Question

Find the number of solutions for; $\left[\right[x]-x]=sinx$; where [.] denotes the greatest integer function.

• A. 0
• B. 1
• C. 2
• D.

Answer 1

The correct option is D

Here, Let's consider
F(x) = $\left[\right[x]-x]$
g(x) = sin x
Let's draw the graph for these two functions.
F(x) = $\left[\right[x]-x]$...(1)
We know $x=\left[x\right]+\left\{x\right\}$
{.} - is a fractional part function
[.] - is a greatest integer function
$x=\left[x\right]+\left\{x\right\}$
$\left[x\right]=x-\left\{x\right\}$
Substituting the value of $\left[x\right]-x$ in equation 1
F(x) = $[-\{x\left\}\right]$

$0<\left\{x\right\}<1$

From above figure,number of solutions is infinite.

Both the graphs are intersecting $\infty$ number of times. So, number of solution for $\left[\right[x]-x]=sinxis\infty$