff. By writing LogicalProblems in Google, we

can find such sites.

One of the useful link is

http://www.lumosity.com

We can access several Math & Logical Problems

on www.indiabix.com Or even One can play simple games for Charging

Brain: http://www.proprofs.com/games/crazy-taxi/
http://www.lumosity.com/http://www.indiabix.com/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.proprofs.com/games/crazy-taxi/http://www.indiabix.com/http://www.indiabix.com/http://www.indiabix.com/http://www.indiabix.com/http://www.indiabix.com/http://www.lumosity.com/http://www.lumosity.com/http://www.lumosity.com/http://www.lumosity.com/http://www.lumosity.com/http://www.lumosity.com/http://www.lumosity.com/


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Logical Thinking through the Venn

diagram

A Venn diagram is a rectangular area showing

the Sample Space & having some circles inside

(usually overlapped) which are showing the

Events.

S={a,b,c,d,.,n}

A={a,b,c,f,g,h}

B={c,d,e,g,h,i}

C={f,g,h,I,j,k}

a,bc d,e

f

g,h

i

J,kl,m,n

A B

C

S


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Logical Operations ( ) A Union ( ) is consideration of all elements by

showing duplicates; once.

AB = {a,b,c,d,e,f,g,h,i} An Intersection () is the

common elements collection

A B = {c,g,h}

Whats A ??A = not(A) = S A

A= {d,e,i,j,k,l,m,n}

a,bc d,e

f

g,h

i

J,kl,m,n

A B

C

S


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Shading the Venn Diagram

For AB, it should be

A B

C

S

For AB, it should beFor A , it should beFor AB, it should beFor AB , it should be

The Demorgans Law


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Background of Mathematics

In stone ages, Mankind used to count objectsby repeating the same accordingly. i.e.

If anyone had seen 4 birds, he would have

shared his knowledge to someone else like;I Saw Bird Bird Bird and Bird

But that method definitely fails if anyone had

seen 100 birds.. !!!!

That problem led mankind to discover

numbers like 0,1,2,3,9.


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Background of Mathematics

Soon after discovering numbers, data

collection was started.

After plotting the data, humans started

observing trends and hence Mathematical

Functions were derived.

Time Population

Year1 2130

Year2 4230

Year3 14500

Year4 258000

10000

20000

30000

Year1 Year2 Year3 Year4

Population

Population


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Mathematical Functions The whole Science of Mathematics based on 3

types of Functions:1) Algebraic functions

2) Exponential or Log Fn(s)

3) Trigonometric Functions

Polynomial is an algebraic function which

can attain any shape.

-100

-50

0

50

100

-10 -5 0 5 10

f(x)=quad

f(x)=quad

f(x)=cubic

0

10

20

30

40

0 2 4

f(x)=log(x)

f(x)=exp(x)

-2

-1

0

1

2

-10 -5 0 5 10

f(x)=sin(x)


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Types of Numbers
http://www.google.com.pk/imgres?imgurl=http://delightfulness.files.wordpress.com/2011/10/confused-face.jpg&imgrefurl=http://delightfulness.wordpress.com/2011/10/20/cantaloupe-antelopes/&usg=__3siP2aWjcIk4PLsjtKdXzt6niGs=&h=512&w=385&sz=67&hl=en&start=3&zoom=1&tbnid=F2-TLeCqkNb1PM:&tbnh=131&tbnw=99&ei=VrLyT_y4G8T3rQfyt92vCQ&prev=/search?q=Confused+face&hl=en&site=imghp&tbm=isch&itbs=1http://4.bp.blogspot.com/-uycSXyw0-hw/Tdyj3wX-7pI/AAAAAAAAAEU/9HenNvPEVuE/s1600/types_of_numbers(classification).jpg


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Types of Numbers

Rational numbers. Rational numbers are numbers thatact rationally! The decimal ends somewhere, or it has arepeating pattern to it. 2, 3.4, 5.77623, and 4.5, or3.164164164 or 0.666666 etc.

Irrational numbers. Irrational numbers are theopposite of rational numbers. An irrational numbercannot be written as a fraction, and decimal values forirrationals never end and never have a nice pattern to

them. For example, pi, with its never-ending decimalplaces, is irrational. =3.14159265358979 or 2 is alsoan irrational number.


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Types of Numbers

Even numbers. An even number is one that

divides evenly by two, such as 2, 4, 6, 8, 10.

