Under Standing With Meths

 

Essential Mathematics For Decision Making By, By, Yaseen Ahmed Meenai Meenai Faculty, DMS-FCS-IBA

 

Decision Making Skills needs Maths

 

Where is The Decision System? •

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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; thanks to our frontal lobes. Such as loosing our empathy Frontal Lobe of the brain completes in 20 ’s of an average human.

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

1- Attention deficits 22-Disinhibition Disinhibition;social ;social inappropriateness http://kaanyucel.wordpress.com/2010/01/10/frontal-lobe-syndromesanother-person-in-the-same-body/   another-person-in-the-same-body/

 

How we can accelerate the frontal Lobe Usage?? •

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We can enhance our Decision System by means of exercising the Logical Thinking  practices. There is a difference b/w ‘Thinking’ and ‘Logical Thinking’ 

By Thinking Logically; we can minimize the TIME and EFFORT for completion of any assigned task….

 

Logical Thinking motiva motivation tion •

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Drawing a FISH can help us understanding the logical thinking:

Now, try to re-draw the same fish, but without lifting your pen once it touches the paper and without striking out any of your drew line.

 

Logical Thinking Motivation from WEB •

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There are so many websites containing similar stuff. By writing “Logical  Problems” in Google, we can find such sites. One of the useful http://www.lumosity.com  http://www.lumosity.com 

link

is

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We can access several www.indiabix.com    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.proprofs.com/games/crazy-taxi/  

 

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

S={a,b,c,d,….,n}

B

A

c a,b  g,h

 A={a,b,c,f,g,h}  B={c,d,e,g,h,i} C={f,g,h,I,j,k}

 f   J,k  C

d,e i

l,m,n

 

Logical Operations (   ) A Union (  ) is consideration of all elements by showing duplicates; once. AB = {a,b,c,d,e,f,g,h,i} An Intersection () is the S B common elements collection A c d,e a,b A B = {c,g,h {c,g,h}}  g,h i What’s  A’ ??  f  A’ = not(A) = S – A  J,k  l,m,n d,e,i,j,k,l,m,n}} A’= {d,e,i,j,k,l,m,n C

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Shading the Venn Diagram S B

A

C

For  B’ , itshould should be  For A’ A it  , B, it bebe For A’ B,should it should be

The Demorgan’s Law

 

Background of Mathematics •

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

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If anyone seen 4 to birds, he would shared his had knowledge someone else have 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.

 

Background of Mathematics •

•

Soon after discovering collection was start started. ed.

numbers,

data

After plotting the data, humans started observing trends and hence Mathematical Functions were derived. derived. Population

Time Year1

Population 2130

Year2

4230

20000

Year3

14500

10000

30000

Population

0

Year4

25800

Year1

Year2

Year3

Year4

 

Mathematical Mathematic al Functions Functions •

The Science of Mathematics based on 3 typeswhole of Functions: 1) Algebraic functions 100 50

f(x)=quad

0 - 10

-5

f(x)=quad 0

5

10

f(x)=cubic

-50 -100

2) Exponential or Log Fn(s)

40 30 f(x)=log(x)

20

f(x)=exp(x)

10 0 0

3) Trigonometric Functions •

2

4

f(x)=sin(x) 2

Polynomial is an algebraic function which can attain any shape….

1 0 -10

-5

-1 -2

0

5

10

 

Types of Numbers

 

Types of Numbers •

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Natural numbers. A  natural number is a number that comes naturally. These are 1, 2, 3, 4, 5, 6, 7, and so on into infinity. We use natural numbers to count items and to make lists. Whole numbers. Whole numbers are just all the natural numbers plus a zero: 0, 1, 2, 3, 4, 5, and so on into infinity Integers. Integers can be described as being positive and negative whole numbers: …  – –3,  –2, – 1, 0, 1, 2, 3, . . . .

