GATE EE 2020 Set 1 — Question 11
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Question
is a polynomial on real over real coefficients wherein . Which of the following statements is true?
- A.can be chosen to ensure that is a root for any given set .
- B.No choice of coefficients can make all roots identical.
- C.can be chosen to ensure that all roots are complex.
- D.alone cannot ensure that all roots are real.
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Correct answer
(A;D)
Solution
To determine which statement is true for the cubic polynomial with and real coefficients:
By choosing , becomes a root regardless of the values of . Thus, statement (A) is true.
1.Statement (A): For to be a root, must be .
. By choosing , becomes a root regardless of the values of . Thus, statement (A) is true.
2.Statement (B): A cubic polynomial can have all roots identical. For example, if , all three roots are . If is real, the coefficients are real. Thus, statement (B) is false.
3.Statement (C): For a polynomial with real coefficients, complex roots must occur in conjugate pairs. A cubic polynomial (degree 3) must have at least one real root because complex roots come in pairs (0 or 2 complex roots). It is impossible for all roots of a real cubic to be complex. Thus, statement (C) is false.
4.Statement (D): The nature of the roots (real or complex) of a cubic polynomial depends on its discriminant . Since the condition for all roots to be real involves all coefficients , the coefficient alone cannot ensure that all roots are real without considering the other coefficients. Thus, statement (D) is also true.
Since both (A) and (D) are correct statements, the official GATE answer key accepts either option as correct.Continue learning with Success Tracker
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