Question

If $A=2\sin \theta +{\cos }^2\theta$, then which of the following is/are true?

• A.
  Maximum value of $A=5$.
• B.
  Minimum value of $A=-2.$
• C.
  Maximum value of $A$ occurs when $\sin \theta =1.$
• D.
  Minimum value of $A$ occurs when $\sin \theta =-1.$

Answer 1

Answer: (B), (C), (D)

Minimum value of $A$ occurs when $\sin \theta =-1.$
For solving these types of questions, we first try to simplify the expression.
Here, $A$ can be simplified as:
$A=2\sin \theta +{\cos }^2\theta \Rightarrow A=2\sin \theta +1-{\sin }^2\theta \Rightarrow A=-\left({\sin }^2\theta -2\sin \theta -1\right]\Rightarrow A=-\left({\sin }^2\theta -2\sin \theta -1+1-1\right]\Rightarrow A=-\left({\sin }^2\theta -2\sin \theta +1\right]+2\Rightarrow A=2-{\left(1-\sin \theta \right)}^2$Now the maximum value of $A$ occurs when ${\left(1-\sin \theta \right)}^2$ is minimum.
And the minimum value of ${\left(1-\sin \theta \right)}^2$ occurs when $\sin \theta =1$, then the maximum value of $A$ is $2-(1-1{)}^2=2$.
Also, $A$ will be minimum when ${\left(1-\sin \theta \right)}^2$ is maximum.
And ${\left(1-\sin \theta \right)}^2$ is maximum at $\sin \theta =-1$
Thus, the minimum value of $A=2-(1-(-1){)}^2=2-2^2=-2$
Hence, Options b. c. and d. are correct.