KVPY Sample Paper KVPY Stream-SX Model Paper-18

Let \[F\left( x \right)\] be a functions defined by \[F\left( x \right)=x-\left[ x \right],0\ne x\in R,\] where \[\left( x \right)\]is the greatest integer less than or equal to x, then the number of solutions of \[f\left( x \right)+F\left( \frac{1}{x} \right)=1\] is

A) 0

B) infinite

C) 1

D) 2

Correct Answer: B

Solution :

Given \[F\left( x \right)=x-\left[ x \right]\]so, \[F\left( x \right)+F\left( \frac{1}{x} \right)=1\]

\[\Rightarrow x-\left[ x \right]+\frac{1}{x}-\left[ \frac{1}{x} \right]=1\]

\[\Rightarrow x+\frac{1}{x}=\left[ x \right]+\left[ \frac{1}{x} \right]+1..\left( 1 \right)\]

\[\because \]RHS is an integer.

\[\therefore \]LHS must be integer.

Let \[x+\frac{1}{x}=\left[ x \right]+\frac{1}{x}+1=k,k\in \operatorname{I},\]the equation is \[x+\frac{1}{x}=k\Rightarrow {{x}^{2}}-kx+\operatorname{I}=0,\]

For real \[x,{{k}^{2}}-4\ge 0\Rightarrow k\le -2\,or\,k\ge 2\]and \[k\in \]I, we can get infinitety many values of k and we get solution for each value of\[k\].

Report Error