Odd numbers. An odd number is one that

does not divide evenly by two, such as 1, 3, 5,

7, 9,..

Prime numbers. Its a number which can

either be divided by itself or 1, such as 1, 3, 5,

7, 11, 13, 17


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Warm up with Math Operations

Suppose we have a task to solve the following

expression:

2+2*2/2-2=???

Dont you think that this would be better

using parenthesis ( )?? Look at it again.

2+(2*2)/2-2 =2+4/2-2 =2+2-2 =2 OR

2+2(2/2)-2 =2+2*1-2 =2+2-2 =2


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Warm up with Math Operations Here is another puzzle.

Put the arithmetic notations like ( + , - , x, ) and by adjusting

parenthesis to prove the following:

(5 _ 5 _ 5 _ 5) = 30 ???

Here is the solution:

(5+5/5) x 5 = (5+1) x 5 = 6 x 5 = 30

Remember if we wrongly place those parenthesis:

5+(5/5x5) = 5+(1x5) = 5+5 = 10

Or even without parenthesis:

5+5/5x5 = 5+1x5 =5+5 = 10

Its is due to the BODMAS rule (Brackets OrdersDivision or Multiplication, Addition or Subtraction)

http://www.mathsisfun.com/operation-order-bodmas.html


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BODMAS (Bracket, Order, Division or

Multiplication, Addition or Subtraction)

If we have following expression:

30 5 x 3 = ????

Since Division and Multiplication rank equally

then In this case we always be CORRECT if weproceed from Left to Right i.e.

30 5 x 3 = 6 x 3 = 18

Solution from Right to Left will give you theWrong answer.

30 5 x 3 = 30 15 = 2 (Wrong)


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Example: How do you work out 3 + 6 2 ?

Multiplication before Addition:

First 6 2 = 12, then 3 + 12 = 15

Example: How do you work out (3 + 6) 2 ? Brackets first:

First (3 + 6) = 9, then 9 2 = 18

Example: How do you work out 12 / 6 3 / 2 ?

Multiplication and Division rank equally, so justgo left to right:

First 12 / 6 = 2, then 2 3 = 6, then 6 / 2 = 3


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Oh, yes, and what about 7 + (6 52 + 3) ?

Start inside Brackets, and then use "Orders" First

7 + (6 5

2

+ 3) =7 + (6 25 + 3)Then Multiply 7 + (150 + 3)

Then Add 7 + (153)

7 + 153Brackets completed, last operation is,

Add 160 DONE !


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What is the value of 3 + 6 3 2 ?

A. 7 B. 6 C. 4 D. 1.5

What is the value of 5 3 - 12 4 + 8A. 3 B. 4 C. 14 D. 20

What is the value of 5 4 - 2 3 + 16 4

A. 10 B. 11 C. 18 D. 34What is the value of 30 - (5 23 - 15)?

A. -25 B. 5 C. 15 D. -15


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Equalities and Inequalities

We have certain Symbols to compare two numbers:

a = b means a is equals to b ____a=b_____->

a > b means a is greater then b ____b___a___->

a < b means a is less than b ____a___b___->

a b means a is greater than or equals to b

a b means a is less than or equals to b

a b means a is equals to b

a b means a is approximately equals to b


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Terms, Expressions and Equations

A Mathematical Term could be a number or a general

value. For e.g. 2 is a term, 2a is a term as well (where

a is any unknown number)

A group of terms can make an expression, for e.g.2a+3, 3a-2b etc. are examples of expressions.

The difference between Mathematical Expression

and a Mathematical Equation is the equals to sign

which is absent in Expressions and present in

Equations.

For e.g., 2a+3b is an expression and 2a+3b=5 is an

equation.


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Solving Expressions (pg. 37)

Evaluate the following if, p=2 , q=3 and r= - 4 :

(i) p q + r

Ans. Is 2

(ii)p q / r

Ans. Is -3/2

(iii) (p + q) / r

Ans. Is -5/4

(iv)p + q/r

Ans. Is +5/4

(v) (p q)

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

(1/p + 1/q)

Ans. is -6/5

(vi) (1/p 1/r )

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

(p + r)

Ans. is -3/8


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Problems with Inequalities (pg. 35)

Determine values ofx which satisfy the following:

(i) x+1 > 0

Ans. Is x > -1

(i) X - 2 > 0

(ii) Ans. Is x > +2

(iii) 1 2x 0Sol

1 +2x => 1/2 xx+1/2

(iv) (x+1)(x-2) > 0 ??

