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BASICSNumber System

N⊊W⊊I or Z⊊Q⊊R⊊C

WHERE:- N⇒natural number

W⇒whole number

I⇒integers

Q⇒rational number

R⇒real number

C⇒complex number

1. Natural Number:N ≡ {1, 2, 3, 4………}

# Prime Number: All natural number except 1 which is dividable by 1 & itself only.

e.g.: 2, 3, 5, 7, 11, 13, 17, 19…………….

#Composite Number: All natural number except 1 which is not a prime number.

#Co-Prime Number: Pair of two natural number (there are prime, are not compulsory) are called co-prime number when its H.C.F is 1 and its L.C.M is equal to its multiplication.

e.g.: (1, 2), (2, 3), (3, 4), (4, 9), (5, 6) etc

#Twin Primes Number: Those pair of prime number having difference 2.

e.g.: (3, 5), (5, 7), (11, 13), (17, 19) …………

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2. Whole Number:W≡ {0, 1, 2, 3, 4 ……..}

3. Integer:I or Z ≡ {…… -3, -2, -1, 0, 1, 2, 3 ………}

NOTE: - 0 is not a positive nor negative.

# Even Number: Which integer is dividable by 2 .

NOTE: - 0 is even number.

# Odd Number: Which integer is not dividable by 2.

4. Rational Number: Those real number who can be written in the form of p/q where p & q ∈ I and q≠0.

Q = {p/q: p, q ∈ I and q≠0}

NOTE: - Fraction part of every rational number is terminating or repeating in a definite period.

Question: Convert rational number 4.02⃑3 into p/q form.

Sol: Let x=4.02⃑3 x=4.023232……

10x=40.2323 ----------(1) 1000x=4023.23… ----------(2)

eq (2)-(1) 990x=3983

Or, x=3983990

Extra example: 5.67⃑=567−56990

5. Irrational Number: Those real number who can’t be written in the form of p/q where p & q ∈ I and q≠0.

NOTE: - Fraction part of every irrational number is not terminating nor repeating in a definite period.

e.g.: √2=1.414….π=3.14….√3=1.732… e=2.71….

Problem: Is √-1 an irrational number?

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6. Real Number: Set of all rational and irrational number is called real number.

* Real Number Line:

-∞<--------------------------------------------------|--------------------------------------------------->∞ 0All real number have been particular place on the real number line.6. Complex Number:# Imaginary Number:

√-1=ι (iota)

√-2=√2ιNOTE: -

ι1= ι ι5= ιι2=-1 ι6= -1 ι3=- ι ι7=- ι ι4=1 ι8=1

Question: Evaluate ι4087.Sol: (ι4)1021 ι3

=1 × (-ι)= (-ι)

Question: Evaluate ι1+ι2+ι3+ι4+ι5+…………..ι100.Sol: 0.

Complex Number: Which number(z) is written in form of z=a+ιb, ∈a, b ∈ R and ι=√-1 is called complex number.e.g.: 3+4ι, 50=50+0ι, √3ι etc.

NOTE:- # In z=a+ιb, ‘a’ is called real part of z and ‘b’ is called imaginary part of z. #If b=0, z is a real number. #If a=0, z is called pure imaginary number.

(It is roung English of sayugmi samisar sankhya) Conjugate of Complex Number: If z=a+ιb is a complex number, its co-pair complex number is z=z=a-ιb.

Exercise: Find Co-pair Complex Number of given Complex Number

(i)ι-5 (ii)2+3ι1−ι

Question: Statement1. a3+b3+c3-3abc = (a + b +c) (a2+b2+c2- ab- bc - cd).Statement2. If a, b, c are three real numbers and a3+b3+c3=3abc then a + b +c=0 compulsory.

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(a)Statement1 & Statement2 both are correct and Statement1 is a complete explanation of Statement2.

(b) Statement1 & Statement2 both are correct but Statement1 is not a complete explanation of Statement2.

(c) Statement1 is correct but Statement2 is not correct.(d) Statement1 is not correct but Statement2 is correct.

Sol: (c)

Sum-Difference Ratio

If ab=cd then a+ba−b=

c+dc−d

Proof: ab

=cd

or, ab

+1=cd

+1

a+bb

=c+dd

------------------ (1)

now, ab

-1=cd

-1

a−bb

=c−dd

------------------ (2)

eq(1)/(2) ;

a+ba−b

=c+dc−d

Hence proved

Question: Solve for real x

3x4+ x2−2x−33 x4−x2+2x+3

= 5 x4+2x2−7 x+35x4−2 x2+7 x−3

NOTE: - (i) At the time of equation solving, don’t cancel same factor of L.H.S & R.H.S. If we do that, we can loss some solution.

(ii) At the time of equation solving, don’t squaring both side as so as possible. If we do that, we can get some fouls solution. If it is compulsory then please check your solution to put it in primarily equation.

Question: Find value of x

X=√6+√6+√6+…∞

Sol: x=√6+x

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or, x2=6+x [squaring both side]

or, x2-x-6=0 or, (x-3)(x+2)=0 or, x=3 & x=-2 x=-2 is not an acceptable solution ∴x=3 is only one solution of given equation.