 

Types of Numbers •

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Rational numbers. Rational numbers are numbers that act rationally! The decimal ends somewhere, or it has a repeating pattern to it. 2, 3.4, 5.77623, and  –4.5, or 3.164164164… or 0.666666 etc. Irrational numbers. Irrational numbers are the opposite of rational numbers. An irrational number cannot be written as a fraction, and decimal values for irrationals never end and never have a nice pattern to them. For example, pi, with its never-ending decimal places, is irrational. =3.14159265358979 or 2 is also an irrational i rrational number. number.

 

Types of Numbers •

Even numbers. An even number is one that divides evenly by two, such as 2, 4, 6, 8, 10….

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Odd numbers. An odd number is one that does not divide evenly by two, such as 1, 3, 5, 7, 9,…..

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Prime numbers.  It’s a number which can either be divided by itself or 1, such as 1, 3, 5, 7, 11, 13, 17… 

 

Warm up with Math Operations •

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Suppose we have a task to solve the following expression: 2+2*2/2-2=??? Don’t you think that this would be better using parenthesis ( )?? Look at it again…. 2+(2*2)/2-2 2+2(2/2)-2

=2+4/2-2 =2+2-2 =2+2*1-2 =2+2-2

=2 OR =2

 

Warm up with Math Operations •

Here is another puzzle…. Put the arithmetic notations like ( + , - , x, ) and by adjusting parenthesis to prove the followi following: ng: (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 Orders Division or Multiplication, Addition or o r Subtraction)

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

 

BODMAS (Brack (Bracket, et, Order, Order, Division Division or Multiplication, Addition or Subtraction) •

If we have following expression: 30  5 x 3 = ???? Since andweMultiplication rank equally then InDivision this case always be CORRECT if we proceed from Left to Right…  i.e. 30  5 x 3 =6x3 = 18 Solution from Right to Left will give you the Wrong Wr ong a answer nswer…. 30  5 x 3 = 30  15 = 2 (Wrong) (Wrong)

 

BODMAS (Brack (Bracket, et, Order, Order, Division Division or Multiplication, Addition or Subtraction) 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

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Example: How do you work out 12 / 6 × 3 / 2 ? Multiplication and Division rank equally, so just go left to right: First 12 / 6 = 2, then 2 × 3 = 6, then 6 / 2 = 3  •

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BODMAS (Brack (Bracket, et, Order, Order, Division Division or Multiplication, Addition or Subtraction) Oh, yes, and what about 7 + (6 × 52 + 3) ?   Start inside Brackets, Brackets, and then use "Orders" "Orders" First Then Multiply

7 + (6 × 52 + 3) =7 + (6 × 25 + 3) 7 + (150 + 3)

Then Add

7 + (153) 7 + 153

Brackets completed, last operation is, Add

160

DONE !

 

BODMAS (Brack (Bracket, et, Order, Order, Division Division or Multiplication, Addition or Subtraction) 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 + 8 A. 3 B. 4 C. 14 D. 20 What is the value of 5 × 4 - 2 × 3 + 16 ÷ 4 A. 10

B. 11 ½

C. 18

D. 34

What is the value of 30 - (5 × 23 - 15)? A. -25

B. 5

C. 15

D. -15

 

Equalities Equa lities and Inequalities We have certain Symbols to compare two numbers: a = b means ‘a’ is equals to ‘b’

____a=b ____ a=b _____->

a > b means ‘a’ is greater then ‘b’  a < b means ‘a’ is less than ‘b’ 

 ____b___a___->    ____b___a___->  ____a___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 approximately equals to ‘b’ 

 

Terms, Expressions and Equations •

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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 (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.

 

Solving Expressions (pg. 37) Evaluate the following if if,, p=2 , q=3 and r= - 4 :

(v)

(i) p q + r   Ans. Is (ii) p (ii)  p q / r  

2

Ans. Is

-3/2

(iii) ( p  p + q) / r   Ans. Is -5/4

(p – q) -------------(1/p + 1/q)  1/q) 

Ans. is

(vi) (1/p – 1/r ) --------------(p + r)

(iv) p (iv)  p + q/r   Ans. Is

-6/5

+5/4

Ans. is

-3/8

 

Problems with Inequalities (pg. 35) Determine values of ‘x’ which satisfy the following:

(i) x+1 > 0   Ans. Is

x > -1 -1

0

+1

….. 

