Paragraph: Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3$. Let $s$ be the sum of all distinct real roots…

Paragraph: Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.Question: The area bounded by the curve $y=f(x)$ and the lines $x=0, y=0$ and $x=t$, lies in the interval
  1. $\left(\frac{3}{4}, 3\right)$
  2. $\left(\frac{21}{64}, \frac{11}{16}\right)$
  3. $(9,10)$
  4. $\left(0, \frac{21}{64}\right)$

Solution

$\int_0^{1 / 2} f(x) d x < \int_0^t f(x) d x < \int_0^{3 / 4} f(x) d x$ Now, $\int f(x) d x$
$ \begin{gathered} =\int\left(1+2 x+3 x^2+4 x^3\right) d x \\ =x+x^2+x^3+x^4 \\ \Rightarrow \quad \int_0^{1 / 2} f(x) d x=\frac{15}{16}>\frac{3}{4} \\ \int_0^{3 / 4} f(x) d x=\frac{530}{256} < 3 \end{gathered} $

Asked in: JEE Advanced 2010 (Paper 2)

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