Reminder Theorem

If any polynomial P(x) is dived by (x-α) then its reminder is P(α).P(x)=(x-α).θ(x) + R

P(α)=R

If put x=α, P(α) is became to zero then (x-α) is a factor of P(x).

Interval1. X ≡ {a, b} ⇒ A set ‘X’ which members are ‘a’ and ‘b’.2. X ≡ (a, b) ⇒ A set ‘X’ which member are all real number between ‘a’ and ‘b’ not including ‘a’ & ‘b’. 3. X ≡ [a, b] ⇒ A set ‘X’ which members are all real number between ‘a’ and ‘b’ including ‘a’ & ‘b’.4. X ≡ [a, b) ⇒ A set ‘X’ which members are all real number between ‘a’ and ‘b’ including ‘a’ but not ‘b’.5. X ≡ (a, b] ⇒ A set ‘X’ which members are all real number between ‘a’ and ‘b’ including ‘b’ but not ‘a’.

Modulus Function

f(x) = |x|

NOTE: - √4=2 [√4=±2 is wrong] -√4=-2 ±√4=±2

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Attention Please: √ x2 = x is wrong. √ x2 = |x| is right.

Definition:

y = |x| = { x ;when x ≥0−x ; when x<0}

Graph

y

x’ -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x

y’

Question: Solve the equation |x-5|=7.

Sol: 1st method CASE1: x≥5 CASE2: x<5 x-5=7 -( x-5)=7 ⇒ x=12 ⇒5-7=x ⇒ x=-2

2nd method |x-5|=7⇒x-5=±7 ⇒ x= ±7+5⇒ x=-2,12

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Question: Solve the equation |x|- |x-2|=2.

Sol: -∞ | | ∞ i 0 ii 2 iii

Case i: x<0 Case ii: 0≤x<2 Case iii: x≥2 (-x)+(x-2)=2 x-[−( x−2) ]=2 x-(x-2)=2⇒-2=2 fouls ⇒x+x-2=2 ⇒2=2∴equation have no solution ⇒2x=4 always true in this case ⇒x=2 all real numbers which is satisfied rejected because x<2 in this case are the solution of this equation. this case∴complete solution of given equation is x∈[2,∞)

Greater Integer FunctionGreater Integer Function Fractional Part Function

i. y=f(x)=[x] i. y=f(x)={x}ii. Definition: On the real number line,

starting from ‘x’ and going on left side of it, which is the largest integer is called greater integer of ‘x’ represented by [x].

ii. Definition: x=[x]+{x} {x }=x−[x ]

iii. Its value is always an integer. iii. its value is always in interval (0,1].

NOTE: - All real numbers are written in the form of X = I + f

Where: - ‘I’ is integer part & ‘f’ is fractional part

Logarithm All positive numbers can be written in power form.

Power form:N = ax

Logarithmic form:

log aN = x

Where: - N ⇒ real positive number a ⇒ is called ‘base’ &b ⇒ is called ‘power’

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NOTE: - log aN is define if and only if(i) N > 0(ii) a > 0(iii) a ≠ 1

# Fundamental Identities

(i) log aa = 1 Proof: If ax=N ⇔ x=log aN

(ii) log a1a

= -1 ∵log aN=log aN(iii) log a1 = 0 ⇒ N=a logaN(iv) a logaN = N

Problem: If N=1.2.3.4.5………….2010 then find the value of expression1

log2N +

1log3N

+ 1

log4N +

1log5N

+ ……………………1

log2010NAns: 1

# Important Property

i) log x ab = log x a + log xb

ii) log xab

= log x a - log xb

iii) log aMα = αlog aM

iv) log α βM = 1αlogαM

v) log ab = 1

logba

vi) log ab = logc b

logc a

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vii) a logb c = c logba

Logarithmic Equation

Question: Solve the following equationi) x2 - 7 log7 x- 12 = 0

Sol: x2 - x -12 = 0 (x – 4)(x + 3)=0 x = 4; x = -3 (rejected, log is not define for this value of x)

i.e. Given equation have only one solution x=4.

ii) log 2log 4 log5 x = 0

iii) xlog x+74 = 10log x+1

iv) ( log x )2 - 5log x + 4 = 0

v) 2log

214a – (a¿¿2+1)3 log273¿ – 2a = 0

Ans: i) x=625ii) x=10, 10−4

iii) x=10, 104

Characteristic & Mantissa

[ log aN ] is called Characteristic of log of N to the base a.

{ logaN } is called Mantissa of log of N to the base a.

NOTE: -1 If Characteristic of any number is ‘p’ when base of log is 10, then number of digit in this number is (p+1).

NOTE: -2 If Characteristic of any number is ‘-q’ with taking log with base 10, then number of zero just after point in this number is (q-1).

Graph of Logarithmic Function

f(x) = y = log a x

Case I: a>1 Case II: a<1

y y

x’ | x x’ | x 1 1

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y’ y’

NOTE: - If the number and the base of a log are situated same side of unity then value of that log is positive.

Domain: x ∈ R+¿ ¿ or (0, ∞)Range: y ∈ R or (-∞, ∞)