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TIFR-2016-Maths-A: 6

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Let $V$ be the vector space over $\mathbb{R}$ consisting of polynomials $p\left ( t \right )$ over $\mathbb{R}$ of degree less than or equal to $4$. Let $D:V\rightarrow V$ be the linear operator that takes any polynomial $p\left ( t \right )$ to its derivative ${p}'\left ( t \right )$. Then the characteristic polynomial $f\left ( x\right )$ of $D$ is 

  1. $x^{4}$
  2. $x^{5}$
  3. $x^{3}\left ( x-1 \right )$
  4. $x^{4}\left ( x-1 \right ).$
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