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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?
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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.
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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?
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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?.
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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
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