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GATE 2020 EE – Question 11

Engineering Mathematics · Linear Algebra: Systems of linear equations · 1 mark · Multiple choice

$ax^3+bx^2+cx+d$ is a polynomial on real $x$ over real coefficients $a,b,c,d$ wherein $a\ne0$. Which of the following statements is true?

  1. $d$ can be chosen to ensure that $x=0$ is a root for any given set $a,b,c$.
  2. No choice of coefficients can make all roots identical.
  3. $a,b,c,d$ can be chosen to ensure that all roots are complex.
  4. $c$ alone cannot ensure that all roots are real.

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Show answer and explanation

Correct answer: (A) $d$ can be chosen to ensure that $x=0$ is a root for any given set $a,b,c$.

Explanation

Choosing $d=0$ makes $x=0$ a root for any $a,b,c$, so A is true. A real cubic always has at least one real root (so C is false), and identical roots are possible (for example $(x-1)^3$), so B is false. D is also a true statement, and the official key accepts both A and D.