GATE EE 2021 Set 1 — Question 12
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Engineering Mathematics → Complex Variables
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Question
Let , where is a complex number. Which one of the following is true?
Correct answer
(D) All the roots cannot be real
Solution
Consider the polynomial .
1.Option (A): For to hold for all , the coefficients of the polynomial must be real. Since the coefficients and are complex, this statement is false.
2.Option (B): The sum of the roots of a polynomial is given by . Here, the sum is , which is not a real number. False.
3.Option (C): Complex roots come in conjugate pairs only if the coefficients of the polynomial are real. Since the coefficients are complex, this is false.
4.Option (D): If all roots were real, their sum would necessarily be a real number. Since the sum of the roots is , which is complex, at least one root must be complex. Therefore, all roots cannot be real. This is true.
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