Number of solutions of the equation 3 x+ x =4[x] will be (where [] represent greatest integer and fractional part respectively)

Question

Number of solutions of the equation 3 x+ x =4[x] will be (where [] represent greatest integer and fractional part respectively) (A) 0 (B) 2 (C) 4 (D) ∞

Tardigrade Admin · Accepted Answer

Θ 3[x]+3 x + x =4[x] ⇒[ x ]=4 x Θ 0 ≤ x <1 ⇒ 0 ≤[x]<4 ⇒[ x ]=0,1,2,3 ⇒ x =0, (1/4), (2/4), (3/4) ⇒ x =0, (5/4), (10/4), (15/4) ∴ text Number of solutions =4

Q. Number of solutions of the equation $3 x + {x} = 4 [x]$ will be (where [] \& \{ \} represent greatest integer and fractional part respectively)

0

2

4

$\infty$

$Θ3 [x] + 3 {x} + {x} = 4 [x]$
$\Rightarrow [x] = 4 {x}$
$Θ0 \leq {x} < 1 \Rightarrow 0 \leq [x] < 4$
$\Rightarrow [x] = 0, 1, 2, 3$
$\Rightarrow {x} = 0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}$
$\Rightarrow x = 0, \frac{5}{4}, \frac{10}{4}, \frac{15}{4}$
$∴ Number of solutions = 4$

Q. Number of solutions of the equation 3x+{x}=4[x] will be (where [] \& \{ \} represent greatest integer and fractional part respectively)

0

2

4

∞

Θ3[x]+3{x}+{x}=4[x] ⇒[x]=4{x} Θ0≤{x}<1⇒0≤[x]<4 ⇒[x]=0,1,2,3 ⇒{x}=0,41​,42​,43​ ⇒x=0,45​,410​,415​ ∴ Number of solutions =4

Q. Number of solutions of the equation $3 x + {x} = 4 [x]$ will be (where [] \& \{ \} represent greatest integer and fractional part respectively)

$\infty$

$Θ3 [x] + 3 {x} + {x} = 4 [x]$
$\Rightarrow [x] = 4 {x}$
$Θ0 \leq {x} < 1 \Rightarrow 0 \leq [x] < 4$
$\Rightarrow [x] = 0, 1, 2, 3$
$\Rightarrow {x} = 0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}$
$\Rightarrow x = 0, \frac{5}{4}, \frac{10}{4}, \frac{15}{4}$
$∴ Number of solutions = 4$