(i) X - 2 > 0   (ii) Ans. Is x > +2

(iii) 1 – 2x   0  Sol  1   +2x => 1/2   x   x   +1/2

0

+1

+2

….. 

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

+1

+2

….. 

 

Problems with Inequalities (pg. 35) Determine values of ‘x’ which satisfy the following:

(iv) (x+1)(x-2) > 0 

f(x) 16 14

Sol.  x 2-x-2 > 0

12 10 8 6

Therefore,

4 2 0

 x < -1 or x > +2

-4

-2

-2

0

2

4

6

-4

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

 

Problems with Inequalities (pg. 35) Determine values of ‘ x   x ’ which satisfy the following:

(iv) (x + 1) -------- < 0  (x – 2) Since we kno know w,

The best matching possibility is:

+ve / -ve < 0 Therefore,  x+1 > 0 and  x > -1 and

x-2 < 0 x < +2

+ve / +ve > 0 -ve / -ve > 0 +ve / -ve < 0 -1

-ve / +ve < 0

0

+1

+2

….. 

 

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 it’s practical implement implementation??  ation??  So while computing distance or the error margin, we mostly ignore the sign…. For e.g. |Theoretical Value –  Empirical Value | = error 

 

Problems with Modulus (pg. 36) Evaluate Ev aluate the ffollowing ollowing 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 | - |x – 1|| + |2 – x|  Ans is 6, 2 , 5/2, 2

 

Problems with Modulus (pg. 36) Evaluate Ev aluate the ffollowing ollowing when x = - 2, 0 , +1/2 and +2:

(v) || x + 1 | + |x – 1|| - |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

 

Inequalities with Modulus There are three possible cases in this situation s ituation

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

….. 

 

Problems with inequalities and Modulus (pg. 36) 5. Determine all values of ‘x’ such that: (v) |3x - 1|  2 st 

This is the 1 case, we will have following two cases; and (3x - 1)   -2 (3x - 1)   + 2 Therefore the two solutions will be: 3x   2 + 1  x  3/3 = 1

and and

3x  -2 + 1 x    -1 / 3 Outer range

-1/3

+1

….. 

 

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 3x=18-3

 x=15/3

=> x=5

=> 3x=15 therefore, 5+6+7=18 

 

Word Problems (Numbers) Even and Odd Integers:

Any number which can be divided by 2 is an Even number and which cannot be divided by  2 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.

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Sample Question2: The sum of three consecutive Even integers is 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 10+12+14=36

=> 3x=30

=> x=10. Therefore,

 

Logarithms (pg. 38) •

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What is the Mathematics? 

concept

of

Logarithms

in

Why Logarithms are necessary? What is the significance of this concept in Mathematics?  In order to scale the magnitude, we usually use log. In order to solve expressions in which subject is in the power; only log can help us. If Exponential form is 10 2=100 then the log form will be log10 (100)=2

 

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

10 2=100

The log form will be

log10 (100)=2

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•

then

For e.g. we can can solve x+3 = 5. i.e. x=2 But we cannot carried out the solution for 3 x =5 without Log. Therefore by taking log on both sides; log(3 x  )=log(5) => xlog(3)=log(5)  x=log(5)/log(3) will be the answer. answer.

 

Properties of Logarithms 1) loga(a)=1

e.g.

log10 (10)=1

2) log(a x  )=xlog(a) e.g.

log(100) = log(10 2 )=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 17. Given that log(4.78)=0.6794, then Evaluate; i)

log(47.8)

Sol. Required value is ‘x’ such that,

log(47.8)=x 

Therefore,  According ing to the given value in the question:  Accord Log(4.78)=0.6794 and we also knew that log(10)=1 so, Log(47.8)=log(4.78 x10)

 

Questions of Logarithms Log(47.8)=log(4.78 x10)  According ding to the third property: property:  Accor 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.

