Functions Ans 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 $(-\infty, \infty)$