cmi-entrance 2024 Q6

cmi-entrance · India · ugmath 2 marks Sign Change & Interval Methods
Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Find an integer $n$ with the least possible absolute value such that $p(x)$ has a real root between $n$ and $n+1$. Write this number along with your reason as per the given instruction. [2 points]
Instruction for (6): Write two numbers separated by a comma: value of $n$, number of the theorem below that justifies this answer. E.g., if you think that $n=5$ because of the factor theorem, then type $\mathbf{5,1}$ as your answer with no space, full stop or any other punctuation.
  1. Factor theorem
  2. Mean value theorem
  3. Intermediate value theorem
  4. Fundamental theorem of algebra
  5. Fundamental theorem of calculus 
Consider the polynomial
$$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$
Find an integer $n$ with the least possible absolute value such that $p(x)$ has a real root between $n$ and $n+1$. Write this number along with your reason as per the given instruction. [2 points]

Instruction for (6): Write two numbers separated by a comma: value of $n$, number of the theorem below that justifies this answer. E.g., if you think that $n=5$ because of the factor theorem, then type $\mathbf{5,1}$ as your answer with no space, full stop or any other punctuation.
\begin{enumerate}
  \item Factor theorem
  \item Mean value theorem
  \item Intermediate value theorem
  \item Fundamental theorem of algebra
  \item Fundamental theorem of calculus
\end{enumerate}
Paper Questions
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