Question

For the given graph of $y=5\text{cosec}\frac{x}{2}.$ Select the correct statements.

• A. Period: $4\pi$
• B. Period: $2\pi$
• C. Range: $R-[-5,5]$
• D. Domain: $R-2n\pi \forall n\in Z$

Answer 1

Answer: (A), (D)

Domain: $R-2n\pi \forall n\in Z$

Given: $y=5\text{cosec}x/2$
From the graph of $5\text{cosec}x/2$ as shown:

$5\text{cosec}\frac{x}{2}\in (-\infty ,-5]\cup [5,\infty )$

$\Rightarrow 5\text{cosec}\frac{x}{2}\in R-(-5,5)$

Also, from the graph, $5\text{cosec}\frac{x}{2}$ is not defined for $x=-2\pi ,0,2\pi ,4\pi ,6\pi ,...$
Or all even integral multiples of $\pi$
Or Domain: $R-2n\pi \forall n\in Z$

Also, for finding the period,
We know the period of $\text{cosec}x$ is $2\pi$
Hence, the period of $\text{cosec}\frac{x}{2}$ is $\frac{2\pi }{(\frac{1}{2})}=2\times 2\pi =4\pi$