Question

Consider the functions $f\left(x\right)$ and $g\left(x\right)$, both defined from $R\to R$ and are defined as $f\left(x\right)=2x-x^2$ and $g\left(x\right)=x^n$ where $n\in N$. If the area between $f\left(x\right)$ and $g\left(x\right)$ in first quadrant is $1/2$ then $n$ is not a divisor of :

• A. $12$
• B. $15$
• C. $20$
• D. $30$

Answer 1

The correct option is A $12$

$Area=\underset{a}{\overset{b}{\int }}\left(uppercurve\right)-\left(lowercurve\right)dx$
$=\underset{0}{\overset{1}{\int }}(f\left(x\right)-g\left(x\right))dx$
$=\underset{0}{\overset{1}{\int }}(2x-x^2-x^n)dx$
$={[x-\frac{x^3}{3}-\frac{x^{n+1}}{n+1}]}_0^1$
$\frac{1}{2}=1-\frac{1}{3}-\frac{1}{n+1}$
$\Rightarrow \frac{1}{2}=\frac{2}{3}-\frac{1}{n+1}$
$\Rightarrow \frac{1}{n+1}=\frac{2}{3}-\frac{1}{2}$
$\Rightarrow \frac{1}{n+1}=\frac{1}{6}$
$\Rightarrow n+1=6$
$n=5$
$Hencenisnotadivisorof12$