 

Questions of Logarithms 17. Given that log(4.78)=0.6794, then Evaluate; i)

log(0.478)

Sol. Required value is ‘x’ such that,

log(0.478)=x 

Therefore,  According ing to the given value in the question:  Accord Log(4.78)=0.6794 and we also knew that log(10)=1 so, Log(0.478)=log(4.78 /10)

 

Questions of Logarithms Log(0.478)=log(4.78 /10)  According ding to the third property: property:  Accor 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

0.6794 – 1 =

OR we can also write the ans,

 

Questions of Logarithms 18. Given that log(6 log(6.85)=0.8 .85)=0.8357, 357, then numbers whose log to base 10 is; i)

find

the

1.8357  

Sol. Required value is ‘x’ such that, Therefore, Log(x)=1 + 0.8357   According to the given value:  According 0.8357 = log(6.85) and 1 = log(10) so, log(x)=log(10) + log(6.85)

log(x)=1.8357 

 

Questions of Logarithms log(x)=log(10) + log(6.85)  According ding to the 3rd  Property:  Accor Log(ab)=log(a) + log(b) Therefore, log(x)=log(10 x 6.85) log(x)=log(68.5) Finally,  X=68.5

Ans.

 

Questions of Logarithms 18. Given that log(6 log(6.85)=0.8 .85)=0.8357, 357, then numbers whose log to base 10 is; i)

find

the

3.8357  

Sol. Required value is ‘x’ such that,

log(x)=1.8357 

Therefore, Log(x)=3 + 0.8357

= 1+1+1+0.8357 

 According to the given value:  According 0.8357 = log(6.85) and 1 = log(10) so, log(x)=log(10) + log(6.85)

 

Questions of Logarithms log(x)=log(10) + log(10)+ log(10)+ log(6.85)  According ding to the 3rd  Property:  Accor Log(ab)=log(a) + log(b) Therefore, log(x)=log(10 x10x10x6.85) log(x)=log(6850) Finally,  X=6850

Ans.

 

Questions of Logarithms 20. Given that  log y =  log x +    where   and    are constants and that  y=10 when x=0.1 and y=100  when 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

=  [0 – 1] +   

1

= - + 

------ (i) 

 

Questions of Logarithms Then the 2nd  Information: Y=100 when x=1

log 100 =  log 1 +    2

=  (0) +   

2

=

or 

 = 2 

   in (i) Putting the value of    1

= - + 

1

= - + 2



= 2 – 1

------ (i)  or 

=1

Ans. 

 

Questions of Logarithms 21. By taking log, find the smallest positive integer  ‘n’    for which 4n < 5n-1. Sol. Taking Taking log on both sides, log 4n

< log 5n-1 

Or 

n log4

< (n-1) log 5

n > log5 / (log5 –  log4)

n log4

< n log5 –  log5

n > 7.212

log5

< n log5 –  n log4 Therefore, < n (log5 –  log4) n=8 Ans.

log5

log5/(log5-log4) < n

 

Questions of Logarithms 22. A biological population initially of  size 1000   , doubles its size everyday. The size of the population after  ‘ n’  generation times is N=10 3 2n. Find using logarithms, the number of generations that must  elapse before the size ex exceeds ceeds (i) 105 (ii) 1010. (i)

Sol. N

> 105 

10 3 2n

> 105 

2n 2n

> 105 / 103  > 102

log(2n )

> log (102 )

n log (2) > 2

Now we will take log on both sides or 

n > 2 / log (2) > 6.64  7 

 

Questions of Logarithms 23. The volume ‘V ‘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 and ‘a’ is 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

Or 

a (1.05) t  = 2 a (1.05) t = 2

Now taking log on both sides and solve

 

Questions of Logarithms t 

log (1.05) t log (1.05)

= log 2 = log 2

t

= log 2 / log (1.05)

t

= 14.2 years

Ans.

 

Compound Interest in Financial Maths Following is the compound interest formula : S = P ( 1 + r ) t 

Where, S = Compounded Amount,

P = Principal Amount 

r = Rate of Interest 

Now reconsider question no. 22:

N = 10 3 2n 

Showing population which ‘doubles’  its size everyday.

