### Dictionary Rank of a Word | Permutations & Combinations

PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

### Complex Numbers Chapter 5 Class 11

Complex Numbers Chapter-5 Class-11
Basic points and complete explanation of the complex numbers with examples and diagrams. Addition, subtraction and multiplication of complex numbers. Representation of a complex number into polar form

## Following concepts will be  discussed in this Blog

 Real NumbersNon-Real Numbers or Imaginary Numbers and concept of iota.Complex Number and its general form.Conjugate of a complex Number.Properties (Addition, Subtraction and multiplication) of Complex Numbers. Modulus and Argument of a Complex Number.Polar form of a complex number.Nature of Quadratic Equations and their solutions.Square Root of a Complex Number with example.

Real numbers:
All rational and irrational numbers are called real numbers.
Non-Real Numbers :
The numbers which are not real are called non real numbers. These numbers are also called imaginary numbers.
Imaginary numbers :-
Square root of a negative real number is called the imaginary number.

Complex Number:
All real and non- real numbers are called complex number
OR
A number of the form a + ib is called a complex number. Here a and b are the real numbers and i is called the iota.

For example  Z = a + ib,  where a is called real part and b is called imaginary part.
Value of iota is √-1
i = √-1 ,  i2 = -1,   i3 = - i,   i4 = 1
Geometrical Representation of the values of iota.

Two complex numbers are equal if their real and imaginary parts are equal
If Y and Z are two complex numbers such that
Y = a + ib and Z = c + id,
Then  Y = Z
a = c and  b = d

Addition of two complex numbers
Two complex numbers are added by simply adding their real and imaginary parts
If Y = a + ib and Z = c + id then
Y + Z = (a + c) + i(b + d)
Properties of addition of two complex numbers
i) Addition of two complex numbers holds the closure property. This means that addition of two complex numbers is also a complex number.

ii) Addition of complex numbers is commutative.
Y + Z = Z + Y

iii) Addition of complex numbers holds associative law
(X + Y) + Z = X + (Y + Z)

iv)Existence of additive identity:
0 is the additive identity for the addition of complex number.  Z + 0 = Z

v) Existence of additive inverse :
If Z is a complex number then - Z is called the additive inverse of Z.      Z + ( - Z) = 0
Subtraction of two complex numbers :
Two complex numbers are subtracted by subtracting their corresponding real and imaginary parts.
If Y = a + ib,   Z = c + id then
Y - Z = (a + ib) - (c + id)
= (a - c) + i(b - d)
Multiplication of two complex numbers
If Y = a + ib,  Z = c + id then
YZ = (a + ib)(c + id)
= ac + iad + ibc + i2bd
= ac + i(ad + bc) + (-bd)
= (ac - bd) + i(ad + bc)
Properties of Multiplication of two complex numbers
i) Closure property :
Product of two complex numbers is also a complex number.

ii) Commutative law :
Multiplication of complex numbers is commutative.  YZ = ZY

iii) Associative Law:
Multiplication of complex numbers is associative.    (XY)Z = X(YZ)

iv) Existence of multiplicative identity:
1 is called the multiplicative identity for the product of complex number.
Z x 1 = 1 x Z = Z

v) Multiplicative inverse :
If Z is a complex number then Z-1 is called the inverse of Z.    Z x  Z-1  = 1

vi) Distributive law :
Product of complex numbers holds the distributive property.  X(Y + Z) = XY + XZ

Conjugate of a complex number $If\: Z = a+ib \; is\; a\; complex\; number\; then \; conjugate\; of\; Z\; is \; denoted\; by\; \overline{Z}\; and \; is\; given\; by$$\overline{Z}=\overline{a+ib}=a-ib$
Note:- Conjugate of the conjugate of complex number is the complex number itself. $If\: z=a+ib,\: then\: \overline{z}=\overline{a+ib}=a-ib$$\overline{\overline{z}}=\overline{a-ib}=a+ib=z\: \Rightarrow \overline{\overline{z}}=z$

Modulus of complex number
If Z = a + ib is a complex number then modulus of Z is denoted by |Z| and is given by $\left | Z \right |= \sqrt{a^{2}+b^{2}}$

Argand Plane :
The plane having a complex number assigned to each of its point is called a complex plan or Argand plane.

Every point of the form (x,y) can be represented in the cartesian coordinate plane.

When a complex number is represented in the plan then the plane is called a complex plane or Argand plane.

In Argand plane x- axis is called real axis and y-axis is called imaginary axis.
If Z = x + iy then $\left | Z \right |= \sqrt{x^{2}+y^{2}}$
|Z| is the distance of the point (x,y) from the origin (0,0)

Polar Representation of a complex number z = x + iy

Let point P represent the complex number Z = x + iy. Let directed line OP = |Z| = r and θ is the angle which OP makes with the positive direction of x - axis. $In\; \Delta OAP,\; \; \; \frac{OA}{OP}=cos\theta$$\frac{x}{r}=cos\theta \Rightarrow x=rcos\theta$
$In\; \Delta OAP,\; \; \; \frac{AP}{OP}=sin\theta$$\frac{y}{r}=sin\theta \Rightarrow y=rsin\theta$
Putting x = rcosθ and y = rsinθ in   Z = x + iy we get
Z = rcosθ + irsinθ  or   Z = r(cosθ + isinθ)
This is called polar form of a complex number.

