KVPY Sample Paper KVPY Stream-SX Model Paper-23

Number of values of x satisfying the equations \[5\{x\}=x+[x]\] and \[[x]-\{x\}=\frac{1}{2}\] is

A) 1

B) 2

C) 3

D) 4

Correct Answer: A

Solution :

We have,
\[5\{x\}=x+[x]\]	? (i)
\[[x]-\{x\}=\frac{1}{2}\]	? (ii)
\[\therefore \] \[0\le \{0\}<1\] \[\Rightarrow \] \[[x]-\frac{1}{2}=\{x\}\]
\[0\le [x]-\frac{1}{2}<1\]\[\Rightarrow \]\[[x]=1,\{x\}=\frac{1}{2}\]
\[\therefore \]\[\frac{5}{2}=x+1\]\[\Rightarrow \]\[x=\frac{3}{2}\]only one solution