-1 0 +1 ..

0 +1 +2 ..

0 +1 +2 ..


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(iv) (x+1)(x-2) > 0

Sol.

x2-x-2 > 0

Therefore,

x < -1 or x > +2

-4

-2

0

2

4

6

8

10

12

14

16

-4 -2 0 2 4 6

f(x)

x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

f(x) 4 1.75 0 -1.25 -2 -2.25 -2 -1.25 0 1.75 4


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(iv) (x + 1)

-------- < 0

(x2)

Since we know,

+ve / +ve > 0

-ve / -ve > 0

+ve / -ve < 0

-ve / +ve < 0

The best matching possibility is:

+ve / -ve < 0

Therefore,x+1 > 0 and x-2 < 0

x > -1 and x < +2

0 +1 +2 ..-1


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Problems with Modulus (pg. 36)

What is Modulus?? For e.g. |A| = +A

|-2|= + 2

|+2|= - 2

Wrong..

|+2|is also +2

So, where is its practical implementation??

So while computing distance or the error margin, wemostly ignore the sign.

For e.g. |Theoretical Value Empirical Value | = error


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Evaluate the following when x = - 2, 0 , +1/2 and +2:

(i) | x + 1 |

Ans is 1 , 1 , 3/2 , 3

(ii) | x - 1 |

Ans is 3 , 1 , 1/2 , 1

(iii) | 2 - x |

Ans is 4 , 2 , 3/2 , 0

(iv) || x + 1 | - |x1|| + |2 x|

Ans is 6, 2 , 5/2, 2


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Evaluate the following when x = - 2, 0 , +1/2 and +2:

(v) || x + 1 | + |x1|| - |2 x|

Sol.

For x= - 2,

|| - 2 + 1 | + |- 2 1|| - |2 ( - 2)|

|| - 1| + |- 3|| - |2 + 2|

| + 1 + 3| - |4|

| +4| - |4| = +4 4 = 0

Ans are 0, 0 , 1/2, 4


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Inequalities with Modulus

There are three possible cases in this situation

Case No. 1:

For e.g. if |x+a| > 3 is required to be solved

In case of the greater-than sign, we always have

following two possibilities;

(x+a) > + 3 and (x+a) < -3

Therefore the two solutions will be:x > 3 a and x < -3 a

Outer range

3 - a ..-3 - a


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Problems with inequalities and

Modulus (pg. 36)

5. Determine all values ofxsuch that:

(v) |3x - 1| 2This is the 1stcase, we will have following two cases;

(3x - 1) + 2 and (3x - 1) -2Therefore the two solutions will be:

3x2 + 1 and 3x -2 + 1x 3/3 = 1 and x -1 / 3

Outer range

+1 ..-1/3


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Inequalities with Modulus

Case No. 2:

For e.g. if |x+a| < 3 is required to be solved

In the case of less-than sign, it will create a bound;

- 3 < (x+a) < + 3

Therefore the two solutions will be:

- 3 < x + a and x + a < + 3

- 3 a < x and x < + 3 a

x > - 3 a and x < + 3 a

Inner range

3 - a ..-3 - a


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Modulus (pg. 36)


(ii) |x + 1| 2This is the 2ndcase, it will create a bound;

- 2 (x+1) + 2Therefore the two solutions will be:

- 2 x + 1 and x + 1 + 2- 2 1 x and x+ 2 1

x - 3 and x + 1Inner range

+1 ..-3


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Modulus (pg. 36)


(iv) |2x + 1| = 2

This is the 3rd case, we will have following two

possibilities;

2x + 1 = -2 or 2x + 1 = + 2

2x = - 2 - 1 and 2x = + 2 1

2x = - 3 and 2x = + 1x = - 3/2 and x = + 1/2

two exact values

+1/2 ..-3/2


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A Factorial !

What is a Factorial sign in Mathematics??

If an expression is written with a Factorial sign like 4!,

The solution will be 4 x 3 x 2 x 1 = 24

But where is the application of Factorial in our real

life??

Suppose if we have 3 persons A, B and C and we have to

stand them in a line.

