JEE Exam > JEE Questions > Which of the following statements are true wh...
Start Learning for Free
Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0?
Most Upvoted Answer
Which of the following statements are true where ϕ(x) is a polynomial....
True Statements Regarding Roots of Polynomials
There are four statements given in the question related to the roots of a polynomial function ϕ(x). Let's analyze each statement one by one to determine which ones are true.
A. Between any two roots of ϕ(x)=0 lies at least one root of ϕ(x)γϕ(x)=0.
This statement is not true in general. For example, consider the polynomial ϕ(x) = x^2 - 1. It has two roots x = 1 and x = -1. However, the function ϕ(x)γϕ(x) = x^2 - 1 is positive for all x except x = 0. Therefore, there is no root of ϕ(x)γϕ(x) between x = 1 and x = -1.
B. Between any two roots of ϕ(x)=0 lies at least one root of xϕ(x)γϕ(x)=0.
This statement is true. Suppose ϕ(x) has two roots x1 and x2, where x1 < x2.="" then,="" by="" the="" intermediate="" value="" theorem,="" there="" exists="" a="" value="" c="" between="" x1="" and="" x2="" such="" that="" ϕ(c)="0." multiplying="" both="" sides="" of="" this="" equation="" by="" c,="" we="" get="" cϕ(c)="0." therefore,="" there="" exists="" a="" root="" of="" xϕ(x)γϕ(x)="" between="" x1="" and="" />
C. Between any two roots of ϕ(x)=0 lies at least one root of (x2+1)ϕ(x).
This statement is not true in general. For example, consider the polynomial ϕ(x) = x^2 - 1. It has two roots x = 1 and x = -1. However, the function (x^2+1)ϕ(x) is positive for all x. Therefore, there is no root of (x^2+1)ϕ(x) between x = 1 and x = -1.
D. Between any two roots of ϕ(x)=0 lies at least one root of ϕ'(x)xϕ(x)=0.
This statement is true. Suppose ϕ(x) has two roots x1 and x2, where x1 < x2.="" then,="" by="" rolle's="" theorem,="" there="" exists="" a="" value="" c="" between="" x1="" and="" x2="" such="" that="" ϕ'(c)="0." multiplying="" both="" sides="" of="" this="" equation="" by="" cϕ(c),="" we="" get="" ϕ'(c)cϕ(c)="0." therefore,="" there="" exists="" a="" root="" of="" ϕ'(x)xϕ(x)="" between="" x1="" and="" />
In summary, the true statements are:
- Between any two roots of ϕ(x)=0 lies at least one root of xϕ(x)γϕ(x)=0.
- Between any two roots of ϕ(x)=0 lies at least one root of ϕ'(x)xϕ(x)=0.
There are four statements given in the question related to the roots of a polynomial function ϕ(x). Let's analyze each statement one by one to determine which ones are true.
A. Between any two roots of ϕ(x)=0 lies at least one root of ϕ(x)γϕ(x)=0.
This statement is not true in general. For example, consider the polynomial ϕ(x) = x^2 - 1. It has two roots x = 1 and x = -1. However, the function ϕ(x)γϕ(x) = x^2 - 1 is positive for all x except x = 0. Therefore, there is no root of ϕ(x)γϕ(x) between x = 1 and x = -1.
B. Between any two roots of ϕ(x)=0 lies at least one root of xϕ(x)γϕ(x)=0.
This statement is true. Suppose ϕ(x) has two roots x1 and x2, where x1 < x2.="" then,="" by="" the="" intermediate="" value="" theorem,="" there="" exists="" a="" value="" c="" between="" x1="" and="" x2="" such="" that="" ϕ(c)="0." multiplying="" both="" sides="" of="" this="" equation="" by="" c,="" we="" get="" cϕ(c)="0." therefore,="" there="" exists="" a="" root="" of="" xϕ(x)γϕ(x)="" between="" x1="" and="" />
C. Between any two roots of ϕ(x)=0 lies at least one root of (x2+1)ϕ(x).
This statement is not true in general. For example, consider the polynomial ϕ(x) = x^2 - 1. It has two roots x = 1 and x = -1. However, the function (x^2+1)ϕ(x) is positive for all x. Therefore, there is no root of (x^2+1)ϕ(x) between x = 1 and x = -1.
D. Between any two roots of ϕ(x)=0 lies at least one root of ϕ'(x)xϕ(x)=0.
This statement is true. Suppose ϕ(x) has two roots x1 and x2, where x1 < x2.="" then,="" by="" rolle's="" theorem,="" there="" exists="" a="" value="" c="" between="" x1="" and="" x2="" such="" that="" ϕ'(c)="0." multiplying="" both="" sides="" of="" this="" equation="" by="" cϕ(c),="" we="" get="" ϕ'(c)cϕ(c)="0." therefore,="" there="" exists="" a="" root="" of="" ϕ'(x)xϕ(x)="" between="" x1="" and="" />
In summary, the true statements are:
- Between any two roots of ϕ(x)=0 lies at least one root of xϕ(x)γϕ(x)=0.
- Between any two roots of ϕ(x)=0 lies at least one root of ϕ'(x)xϕ(x)=0.
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
Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0?
Question Description
Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0?.
Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0?.
Solutions for Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? 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 Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? defined & explained in the simplest way possible. Besides giving the explanation of
Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0?, a detailed solution for Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? has been provided alongside types of Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? theory, EduRev gives you an
ample number of questions to practice Which of the following statements are true where ϕ(x) is a polynomial. A. between any two roots of ϕ(x)=0 lies atleast one root of ϕ(x) γϕ(x)=0 B. between any two roots of ϕ(x)=0 lies atleast one root of xϕ(x) γϕ(x)=0 C. between any two roots of ϕ(x)=0 lies atleast one root of (x2 1)ϕ(x) ϕ(x)=0 D. between any two roots of ϕ(x)=0 lies atleast one root of ϕ'(x) xϕ(x)=0? 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
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup