Best Mathematics contents for all engineering entrance exams

Theory of equation // quadratic equation for jee mains & advance

2019-12-20T13:35:00.002+05:30

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Important question for jee mains 2020

solution

some other important questions

Foundamental Theorem of Algebra ( non real coiffcient of polynomials)

The fundamental theorem states that every non-constant single variable polynomial with complex coefficients has at least one complex roo

Thefundamental theorem of algebra states that every non-constant, single- variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero as its coefficient.

The fundamental theorem is also stated as follows: every non-zero, single-variable, degree n

polynomial withcomplex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division.:

multiplicity the number of values for which a given condition holds

Some polynomials with real coefficients, like

x^{2} + 1

have no real zeros. As it turns out, every polynomial with a complex coefficient has a complex zero. Every polynomial of odd degree with real coefficients has a real zero

The Complex Conjugate Root Theorem

The complex conjugate root theorem says that if a complex number

a + b i

is a zero of a polynomial with real coefficients, then its complex conjugate

a - b i

is also a zero of this polynomial ( condition is that all coiffcient of polynomial are real

Now let our real polynomial admits a root

a + b i

with

b \neq 0

. By dividing with the real polynomial

(x - (a + b i)) (x - (a - b i)) = (x - a)^{2} + b^{2}

, we obtain another real polynomial, for which the complex conjugate root theorem again applies. In this way, we see that the total multiplicity of non-real complex roots of a polynomial with real coefficients must always be even.

This last remark, together with the alternative statement of the fundamental theorem of algebra, tells us that the parity of the real roots (counted with multiplicity) of a polynomial with real coefficients must be the same as the parity of the degree of said polynomial. Therefore, a polynomial of even degree admits an even number of real roots, and a polynomial of odd degree admits an odd number of real roots (counted with multiplicity). In particular, every polynomial of odd degree with real coefficients admits at least one real root.

then one example comes that if we take an example of a polynomial with non-real coefficients, but with real roots.

A trivial example to provide is

P (x) = i x - i

, which has non-real coefficients but a real root

x = 1

. However, I feel that this is a ' cheat example' if that makes sense because the coefficients just cancel out anyway Is there a 'less cheating' example out there
write your comment in commentbox

electrostatics full

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notes

Charge
charge is the property of matter that is responsible for electrical phenomena by which two objects attract or repel each other
There is two type of charges
1(+)Ve positive charge
2 (–)ve negative charge
electrostatics it is branch of physics which deals phenomena and property of electric charges at rest

The substances are divided into four categories

Conductors, insulator, semiconductor , dielectrics

charge
charge is the property of matter that is responsible for electrical phenomena
charge is quantized q = ne

Coulomb’s Law:- according to coulombs law the electrostatic force or repulsion b/w two charged body is directly proportional to product of their charges and inversely proportional to the square of the distsnce between thgeem

Coulomb's law in vector form

principle of superposition

according to principle of superposition total force on charge due to another number of other charges at rest is the vector sum of all the forces due to other charges

Electric field

Electric field line

an electric field line defined as a path of straight or curved electric field

properties of electric field line

1 electric field line is continuous but do not form closed loops

2 tangent on the curve of electric field line is direction of electric field

3 electric field line is always normal to surface of conductor

4 the magnitude of the field is represented by density of field line

5 electric field lines can not intersect each other

Electric field on the axis of charged ring

Electric Dipole .

an electric dipole consist of a pair of equal and opposite charges separated by some small distance

It is vector quantity
It'ssi unit is c/m
It is denoted by p , P =Qx2a

Electric field at axial position of dipole

Electric field at equitorial position iof dipole

Electric field at any point from of dipole

Torques

Potential energy .

Playing with graph

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One of the best book to understand the graphical problems diff type of f(x)

★Alzebric function .

Polynomials, rational, irrational, modulus,Signum, greatest integers ★Trigonometric functions

★Exponential functions

★Logarithmic functionsG

★Geometrical curves.

★ITF(inverse trigonometric function).

Basic Trigonometry

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Trigonometr

Integration for board exams

2019-07-11T17:43:00.001+05:30

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Complex number

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Complex number

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Quadratic equation

2019-07-10T14:31:00.001+05:30

Quadratic equation

Sequence and series R D Objective #iit #math

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Sequence and series

HC verma vol 2 physics in PDF free

2019-07-01T07:13:00.003+05:30

electrostatics

Gauss law

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Gauss law and capacitors

1 Electrostatics 1

2 Gauss law ( electrostatics 2 )

3 Capacitors.

Chemistry ch2 Solution notes pdf

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Definite integrals for entrance exams pdf

2019-06-26T10:27:00.000+05:30

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👉 Application of integration 1 .

👉AOI2

👉Aoi3.

👉 2. 1Definite integrals

👉 2.2Definite integrals

👉 2.3definite integrals

👉 1.1Basic indefinite integrals

👉 1.2 indefinite integrals

👉 1.3 indefiniteintegrals

Indefinite integrals I A Moron classic series

2019-06-25T16:00:00.000+05:30

Indefinite integrals

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👉 Indefinite integral1

👉 Indefinite integrals 2

👉 Indefinite integrals 3

Electrostatics HC verma in PDF

2019-06-24T13:33:00.001+05:30

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Kinematics 1

Kinematics/laws of motion/rotational motion dc pandey in pdf

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1 Kinematics 1.

Solution

2 Kinematics 2 .

3 Laws of motion.

4 Work energy power.

5 Circular motion .

6 Center of mass.

7 Rotation we .

All the pages are in reverse order

Fluid mechanics

2019-06-20T10:46:00.001+05:30

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1 Fluid mechanics

2 Mechanical properties of matter

3 Heat and temperature

4 Kinetic theory of gases

5 Calorimetry

6 Lawof thermodynamics

7 Heat transfer

8 Heat transfer 2

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Algebra of Vectors R D objective

2019-06-18T16:08:00.002+05:30

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Vector triple product

Physics(IE Irodov) mechanics

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IEIrodov.👉👉Mechanics

differentiatial equation

2019-06-15T19:52:00.002+05:30

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theory
DIFFERENTIATIAL EQUATION

Application of integration

2019-06-14T15:27:00.000+05:30

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Application of Integration with solutio

Electrostatics DC pandey in pdf

2019-06-13T09:03:00.000+05:30

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Indefinite integrals

2019-06-09T19:46:00.000+05:30

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Differentiation (all types of questions )

2019-06-09T00:40:00.000+05:30

Application of Derivative

2019-06-05T14:43:00.001+05:30

Application of Derivatives. The derivative is defined as something which is based on some other thing. In Mathematics, Derivative is an expression that gives rate of change of a function with respect to independent variable. it have various applications in Mathematics, Science, and Engineering.

limit, Continuity and differentiability

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