We can also re-write the same as:

N = 1000 (1+1) n

It’s the same formula

S =

P

(1+r) n

 

Compound Interest in Financial Maths Now comparing the same compound interest formula with the model given in question no. 23:

The volume ‘V ‘V’  ’  of 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.

 

Questions of Logarithms 24. In the study of hearing the loudness ‘L ‘L’ ’  is expressed in terms of the intensity ‘I’  by the equation: L = 10 log (I / I o ) Where Io is approx. 10 -12 Watt/m2. Express ‘I’  in terms of L and determine ‘I’  when L=60. Sol.

We can use the property of Log here, L log(10) = 10 log (I / Io )

[10 L ] 1/10

= [ (I/Io )10 ] 1/10 

log (10) L = log (I/Io ) 10 

10 L/10

= I / Io 

Taking Antilog on both sides, 10 L = (I/Io ) 10 

Therefore,

Taking 10th Root (power 1/10)

Finally, L=60 and Io = 10-12

I = 10 L/10 Io  -6

on both sides,

I = 10

 

Eulers number or Napier’s Constant  •

This is ‘e’  

•

e is said to be the life’s function

•

•

•

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

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 Math constant. Mathematician ematician used it) Following is the function of e: e = (1 + x) 1/x 

but

Lim x     0

If we put x=0 x =0 then e = inf. inf. So we have to ‘poke’  the 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 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) •

Determine the equation of straight line passing through a point (4,-1) having slope=-3 a.

•

y= - 3x+11

y=2x-3

d. y=3x-2  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. y= 3x+11 c.

b. x-3y= - 8

c. 2x-3y=4

d. x-2y=5  x-2y=5 

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

a.

b.

c.

d. d.  

 

Practice Questions (Maths) •

The roots of the quadratic equation are;  y  ( x  1) 2  9 a. (2,-4)

•

b.

(-4,2)

c.

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

•

(-4,-8)

c. -17.77

d. 32.00  32.00 

An object projected vertically upwards at ttime ime t=0  with a velocity of  14 meters/second  reaches a height ‘y ’ in meters 2

given by by,,  y  14 t   4.9t  , when y=10 , the value of ‘t ’ will be; a. 10/7 b. 12/7 c. 5/9 d. -10/5  -10/5 

 

Practice Questions (Maths) •

The value of ‘ x   x ’ in the equation (a  b) x  c a.  2b b. (a  bc) c . ( ab  c) (a  c)

c •

d .

( a  1  b)

is

(c  d ) ( a  2b  c )

, where ‘’ and ‘’ are

log 10  y    log 10 x   

Given that

 (b  c) x  d 

constants and that y=10  y=10 when when x=0.1  x=0.1 and y=100  y=100 when when x=1  x=1,, then the values of (,) will be; a. (1,2) •

b. (-1,2)

 x ’ in If ‘y ’ is a function of ‘ x  will be ;   y a. log10        

 

b.

c. (2,1)

d. (-1,-1)

 x ’ in terms of ‘y ’  y   10  x , then ‘ x 

log 10      y     

c. log

10

   y 

d. log 10  y 

 

Practice Questions (Maths) •

The expression (5  xy 1

a.

 xy

b.

The expression a.

•

1 4  x

) x 1 y 2

could be simplified as;

1 5

4

 x  y

c.

13

5

  ( x 4)    x3 2   x  ( x )  2

•

3

b.

3

3

 x

4

 xy

d. none of these these  

2

could be simplified as; c.

x

The value of ‘y ’ in the radical equation  y  3

a. 40

b. 2

c. 32

d. none of these these  

 y  3 is; d. 4 

 

Practice Questions (Maths) •

•

Using quadratic formula on 2  6 2  y  9 y 2 a. 1 2 b. 2 3 c. 2 The value of ‘w’ in

6w  7

 2w 1

a. 3 •

b. 14



6w  1



0

the root will be; d.

3

 1 is;

2w c. zero

The value of log ln 7  e 2  7  ln 7  e 2  7

d. none of these

is;

2

a. 2

b. 14

2



c. zero

d. -5