Complex numbers for IIT-JEE MAINS /ADVANCED

Complex numbers basic concepts-Defining

A number in the form of a + ib, where a, b are real numbers and is called a complex
Number. A Complex Number can also be defined as an ordered pair of real numbers a and b any
may be written as (a, b), where the first number denotes the real part and the second number
denotes the imaginary part. If z = a + ib, then the real part of z is denoted by Re (z) and the
imaginary part by Im(z). A complex number is said to be purely real if Im(z)= 0, and is said to
be purely imaginary if Re(z) = 0. The complex number 0 = 0 + i0 is both purely real and imaginary.
Two complex numbers are said to be equal if and only if their real parts and imaginary parts are
separately equal i.e. a + ib = c + id implies a = c and b = d. However, there is no order relation
between complex numbers and the expressions of the type a + ib < (or > ) c + id are meaningless

Geometrical representation of complex numbers

A complex number z = x + iy, written as an ordered pair (x, y), can be represented by a point P whose Cartesian coordinates are (x, y) referred to axes OX and OY, usually called the real and the imaginary axes.

complex numbers 1

The plane of OX and OY is called the Argand diagram or the complex plane. Since the origin O lies on both OX and OY, the corresponding complex number z = 0 is both purely real and purely imaginary.

 

Modulus and Argument of a Complex Number

We define modulus of the complex number z = x + iy as

complex numbers 2

complex numbers 3
We can define the argument of a complex Number also as any value of the which satisfies the
complex numbers 4
is the angle which OP makes with the positive x-axis.

Trigonometric ( or polar ) from a Complex Number

complex numbers 5

complex numbers 6

Remark : Method of finding the principal value of the argument of a complex number z = x + iy.

complex numbers 7

Unimodular Complex Number

complex numbers 8

Algebraic Operations with Complex Number

complex numbers 9

Geometrical Meaning of Product and Division

Let z1 = x1 + i y1 and z2 = x2 + i y2 be two complex numbers represented by the points P1(x1, y1) and P2(x2, y2) respectively. By definition z1 + z2 should be represented by the point (x1 + x2 , y1 + y2 ). This point is the vertex which completes the parallelogram with the line segments joining the origin with OP1 and OP2 as the adjacent sides.
| z1 + z2 | = OP.

complex numbers 10
lso by definition z1 – z2 should be represents by the point (x1 – x2, y1 – y2). This point is the vertex
which completes the parallelogram with the line segments joining the origin with OP1 and
(where the point represents ; the point can be obtained by producing the directed
line P2O by length | z2 | ) as the adjacent sides.
| z1 – z2 | = OQ = P2 P1.
Remarks
(a) In any triangle, sum of any two sides is greater than the third side and difference of any two

complex numbers 11

Construction for the point representing the product z1 z2

complex numbers 12

complex numbers 13

.

Construction for the point representing the quotient z1/z2

Draw the triangle OQ1P directly similar to the triangle OQ2L
The P represents the quotient z1/z2 .

complex numbers 14

complex numbers 15

 

Square Root of a Complex Number

How we get the square root of complex number

complex-numbers-16

complex numbers 17

Conjugate of a Complex Number

The conjugate of the complex number z = a + ib is defined to be a – ib and i denoted by . In other words is the mirror image of z in the real axis.

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Properties of Conjugate, Modulus, Argument

complex numbers 20

complex numbers 21

complex numbers 22
Modulus of a Complex Number

Distance of a complex number z from origin is called the modulus of the complex number z and it is denoted by |z| .

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complex numbers 24

Properties of Arguments

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Complex Numbers Represented By Vectors
It can be easily seen that multiplication by real numbers of a complex number is subjected to the same rule as the vectors . The addition or the subtraction of two complex numbers is also the same as the addition or the substraction of two vectors. This fact is fundamental in theory and very useful in practice.
It should be noticed that if a number z is represented by points P and OP by a vector then
| z | is the length OP and arg(z) is the angle which the directed line OP makes with directed OX.
Please note that if z = x + iy and P is the point (x, y), a one–to–one correspondence exists between the number z and any of the following : (i) the point P; (ii) the displacement ; (iii) the vector
(or directed length ) .
Any one of these three things may therefore be said to represent z, or to be represented by z.

 

De-Moiver’s Theorem

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Application of De Moiver’s Theorem

This is a fundamental theorem and has various applications. Here we will discuss few of these
which are important from the examination point of view

The nth Roots of Unity

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Cube Roots of Unity

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Logarithm of Complex Number

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CONCEPT OF ROTATION

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complex numbers 33

complex numbers 34

Geometrical Application

Section Formula

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Equation of a straight line

Equation of straight line with the help of coordinate geometry

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Equation of straight line with the help of rotation formula
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General equation of the line

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complex numbers 40

complex numbers 42

 

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Equation of Perpendicular Bisector

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arg (z – z0) =p  represents a line passing through z0 with slope (making angle p with the positive direction of x-axis).

Equation of a circle

Consider a fixed complex number z0 and let z be any complex number which moves in such a way
that it’s distance from z0 is always equals to ‘r’. This implies z would lie on a circle whose centre is
z0 and radius r. And its equation would | z – z0 | = r

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It represents the general equation of a circle in the complex plane.

complex numbers 46

complex numbers 47

complex numbers 48

complex numbers 49

Note :
(i) If we take ‘C’ to be mid–point of A2A1, it can be easily proved that CA.CB = (CA1)2 i.e. | z1 – z0 | | z2 – z0 | = r2, where the point C is denoted z0 and r is the radius of the circle.

(ii) If | z1 – z0 | = | z2 – z0 | hence P(z) would lie on the right bisector of the line A(z1)
and B(z2). Note that in this case z1 and z2 are the mirror images of each other with respect
to the right bisector.

Equation of tangent a given circle at a given point

complex numbers 50

Some important results

complex numbers 51

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Quadratic equations for IIT-JEE MAINS /ADVANCED

BASIC RESULTS RELATED TO A QUADRATIC EQUATIONS

quadratic 1

The quantity D

quadratic 2

  • The quadratic equation has real and equal roots if and only if D = 0

 

  • The quadratic equation has real and distinct roots if and only if D > 0

 

  • The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 . If p + iq (p and q being real) is a root of the quadratic equation where i = , then p – iq is also a root of the quadratic equation.

 

CONCEPT OF IDENTITY

If the quadratic equation is satisfied by more than two distinct numbers (real or complex), then it becomes an identity i.e. a = b = c = 0. For example

quadratic identity-1

is satisfied by three value of x which are a, b and c. Hence this is an identity in x.

quadratic identity

 

HOW TO SOlVE RATIONAL INEQUALITIES

In order to solve inequalities of the form

quadratic.1

we use the following method:

If x1 and x2 (x1 < x2) are two consecutive distinct roots of a polynomial equation, then within this
interval the polynomial itself takes on values having the same sign. Now find all the roots of the polynomial equations P(x) = 0 and Q(x) = 0. Ignore the common roots and write

quadratic equation 2
Where a1, a2, . . . . . an, b1, b2, . . . . . , bm are distinct real numbers. Then f(x) = 0 for x = a1,
a2, . . . . . , an and f(x) is not defined for x = b1, b2, . . . . . , bm. Apart from these (m + n) real
numbers f(x) is either positive or negative. Now arrange a1, a2, . . . . . , an, b1, b2, . . . . . , bm in an
increasing order say c1, c2, c3, c4, c5, . . . . . , cm+n. Plot them on the real line. Draw a curve starting
from right of cm+n along the real line which alternately changes its position at these points. This curve
is known as the wavy curve.
quadratic equation 3
The intervals in which the curve is above the real line will be the intervals for which f(x) is positive
and the intervals in which the curve is below the real line will be the intervals in which f(x) is
negative.

Other way of understanding the same concept

  • Write the inequality in the correct form. One side must be zero and the other must be in product form
  • Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and change them in product of linear forms
  • Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to and mention them on line in increasing form.
  • Perform the sign analysis. and make wave that give the sign of product
  • What is needed acording to that write final answers
  • Problems related to rational inequalities with video solutions

 

CONDITIONS OF COMMON ROOTS FOR TWO QUADRATIC EQUATIONS

For two quadratic equations when one root is common / When both roots are common are given

quadratic equation - common roots-1

HOW TO FIND  RANGE OF A RATIONAL EXPRESSION IN X, WHERE X IS REAL

   Put the given rational expression equal to y and form the quadratic equation in x.

Find the discriminant D of the quadratic equation obtained in step 1.

  Since x is real, therefore, put D   GREATER THAN OR EQUAL TO 0. We get an inequation in y.

