GATE EE 2020 Set 1 — Question 11
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General Aptitude → Quantitative Aptitude
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Question
is a polynomial on real over real coefficients wherein . Which of the following statements is true?
Solution
Let with real coefficients and .(A) . Choosing makes a root whatever the values of . So (A) is true.(B) has real coefficients for any real , and all three of its roots equal . So (B) is false.(C) Complex roots of a polynomial with real coefficients occur in conjugate pairs, so a cubic always has at least one real root. So (C) is false.(D) Whether all roots are real depends on all four coefficients, through the discriminant(all roots are real iff ). Fixing alone does not settle this: for any chosen , take , and large; then and two roots are complex. So no choice of by itself ensures that all roots are real, and (D) is also a true statement.Both (A) and (D) are correct statements.Answer: (A) or (D)
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