Q66 / 160 · TS EAMCET 2015 · Mathematics
If $A=\sin ^2 \theta+\cos ^4 \theta$, then for all values of $\theta, A$ lies in the interval
- A$[1,2]$
- Correct answerB$\left[\frac{3}{4}, 1\right]$
- C$\left[\frac{1}{2}, \frac{3}{4}\right]$
- D$\left[\frac{3}{4}, \frac{19}{16}\right]$
Solution
We have,
$$
\begin{aligned}
A & =\sin ^2 \theta+\cos ^4 \theta \\
& =1-\cos ^2 \theta+\cos ^4 \theta \\
& =1-\cos ^2 \theta\left(1-\cos ^2 \theta\right) \\
& =1-\cos ^2 \theta \sin ^2 \theta \\
& =1-\frac{1}{4} \sin ^2 2 \theta
\end{aligned}
$$
We know that,
$$
\begin{array}{rlrl}
& 0 & \leq \sin ^2 2 \theta \leq 1 \\
\Rightarrow & & \frac{-1}{4} & \leq \frac{-1}{4} \sin ^2 2 \theta \leq 0 \\
\Rightarrow & 1-\frac{1}{4} & \leq 1-\frac{1}{4} \sin ^2 2 \theta \leq 1 \Rightarrow \frac{3}{4} \leq A \leq 1 \\
\therefore & & A & \in\left[\frac{3}{4}, 1\right]
\end{array}
$$
Hence, the correct answer is option: 'B'.