      Solve the above inequation for y. The range of y so obtained determines the range attained by the given rational expression

RANGE WITH HELP OF QUADRATIXC EQUATION

 

QUADRATIC EXPRESSION-LOCATION OF ROOTS

The expression ax2 + bx + c is said to be a real quadratic expression in x where a, b, c are real and
a 0. Let f(x) = ax2 + bx + c, where a, b, c R (a 0). f(x) can be re-written as

quadratic expression 1
is the discriminant of the quadratic expression.

Therefore y = f(x) represents a parabola whose axis is parallel to the y-axis, with vertex at
A .

quadratic expression 2
Note that if a > 0, the parabola will be concave upwards and if a < 0 the parabola will be concave
downwards and it depends on the sign of D that the parabola cuts the x-axis at two points
(D > 0), touches the x-axis (D = 0) or never intersects with the x-axis (D< 0).
This gives rise to the following cases:
quadratic expression 3
In this case the parabola always remains concave upwards and above the x-axis

quadratic expression 4

quadratic expression 5
In this case the parabola cuts the x-axis at two points and and remains concave
upwards.

(iv)

quadratic expression 6

a < 0 and D< 0
f(x) < 0xR.
In this case the parabola remains concave downwards and always below the x-axis.

quadratic expression 7
In this case the parabola touches the x-axis and remains concave downwards.

quadratic expression 9

RELATION BETWEEN THE ROOTS OF A POLYNOMIAL EQUATION OF DEGREE N AND coefficient

relation between coefficient nd roots 1

relation between coefficient nd roots 2

IMPORTANT POINTS

A polynomial equation of degree n has n roots (real or imaginary).

If all the coefficients are real then the imaginary roots occur in pairs i.e., number of imaginary roots is always even.

If the degree of a polynomial equation is odd then the number of real roots will also be odd. It follows that at least one of the roots will be real.

Polynomial in x of degree n can be factorized into a product of linear/quadratic form.

I

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Probability for IIT-JEE MAINS /ADVANCED

Probability is very important topic for IIT JEE MAINS / ADVANCED.

Its the topic for which understanding is must .

For JEE MAINS its   weightage is 3 to 4%

and for  JEE ADVANCED its weightage is 11 TO 14 %

probability CONCEPTS 1
probability CONCEPTS 1
probability CONCEPTS 2
probability CONCEPTS 2
PROBABILITY CONCEPTS -7
PROBABILITY CONCEPTS –3
PROBABILITY CONCEPTS -8
PROBABILITY CONCEPTS –4
PROBABILITY CONCEPTS -9
PROBABILITY CONCEPTS –5
PROBABILITY CONCEPTS -10
PROBABILITY CONCEPTS –6
PROBABILITY CONCEPTS -11
PROBABILITY CONCEPTS –7

PROBABILITY CONCEPTS -13
PROBABILITY CONCEPTS -13
PROBABILITY CONCEPTS -14 ONE
PROBABILITY CONCEPTS -14 ONE
PROBABILITY CONCEPTS -14 TWO
PROBABILITY CONCEPTS -14 TWO
PROBABILITY CONCEPTS -15
PROBABILITY CONCEPTS -15
PROBABILITY CONCEPTS -16 ONE
PROBABILITY CONCEPTS -16 ONE
PROBABILITY CONCEPTS -16 TWO
PROBABILITY CONCEPTS -16 TWO

L
LEARN IIT JEE MATHS WITH VIDEO LECTURES OF KAMAL SIR

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Functions-Differential Calculus-Objective-Solved-Problems-Maths-IIT-JEE-MAINS-ADVANCED

To understand this problem of Function  you must know : The concept of composite function

P-Function-Objective-S-130-1
P-Function-Objective-S-130-1

SOLUTION

S-Function-Objective-S-130-1
S-Function-Objective-S-130-1

PROBLEM

To understand this problem of Function  you must know : The concept of periodic functions , order pair relation

P-Function-Objective-S-131-1
P-Function-Objective-S-131-

SOLUTION

S-Function-Objective-S-131-1
S-Function-Objective-S-131-1

PROBLEM

To understand this problem of Function  you must know : The concept of mod  functions and simple trigonometric calculations

P-Function-Objective-S-132-1
P-Function-Objective-S-132-1

SOLUTION

S-Function-Objective-S-132-1
S-Function-Objective-S-132-1

PROBLEM

To understand this problem of Function  you must know : The concept of even and odd functions and how to deal functional equations

P-Function-Objective-S-133-1
P-Function-Objective-S-133-1

SOLUTION

S-Function-Objective-S-133-1
S-Function-Objective-S-133-1

PROBLEM

 

