The equation x^2-4 x+[x]+3=x[x], where [x] denotes the greatest i… - Vidyadip

MCQ

The equation $x^2-4 x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:

A
exactly two solutions in $(-\infty, \infty)$
B
no solution
C
a unique solution in $(-\infty, 1)$
✓
a unique solution in $(-\infty, \infty)$

✓

Answer

Correct option: D.

a unique solution in $(-\infty, \infty)$

d
$x^2-4 x+[x]+3=x[x]$

$\Rightarrow x^2-4 x+3=x[x]-[x]$

$\Rightarrow(x-1)(x-3)=[x] .(x-1)$

$\Rightarrow x=1 \text { or } x-3=[x]$

$\Rightarrow x-[x]=3$

$\Rightarrow\{x\}=3 \text { (Not Possible) }$

Only one solution $x=1$ in $(-\infty, \infty)$

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