JEE Exam > JEE Questions > If alpha beta gamma are the roots of the equa...
Start Learning for Free
If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?
Most Upvoted Answer
If alpha beta gamma are the roots of the equation x cube Pnot X squa...
Solution:
Given equation: x^3 + Px^2 + Qx + R = 0
Let's assume that alpha, beta, and gamma are its roots.
Sum of roots:
α + β + γ = -P
Product of roots:
αβγ = -R
Now, let's find the value of 1 - (α^2) * 1 - (β^2) * 1 - (γ^2)
1 - (α^2) * 1 - (β^2) * 1 - (γ^2) = (1 - α^2) * (1 - β^2) * (1 - γ^2)
We can write (1 - α^2) as (1 + α) * (1 - α), and similarly for (1 - β^2) and (1 - γ^2).
So,
(1 - α^2) * (1 - β^2) * (1 - γ^2) = (1 + α) * (1 - α) * (1 + β) * (1 - β) * (1 + γ) * (1 - γ)
= [(1 + α) * (1 + β) * (1 + γ)] * [(1 - α) * (1 - β) * (1 - γ)]
Now, let's find the value of (1 + α) * (1 + β) * (1 + γ) and (1 - α) * (1 - β) * (1 - γ)
(1 + α) * (1 + β) * (1 + γ) = 1 + (α + β + γ) + (αβ + βγ + γα) + αβγ
= 1 - P + Q - R
Similarly,
(1 - α) * (1 - β) * (1 - γ) = 1 - (α + β + γ) + (αβ + βγ + γα) - αβγ
= 1 + P + Q + R
Now,
(1 - α^2) * (1 - β^2) * (1 - γ^2) = [(1 - α) * (1 - β) * (1 - γ)] * [(1 + α) * (1 + β) * (1 + γ)]
= (1 + P + Q + R) * (1 - P + Q - R)
= (1 - R^2)
Therefore, 1 - (α^2) * 1 - (β^2) * 1 - (γ^2) = 1 - R^2
Hence, the value of 1 - (α^2) * 1 - (β^2) * 1 - (γ^2) is equal to 1 - R^2.
Given equation: x^3 + Px^2 + Qx + R = 0
Let's assume that alpha, beta, and gamma are its roots.
Sum of roots:
α + β + γ = -P
Product of roots:
αβγ = -R
Now, let's find the value of 1 - (α^2) * 1 - (β^2) * 1 - (γ^2)
1 - (α^2) * 1 - (β^2) * 1 - (γ^2) = (1 - α^2) * (1 - β^2) * (1 - γ^2)
We can write (1 - α^2) as (1 + α) * (1 - α), and similarly for (1 - β^2) and (1 - γ^2).
So,
(1 - α^2) * (1 - β^2) * (1 - γ^2) = (1 + α) * (1 - α) * (1 + β) * (1 - β) * (1 + γ) * (1 - γ)
= [(1 + α) * (1 + β) * (1 + γ)] * [(1 - α) * (1 - β) * (1 - γ)]
Now, let's find the value of (1 + α) * (1 + β) * (1 + γ) and (1 - α) * (1 - β) * (1 - γ)
(1 + α) * (1 + β) * (1 + γ) = 1 + (α + β + γ) + (αβ + βγ + γα) + αβγ
= 1 - P + Q - R
Similarly,
(1 - α) * (1 - β) * (1 - γ) = 1 - (α + β + γ) + (αβ + βγ + γα) - αβγ
= 1 + P + Q + R
Now,
(1 - α^2) * (1 - β^2) * (1 - γ^2) = [(1 - α) * (1 - β) * (1 - γ)] * [(1 + α) * (1 + β) * (1 + γ)]
= (1 + P + Q + R) * (1 - P + Q - R)
= (1 - R^2)
Therefore, 1 - (α^2) * 1 - (β^2) * 1 - (γ^2) = 1 - R^2
Hence, the value of 1 - (α^2) * 1 - (β^2) * 1 - (γ^2) is equal to 1 - R^2.
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?
Question Description
If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?.
If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?.
Solutions for If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? defined & explained in the simplest way possible. Besides giving the explanation of
If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?, a detailed solution for If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? has been provided alongside types of If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? theory, EduRev gives you an
ample number of questions to practice If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup