The equation x 2 - 4 x + [ x ] + 3 = x [ x ] , where [x] denotes the greatest integer function, has [2023]

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Q.
The equation $x^{2} - 4 x + [x] + 3 = x [x],$ where [x] denotes the greatest integer function, has [2023]

1 no solution

2 exactly two solutions in $(- \infty, \infty)$

3 a unique solution in $(- \infty, 1)$

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

Ans.
(4)

$x^{2} - 4 x + [x] + 3 = x [x]$

$\Rightarrow x^{2} - x [x] - (x - [x]) + 3 (x - 1) = 0$

$\Rightarrow x (x - [x]) - 1 (x - [x]) - 3 (x - 1) = 0$

$\Rightarrow (x - 1) (x - [x]) - 3 (x - 1) = 0$

$\Rightarrow (x - 1) (x - [x] - 3) = 0$

$\Rightarrow (x - 1) ({x} - 3) = 0 \Rightarrow x = 1 as {x} \neq 3$

Solution

Q.
The equation $x^{2} - 4 x + [x] + 3 = x [x],$ where [x] denotes the greatest integer function, has [2023]

1 no solution

2 exactly two solutions in $(- \infty, \infty)$

3 a unique solution in $(- \infty, 1)$

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

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Solution

Q. The equation x2-4x+[x]+3=x[x], where [x] denotes the greatest integer function, has [2023]

1 no solution

2 exactly two solutions in (-∞,∞)

3 a unique solution in (-∞,1)

4 a unique solution in (-∞,∞)

Ans. (4) x2-4x+[x]+3=x[x] ⇒x2-x[x]-(x-[x])+3(x-1)=0 ⇒x(x-[x])-1(x-[x])-3(x-1)=0 ⇒(x-1)(x-[x])-3(x-1)=0 ⇒(x-1)(x-[x]-3)=0 ⇒(x-1)({x}-3)=0⇒x=1 as {x}≠3

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Q.
The equation $x^{2} - 4 x + [x] + 3 = x [x],$ where [x] denotes the greatest integer function, has [2023]

2 exactly two solutions in $(- \infty, \infty)$

3 a unique solution in $(- \infty, 1)$

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