Functions Question 1
Question 1 - 24 January - Shift 1
The equation $x^{2}-4 x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:
(1) exactly two solutions in $(-\infty, \infty)$
(2) no solution
(3) a unique solution in $(-\infty, 1)$
(4) a unique solution in $(-\infty, \infty)$
Show Answer
Answer: (4)
Solution:
Formula: Greatest Integer function, Properties of Fractional part function
$x^{2}-4 x+[x]+3=x[x]$
$\Rightarrow x^{2}-4 x+3=x[x]-[x]$
$\Rightarrow(x-1)(x-3)=[x] .(x-1)$
$\Rightarrow x=1$ or $x-3=[x]$
$\Rightarrow x-[x]=3$
$\Rightarrow{x}=3$ (Not Possible)
Only one solution $x=1$ in $(-\infty, \infty)$
