Question

Observe the following statements
Assertion ($A$): The general solution of $\sin x=-1$ is $n\pi +(-1{)}^n\frac{3\pi }{2}$.
Reason ($R$): The principal value of $\sin x=k$ lies in $[-\frac{\pi }{2},\frac{\pi }{2}]$.

• A. Both $A$ and $R$ are true and $R$ is correct explanation of $A$.
• B. Both $A$ and $R$ are true and $R$ is not the correct explanation of $A$.
• C. $A$ is true but $R$ is false.
• D. $A$ is false but $R$ is true.

Answer 1

The correct option is D $A$ is false but $R$ is true.

The general solution of $\sin x=\sin \alpha$, is
$x=n\pi +{(-1)}^n\alpha$
So for $\sin x=-1=\sin \left(\frac{-\pi }{2}\right)$
Hence, the general solution is,
$x=n\pi +{(-1)}^n\left(\frac{-\pi }{2}\right)$
Moreover, the principal value of $\sin x$ lies in ($\frac{-\pi }{2},\frac{\pi }{2})$
Hence, A is false and R is true