So the possible ways will be,A, B C: A, C, B : B, A, C: B, C, A : C, A, B and C, B, A

3 ! = 3 x 2 x 1 = 6 ways


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Expression with Factorial (pg. 36)

6. Evaluate the following:(i) 4 ! / 2 !

(4 x 3 x 2 x 1 ) / ( 2 x 1) = 4 x 3 = 12 Ans.

(ii) 6! 4 ! / (5! 3!)(6 x 5! 4 x 3!) / ( 5! X 3!) = 6 x 4 = 24 Ans.

(iii) (5! + 4!) / 3!

By breaking the LCM:

5!/3! + 4!/3! = 5 x 4 + 4 = 24 Ans.

(vi) (n + 1)! / (n 1)! = ???? :-S


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Factorial (A Class Activity)

Once after your CLASS TEACHER says START thenyou all have to Change your Seats.

Do it as soon as you can

Compute the TOTAL Possible arrangements.

N! or NPr N! x time in seconds = Total seconds

(Total Seconds)/60 = Total minutes

(Total Minutes)/60 = Total Hours (Total Hours)/24 = Total days

(Total days)/365 = Total Years required

h i l di ( )


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Mathematical Indices (pg. 37)

13. Express the following numbers in the form 2p x 3q:

(i) 62 x 8

---------

123

Sol.

(2 x 3)2 x 2 x 2 x 2

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

(2 x 2 x 3)3

22 x 32 x 23

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

23 x 23 x 33

25 x 32

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

26 x 33

25 x 32x 2-6 x 3-3

2-1 x 3-1 Answer

h i l di ( 3 )


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Mathematical Indices (pg. 37)

13. Express the following numbers in the form 2p x 3q:

(ii) 63 x 82

---------

12 4

(iii) 64 x 12

---------

4 5

(iv) 610x 49 x 12-6

14. Express the Cube root of

200 divided by thesquare root of 40 in the

form 2p x 5q

W d P bl (N b )


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Word Problems (Numbers)

Integers: All positive and negative whole numbers

including zero are integers:Sample Question1: The sum of three consecutive

integers is 18. What is the smallest of the number?

Sol.Ifxis the smallest integer, then next two consecutive

integers are x+1andx+2then the sum will be;

x+(x+1)+(x+2)=18

3x+3=18 => 3x=18-3 => 3x=15

x=15/3 => x=5 therefore, 5+6+7=18

W d P bl (N b )


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Word Problems (Numbers)

Even and Odd Integers: Any number which can be

divided by 2 is an Even number and which cannot be divided by2 is an Odd number. For e.g. 2,4,6,8, are Even numbers and

1,3,5,7,9. Are Odd numbers.

The difference b/w two even or two odd numers is always 2.

Sample Question2: The sum of three consecutive Even integersis 36. What is the smallest of the number?

Sol.

If x is the smallest Even integer, then next two consecutive

integers are x+2 and x+4 then the sum will be;x+(x+2)+(x+4)=36

3x+6=36 => 3x=30 => x=10. Therefore,

10+12+14=36

L i h ( 38)


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Logarithms (pg. 38)

What is the concept of Logarithms in

Mathematics?

Why Logarithms are necessary? What is the

significance of this concept in Mathematics?

In order to scale the magnitude, we usually uselog.

In order to solve expressions in which subject is in

the power; only log can help us.

If Exponential form is 102=100 then the log form

will be log10(100)=2

L ith


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Logarithms

Notation: Loga(b)=c

(where a is the base, b is any number and c is the answer) What is the concept & use of Logarithms?

If Exponential form is 102=100 then

The log form will be log10(100)=2 For e.g. we can solve x+3 = 5. i.e. x=2

But we cannot carried out the solution for 3x=5

without Log. Therefore by taking log on both

sides;

log(3x)=log(5) => xlog(3)=log(5)

x=log(5)/log(3) will be the answer.

Properties of Logarithms


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Properties of Logarithms

1) loga(a)=1 e.g. log10(10)=1

2) log(ax)=xlog(a)

e.g. log(100) = log(102)=2log(10)=2x1=2

3) log(ab)=log(a)+log(b)

e.g. log(100) = log(10x10)=log(10)+log(10)=1+1=2

4) log(a/b) = log(a)-log(b)

e.g. log(1)=log(10/10)=log(10) log(10) = 1 1 = 0

Questions of Logarithms


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17. Given that log(4.78)=0.6794, then Evaluate;

i) log(47.8)Sol.

