Question

Number of solutions of the equation $(2\mathrm{cosec}x-1+(\mathrm{cosec}x-1=1$ in $(-k\pi ,k\pi )$ is 16 , then possible value of ' $k$ ' is

• A. $\varphi$
• B. $4$
• C. $8$
• D. $16$

Answer 1

Answer: (D)

$(2\mathrm{cosec}x-1+(\mathrm{cosec}x-1=1$
cube both sides
$(3\mathrm{cosec}x-2)+3(2\mathrm{cosec}x-1(\mathrm{cosec}x-1$ $=1$
$(2\mathrm{cosec}x-1(\mathrm{cosec}x-1=(1-\mathrm{cosec}x)$
$(2\mathrm{cosec}x-1=-(\mathrm{cosec}x-1$
$\Rightarrow \mathrm{cosec}x=0$ (neglect)
or
$\mathrm{cosec}x=1\Rightarrow \sin x=1,x=2n\pi +\frac{\pi }{2}$
$\Rightarrow 1$ solution in $(0,2\pi )$
$\Rightarrow k$ solutions in $(-k\pi ,k\pi )\Rightarrow k=16$