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 } \left.=3+\frac{1+\sqrt{29}}{2}=\frac{7+\sqrt{29}}{2}, \right] $