If Z = a + ib, then to convert it into polar form  we put a = rcosθ and b = rsinθ
tanθ = b/a  and     θ = tan-1(b/a)

Sign convention while converting a complex number from Cartesian form to polar form.

 DISCUSSION AND CONCLUSIONSIf (a, b) lie in first quadrant then Argument =   θIf (a, b) lie in second quadrant then Argument = π - θIf (a, b) lie in third quadrant then Argument =  -π + θIf (a, b) lie in fourth quadrant then Argument =  - θIf complex number Z is purely +ve real number then Argument is  0If complex number Z is purely -ve real number then Argument is  πIf complex number Z is purely +ve imaginary number then Argument is  π/2If complex number Z is purely -ve imaginary number then Argument is  - π/2

Method to convert a complex number into a polar form
$z=-\sqrt{3}+i$$r=|z|=\sqrt{\left ( \sqrt{3} \right )^{2}+\left ( 1 \right )^{2}}=2$Let polar form of complex number is $z=rcos\theta +irsin\theta$Putting r = 2 we get$z=2cos\theta +i2sin\theta$Compairing this with the first equation we get $2cos\theta =-\sqrt{3}\; \; and\; \; 2sin\theta =1$$\Rightarrow cos\theta =-\frac{\sqrt{3}}{2}\; \; and\; \; sin\theta =\frac{1}{2}$$tan\theta =\frac{sin\theta }{cos\theta }=\frac{1/2}{-\sqrt{3}/2}=\frac{-1}{\sqrt{3}}$$tan\theta =\frac{-1}{\sqrt{3}}=-tan\left ( \frac{\pi }{6} \right )$$since\; point \; \; (-\sqrt{3},1)\; \; lie \; in \; the\; second\; quadrant$$\Rightarrow Argument = \pi -\theta =\pi -\frac{\pi }{6}=\frac{5\pi }{6}$ Hence required polar form of a complex number is $z=2cos(\frac{5\pi }{6})+i\: 2sin(\frac{5\pi }{6})$
General quadratic equation is  ax+ bx + c = 0
Discriminant (D)  = b2 - 4ac

Nature Of The Roots Of The Quadratic Equation
$If D> 0 \; then \; roots\; are \; real\; and\; unequal\; or\; distinct\; or\; different)$
$If D= 0 \; then \; roots\; are \; real\; and\; equal$
$If D< 0 \; then \; roots\; are \; not\; real$
$If D\geq 0 \; then \; roots\; are \; \; real$
Quadratic Formula For Solving The Quadratic Equations
$x=\frac{-b\pm \sqrt{D}}{2a}\; \; \; or\; \; \; x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$
 Method of finding the square root of a complex number This can be explained by taking an example $Find\; the\; square \; root\; of\; the\; complex\; number\; \; z=\sqrt{3+i4}$$Let\; \; a+ib=\sqrt{3+i4}$Squaring on both side we get,$(a)^{2}+(ib)^{2}+2iab=3+i4$$(a)^{2}-b^{2}+2abi=3+i4$Compairing real and imaginary parts we ger, $(a)^{2}-b^{2}=3 .......(1),\: \: and\: 2ab=4$$Now\: using\: the\: formula\: \: (a^{2}+b^{2})^{2}=(a^{2}-b^{2})^{2}+(2ab)^{2}$$(a^{2}+b^{2})^{2}=(3)^{2}+(4)^{2}=9+16=25$$\Rightarrow (a^{2}+b^{2})=5 ......(2)$Adding eqn(1) and eqn(2) we get$2a^{2}=8\Rightarrow a^{2}=4\Rightarrow a=\pm 2$From eqn(1) - eqn(2) we get$2b^{2}=2\Rightarrow b^{2}=1\Rightarrow b=\pm 1$$\sqrt{3+i4}=a+ib=\pm 2\pm 1i=\pm (2+i)$Here between 2 and i we apply +ve sign because 2ab is positive. If the value of 2ab is -ve then between a and b we apply -ve sign in the square root of the complex number.

Question:$If\: x=-5+2\sqrt{-4}, \: then\: find \: the\: value\: of\\ x^{4}+9\: x^{3}+35\: x^{2}-x+4$
Solution:$x= -5+2\sqrt{-4}$$x+5=2\sqrt{-4}$
Squaring on both sides we get$(x+5)^{2}=\left [ 2\sqrt{-4} \; \right ]^{2}$$x^{2}+5^{2}+2\times x\times 5=4\times (-4)$$x^{2}+25+10x+16=0$$x^{2}+10x+41=0$
Now divide the given equation with this equation we get

Dividend = Divisor X Quotient + Remainder
P(x) = (x2 + 10x + 41)(x2 – x + 4) – 160
= 0 X (x2 – x + 4) – 160 = - 160 Ans

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