Required value is xsuch that, log(47.8)=x

Therefore,According to the given value in the question:

Log(4.78)=0.6794 and we also knew that log(10)=1 so,

Log(47.8)=log(4.78 x10)



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Log(47.8)=log(4.78 x10)

According to the third property:log(ab)=log(a) + log(b)

So,

log(47.8) = log(4.78) + log(10)Therefore,

=0.6794 + 1

And,

log(47.8) =1.6794 Ans.



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17. Given that log(4.78)=0.6794, then Evaluate;

i) log(0.478)Sol.

Required value is xsuch that, log(0.478)=x

Therefore,According to the given value in the question:

Log(4.78)=0.6794 and we also knew that log(10)=1 so,

Log(0.478)=log(4.78 /10)



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Log(0.478)=log(4.78 /10)

According to the third property:log(a/b)=log(a) - log(b)

So,

log(0.478) = log(4.78) - log(10)Therefore,

=0.6794 - 1

And,log(0.478) = - 0.3206 OR we can also write the ans,

0.6794 1 =



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18. Given that log(6.85)=0.8357, then find the

numbers whose log to base 10 is;i) 1.8357

Sol.

Required value is xsuch that, log(x)=1.8357

Therefore,

Log(x)=1 + 0.8357

According to the given value:

0.8357 = log(6.85) and 1 = log(10) so,

log(x)=log(10) + log(6.85)



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log(x)=log(10) + log(6.85)

According to the 3rdProperty:Log(ab)=log(a) + log(b)

Therefore,

log(x)=log(10 x 6.85)log(x)=log(68.5)

Finally,

X=68.5 Ans.



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18. Given that log(6.85)=0.8357, then find the

numbers whose log to base 10 is;i) 3.8357

Sol.

Required value is xsuch that, log(x)=1.8357

Therefore,

Log(x)=3 + 0.8357 = 1+1+1+0.8357

According to the given value:

0.8357 = log(6.85) and 1 = log(10) so,

log(x)=log(10) + log(6.85)



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log(x)=log(10) + log(10)+ log(10)+ log(6.85)

According to the 3rd

Property:Log(ab)=log(a) + log(b)

Therefore,

log(x)=log(10 x10x10x6.85)log(x)=log(6850)

Finally,

X=6850 Ans.



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20. Given that log y = log x + where and areconstants and that y=10 when x=0.1 and y=100when x=1. Determine and.

Sol. According to the given information:

y=10 when x=0.1 we put these values in the eq.

log 10 = log 0.1 + 1 = log (1/10) + 1 = [log1 log10] +1 = [01] + 1 = - + ------ (i)



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Then the 2nd Information:

Y=100 when x=1log 100 = log 1 + 2 = (0) + 2 = or = 2

Putting the value ofin (i)

1 = - + ------ (i)1 = - + 2 = 2 1 or = 1 Ans.



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21. By taking log, find the smallest positive integern

for which 4n

< 5n-1

.Sol. Taking log on both sides,

log 4n < log 5n-1

n log4 < (n-1) log 5

n log4 < n log5 log5

log5 < n log5 n log4

log5 < n (log5 log4)

log5/(log5-log4) < n

Or

n > log5 / (log5 log4)

n > 7.212

Therefore,

n=8 Ans.



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22. A biological population initially ofsize 1000, doubles its size

everyday. The size of the population after n generation

times is N=103 2n.

Find using logarithms, the number of generations that must

elapse before the size exceeds (i) 105 (ii) 1010.

(i) Sol.

N > 105

103 2n > 105

2n > 105 / 103

2n

> 102

Now we will take log on both sideslog(2n) > log (102)

n log (2) > 2 or n > 2 / log (2) > 6.64 7



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23. The volume V of timber in a given tree increases by 5%

every year so that V = a (1.05) t where t is the time in years

andais the volume at time t=0.

i) Calculate the volume of the tree after 5 years?

Sol.

V = a (1.05) 5 = 1.28 a

ii) Find the time taken for the tree to double its volume.

Sol.

Initial volume of the tree V = a

So, doubles the volume means V = 2a Ora (1.05) t = 2 a

(1.05) t = 2 Now taking log on both sides and solve



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log (1.05) t = log 2

t log (1.05) = log 2

t = log 2 / log (1.05)

t = 14.2 years Ans.

Compound Interest in Financial Maths


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Following is the compound interest formula :

S = P ( 1 + r )t

Where,

S = Compounded Amount, P = Principal Amount

r = Rate of InterestNow reconsider question no. 22:

N = 103 2n Showing population which doublesits size everyday.

