Question

If sum of values of $x$ in $[3,\sqrt{11})$ for which fractional part of $x$ is equal to fractional part of $x^2$ is $\frac{m+\sqrt{n}}{2},m,n\in N$, then $(m+n)$ equals

• A. 32
• B. 35
• C. 36
• D. 38

Answer 1

Answer: (C)

$x\in [\sqrt{9},\sqrt{11})$
$\{x\}=\left\{x^2\right\}$
$x-[x]=x^2-\left[x^2\right]$
$x-x^2=[x]-\left[x^2\right]$
$\text{ Case-I: }\text{ when }x\in [\sqrt{9},\sqrt{10})$
$x-x^2=3-9=-6\Rightarrow x^2-x-6=0\Rightarrow (x-3)(x+2)=0$
$x=\{3,-2\}$
$x=-2\text{ is rejected }$
$\therefore x=3$
$\text{ Case-II: }\text{ when }x\in [\sqrt{10},\sqrt{11})$
$x-x^2=3-10=-7\Rightarrow x^2-x-7=0\Rightarrow x=\frac{1+\sqrt{29}}{2}$
$\text{ sum }=3+\frac{1+\sqrt{29}}{2}=\frac{7+\sqrt{29}}{2},]$