Question

$\left[2x\right]-2\left[x\right]=\lambda$ where $[.]$ represents greatest integer function and $\{.\}$ represents fractional part of a real number then

• A. $\lambda =1,\forall x\in R$
• B. $\lambda =0,\forall x\in R$
• C. $\lambda =1,\forall \left\{x\right\}\ge \frac{1}{2}$
• D. $\lambda =0,\forall \left\{x\right\}<\frac{1}{2}$

Answer 1

Answer: (C), (D)

The correct options are
C $\lambda =1,\forall \left\{x\right\}\ge \frac{1}{2}$
D $\lambda =0,\forall \left\{x\right\}<\frac{1}{2}$

Consider the property $\left[2x\right]=\left[x\right]+[x+\frac{1}{2}]$

$\therefore \left[2x\right]-2\left[x\right]=[x+\frac{1}{2}]-\left[x\right]$

$=[\left[x\right]+\left\{x\right\}+\frac{1}{2}]-\left[x\right]$

$=[\left\{x\right\}+\frac{1}{2}]$

If $\left\{x\right\}\ge \frac{1}{2}$

$\Rightarrow \left\{x\right\}+\frac{1}{2}\ge 1$

$\Rightarrow [\left\{x\right\}+\frac{1}{2}]=1$

$\Rightarrow \lambda =1$

If $\left\{x\right\}<\frac{1}{2}$

$\Rightarrow \left\{x\right\}+\frac{1}{2}<1$

$\Rightarrow [\left\{x\right\}+\frac{1}{2}]=0$

$\Rightarrow \lambda =0$