We can also re-write the same as: N = 1000 (1+1) n

Its the same formula S = P (1+r) n



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p

Now comparing the same compound interest formula

with the model given in question no. 23:The volume Vof timber in a given tree increases by

5% every year so that V = a (1.05)t

V = a (1 + 0.05) t

S = P (1 + r ) t

In short, the compound interest formula is also a

Mathematical Model which can be fitted in any growth

related phenomenon.



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Q g

24. In the study of hearing the loudness L is expressed in terms

of the intensityIby the equation:

L = 10 log (I / Io)

Where Io is approx. 10 -12 Watt/m2. Express I in terms of L

and determine Iwhen L=60.

Sol. We can use the property of Log here,

L log(10) = 10 log (I / Io)

log (10) L = log (I/Io)10

Taking Antilog on both sides,

10 L = (I/Io) 10Taking 10th Root (power 1/10)

on both sides,

[10 L ] 1/10 = [ (I/Io)10] 1/10

10 L/10 = I / Io

Therefore,

I = 10 L/10 IoFinally, L=60 and Io = 10

-12

I = 10 -6

Eulers number or Napiers Constant


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p

This is e http://en.wikipedia.org/wiki/E_(mathematical_constant)

e is said to be the lifes function

Whenever we have a life-time distribution or a natural growth

/ data is there, the model should be having e.

For e.g. The well known Normal Distribution based on this

constant. Gauss (The German Mathematician used it)

Following is the function of e:

e = (1 + x) 1/x but Lim x0

If we put x=0 then e = inf.

So we have to pokethe function:By putting x=0.001 (closest to zero) This will give us a result:

e = (1 + 0.001) ^ (1/0.001) = 2.719

Graphing Sense


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p g

Practical Object:

Plot the Graphs/Sketches of the Bivariate

data given in Q. 25 and Q. 27 respectively.

Do it as neat as possible and submit these

graphs by writing your name on the sheet.

Practice Questions (Maths)


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Q ( )

Determine the equation of straight line passing through

a point (4,-1) having slope=-3a. y= - 3x+11 b. y= 3x+11 c. y=2x-3 d. y=3x-2

The correct linear equation given that it is passing

through points (1,3) and (7,5) is;

a. x+3y= - 4 b. x-3y= - 8 c. 2x-3y=4 d. x-2y=5

Which of the following straight line has rising with finite +ve

slope?

a. b. c. d.



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Q ( )

The roots of the quadratic equation are;

a. (2,-4) b. (-4,2) c. (-4,-8) d. (4,2)

The Centigrade temperature oC and Fahrenheit temperature oF

are related by the equation 9C=5F-160, then the value of oC

when F=0 is,

a. 21.33 b. -40.00 c. -17.77 d. 32.00 An object projected vertically upwards at time t=0 with a

velocity of 14 meters/second reaches a height y in meters

given by, , when y=10, the value oft will be;

a. 10/7 b. 12/7 c. 5/9 d. -10/5

9)1( 2 xy

29.414 tty



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( )

The value of x in the equation is

a. b. c. d.

Given that , where and areconstants and that y=10whenx=0.1 and y=100whenx=1, then

the values of (,) will be;a. (1,2) b. (-1,2) c. (2,1) d. (-1,-1)

If y is a function of x in , then x in terms of y

will be ;

a. b. c. d.

c

b2

)(

)(

ca

bca

)1(

)(

ba

cab

)2(

)(

cba

dc

dxcbcxba )()(

xy 1010 loglog

y10log

y10log y10log y10log

xy 10



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( )

The expression could be simplified as;

a. b. c. d. none of these

The expression could be simplified as;

a. b. c. d. none of these

The value of y in the radical equation is;

a. 40 b. 2 c. 32 d. 4

215 3 )( yxxy

xy1

5 134

1

yx5 xy

2

23

3

4

32

)(

)(

x

x

x

x

41x

4x x

33 yy



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( )

Using quadratic formula on the root will be;

a. b. c. 2 d.

The value of win is;

a. 3 b. 14 c. zero d. none of these

The value of is;

a. 2 b. 14 c. zero d. -5

092622 yy

2

1

3

2

3

2

12

16

12

76

w

w

w

w

77ln77lnlog 222 ee

under standing with meths

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