Complex Numbers And Quadratic Equations

Given

%5Calpha%20%2C%5Cbeta%20%2C%5Cgamma

are the roots of the equation

%7B%7Bx%7D%5E%7B3%7D%7D-7x%2B7%3D0

%5Ctherefore

%5CSigma%20%5Calpha%20%3D0%2C%5C%2C%5C%2C%5CSigma%20%5Calpha%20%5Cbeta%20%3D-7%2C%5C%2C%5C%2C%5Calpha%20%5Cbeta%20%5Cgamma%20%3D-7

Now,

%5Cfrac%7B1%7D%7B%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7D%2B%5Cfrac%7B1%7D%7B%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%7D%2B%5Cfrac%7B1%7D%7B%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D%7D%3D%5Cfrac%7B%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7D%7B%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D%7D

%3D%5Cfrac%7B%5CSigma%20%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%7D%7B%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D%7D

..(i)

%5CSigma%20%5Calpha%20%5Cbeta%20%5C%2C%5CSigma%20%5Calpha%20%5Cbeta%20%5C%2C%5CSigma%20%5Calpha%20%5Cbeta%20%5C%2C%5CSigma%20%5Calpha%20%5Cbeta%20%3D%7B%7B(%5CSigma%20%5Calpha%20%5Cbeta%20)%7D%5E%7B2%7D%7D.%7B%7B(%5CSigma%20%5Calpha%20%5Cbeta%20)%7D%5E%7B2%7D%7D

%5CRightarrow

%7B%7B(-7)%7D%5E%7B4%7D%7D%3D(%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%2B2%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%5Cbeta%20%5Cgamma

%2B2%5Calpha%20%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%5Cgamma%20%2B2%5Calpha%20%5Cbeta%20%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D)

(%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%2B2%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B2%5Calpha%20%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%5Cgamma%20%2B2%5Calpha%20%5Cbeta%20%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D)

%3D%5B%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%2B2%5Calpha%20%5Cbeta%20%5Cgamma%20(%5Calpha%20%2B%5Cbeta%20%2B%5Cgamma%20)%5D

%5B%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%2B2%5Calpha%20%5Cbeta%20%5Cgamma%20(%5Calpha%20%2B%5Cbeta%20%2B%5Cgamma%20)%5D

%3D(%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D)

(%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D)

(%5Cbecause%20%5C%2C%5C%2C%5CSigma%20%5Calpha%20%3D%5Calpha%20%2B%5Cbeta%20%2B%5Cgamma%20%3D0)

%3D%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%2B2%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D

%2B2%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%2B2%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D

%3D%5CSigma%20%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%2B2%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D(%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%2B%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D)

%3D%5CSigma%20%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%2B2%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%5B%7B%7B(%5CSigma%20%5Calpha%20)%7D%5E%7B2%7D%7D-2%5CSigma%20%5Calpha%20%5Cbeta%20%5D

%3D%5CSigma%20%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%2B2%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B2%7D%7D%7B%7B%5Cgamma%20%7D%5E%7B2%7D%7D%5B0-2%5Ctimes%20(-7)%5D

%5CRightarrow

%7B%7B(-7)%7D%5E%7B4%7D%7D%3D%5CSigma%20%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%2B2%7B%7B(-7)%7D%5E%7B2%7D%7D(2%5Ctimes%207)

%5CRightarrow

%5CSigma%20%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%3D%7B%7B(-7)%7D%5E%7B4%7D%7D%2B4%7B%7B(-7)%7D%5E%7B3%7D%7D

%5CRightarrow

%5CSigma%20%7B%7B%5Calpha%20%7D%5E%7B2%7D%7D%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%3D%7B%7B(-7)%7D%5E%7B3%7D%7D(-7%2B4)%3D-3%7B%7B(-7)%7D%5E%7B3%7D%7D

On putting this value in E (i), we get

%5Cfrac%7B1%7D%7B%7B%7B%5Calpha%20%7D%5E%7B4%7D%7D%7D%2B%5Cfrac%7B1%7D%7B%7B%7B%5Cbeta%20%7D%5E%7B4%7D%7D%7D%2B%5Cfrac%7B1%7D%7B%7B%7B%5Cgamma%20%7D%5E%7B4%7D%7D%7D%3D%5Cfrac%7B-3%7B%7B(-7)%7D%5E%7B3%7D%7D%7D%7B%7B%7B(-7)%7D%5E%7B4%7D%7D%7D%3D%5Cfrac%7B-3%7D%7B-7%7D%3D%5Cfrac%7B3%7D%7B7%7D

$%5Cfrac%7Bi%2B1%7D%7B%5Ccos%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D-i%5Csin%20%5Cfrac%7B%5Cpi%20%7D%7B4%7D%7D%3D%5Cfrac%7Bi%2B1%7D%7B%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D(1-i)%7D%7D%5Ctimes%20%5Cfrac%7B1%2Bi%7D%7B1%2Bi%7D$
$%3D%5Cfrac%7B%5Csqrt%7B2%7D%7B%7B(1%2Bi)%7D%5E%7B2%7D%7D%7D%7B1-%7B%7Bi%7D%5E%7B2%7D%7D%7D$
$%3D%5Cfrac%7B%5Csqrt%7B2%7D(1-1%2B2i)%7D%7B1%2B1%7D$
$%3D%5Cfrac%7B%5Csqrt%7B2%7D.2i%7D%7B2%7D%3D%5Csqrt%7B2%7Di$
$%3D%5Csqrt%7B2%7D%5Cleft(%20%5Ccos%20%5C%2C%5Cfrac%7B%5Cpi%20%7D%7B2%7D%2Bi%5C%2C%5Csin%20%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%5Cright)$

For a single coordinate point, the formula allows us to produce an endless number of polar coordinates. The equation is expressed as follows:

(r, θ + 2πn) or (-r, θ + (2n+1)π)

n is an integer in this situation.

The value of will be positive if measured in an anticlockwise direction and negative if measured in a clockwise direction. The value of r will also be positive if you lay off the terminal side, but negative if you lay off the prolongation via the origin from the terminal side. The starting side of an angle is referred to as such, while the terminal side is the ray on which the angle measurement terminates.

High-Yield Trend

Chapter Questions
12 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

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Official Solution

Official Solution

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Official Solution

Official Solution

About Complex Numbers And Quadratic Equations - JKCET

Frequently Asked Questions

Why focus on Complex Numbers And Quadratic Equations PYQs?

How to best use this analysis?

Complex Numbers And Quadratic Equations

High-Yield Trend

Chapter Questions 12 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Complex Numbers And Quadratic Equations - JKCET

Frequently Asked Questions

Why focus on Complex Numbers And Quadratic Equations PYQs?

How to best use this analysis?

Chapter Questions
12 MCQs