P-Function-Objective-S-134-1
P-Function-Objective-S-134-1

SOLUTION

S-Function-Objective-S-134-1
S-Function-Objective-S-134-1

PROBLEM

To understand this problem of Function  you must know : The concept of cubic functions

P-Function-Objective-S-135-1
P-Function-Objective-S-135-1

SOLUTION

S-Function-Objective-S-135-1
S-Function-Objective-S-135-1

PROBLEM

To understand this problem of Function  you must know : The concept of composite function

P-Function-Objective-S-136-1
P-Function-Objective-S-136-1

SOLUTION

S-Function-Objective-S-136-1
S-Function-Objective-S-136-1

PROBLEM

P-Function-Objective-S-137-1
P-Function-Objective-S-137-1

SOLUTION

S-Function-Objective-S-137-1
S-Function-Objective-S-137-1

PROBLEM

P-Function-Objective-S-138-1
P-Function-Objective-S-138-1

SOLUTION

S-Function-Objective-S-138-1
S-Function-Objective-S-138-1

PROBLEM

To understand this problem of Function  you must know : The concept of AP and how we deal functional equations

P-Function-Objective-S-139-1
P-Function-Objective-S-139-1

SOLUTION

S-Function-Objective-S-139-1
S-Function-Objective-S-139-1

PROBLEM

P-Function-Objective-S-140-1
P-Function-Objective-S-140-1

SOLUTION

S-Function-Objective-S-140-1

PROBLEM

To understand this problem of Function  you must know : The concept of composite function

P-Function-Objective-S-141-1
P-Function-Objective-S-141-1

SOLUTION

S-Function-Objective-S-141-1
S-Function-Objective-S-141-1

PROBLEM

To understand this problem of Function  you must know : The concept of inverse  function

P-Function-Objective-S-142-1
P-Function-Objective-S-142-1

SOLUTION

S-Function-Objective-S-142-1
S-Function-Objective-S-142-1

PROBLEM

To understand this problem of Function  you must know : The concept of peridic function

P-Function-Objective-S-143-1
P-Function-Objective-S-143-1

SOLUTION

S-Function-Objective-S-143-1
S-Function-Objective-S-143-1

PROBLEM

P-Function-Objective-S-144-1
P-Function-Objective-S-144-1

SOLUTION

S-Function-Objective-S-144-1
S-Function-Objective-S-144-1

PROBLEM

P-Function-Objective-S-145-1
P-Function-Objective-S-145-1

SOLUTION

S-Function-Objective-S-145-1
S-Function-Objective-S-145-1

PROBLEM

To understand this problem of Function  you must know : The concept of greatest integer function

P-Function-Objective-S-146-1
P-Function-Objective-S-146-1

SOLUTION

S-Function-Objective-S-146-1
S-Function-Objective-S-146-1

PROBLEM

P-Function-Objective-S-147-1
P-Function-Objective-S-147-1

SOLUTION

S-Function-Objective-S-147-1
S-Function-Objective-S-147-1

PROBLEM

P-Function-Objective-S-148-1
P-Function-Objective-S-148-1

SOLUTION

S-Function-Objective-S-148-1
S-Function-Objective-S-148-1

PROBLEM

P-Function-Objective-S-149-1
P-Function-Objective-S-149-1

SOLUTION

S-Function-Objective-S-149-1
S-Function-Objective-S-149-1
P-Function-Objective-S-150-1
P-Function-Objective-S-150-1

SOLUTION

S-Function-Objective-S-150-1
S-Function-Objective-S-150-1

PROBLEM

P-Function-Objective-S-152-1
P-Function-Objective-S-152-1
S-Function-Objective-S-151-1
S-Function-Objective-S-151-1

PROBLEM

P-Function-Objective-S-153-1
P-Function-Objective-S-153-1

SOLUTION

 

 

PROBLEM

SOLUTION

S-Function-Objective-S-152-1
S-Function-Objective-S-152-1

PROBLEM

SOLUTION

S-Function-Objective-S-155-1
S-Function-Objective-S-155-1

 

PROBLEM

SOLUTION

S-Function-Objective-S-158-1
S-Function-Objective-S-158-1

PROBLEM

SOLUTION

S-Function-Objective-S-159-1
S-Function-Objective-S-159-1

PROBLEM

To understand this problem of Function  you must know : The concept of periodic functions

SOLUTION

S-Function-Objective-S-160-1
S-Function-Objective-S-160-1

PROBLEM

SOLUTION

S-Function-Objective-S-162-1
S-Function-Objective-S-162-1