Question

Number of solutions of the equation $[y+[y\left]\right]=2\cos x$ is, where $y=\frac{1}{3}[\sin x+[\sin x+\left[\sin x\right]\left]\right]$ and [.] denotes the greatest integer function

• A. $1$
• B. $2$
• C. $3$
• D. None of these

Answer 1

Answer: (D)

None of these

$[y+[y\left]\right]=2\cos x\Rightarrow \left[y\right]=\cos x$
$y=\frac{1}{3}[\sin x+[\sin x+\left[\sin x\right]\left]\right]=\left[\sin x\right]$
$\Rightarrow \left[\sin x\right]=\cos x$
$\therefore$ Number of solution in $[0,2\pi ]$ is $0$
Hence total solution is $0$.
$\because$ both are periodic with period 2$\pi$