Question

If $A=2{\sin }^2\mathrm{\theta }-\cos 2\mathrm{\theta }$, then A lies in the interval
(a) $\left[-1,3\right]$
(b) $\left[1,2\right]$
(c) $\left[-2,4\right]$
(d) none of these

Answer 1

(a) $\left[-1,3\right]$

$$\begin{gathered} A=2{\sin }^2\mathrm{\theta }-\cos 2\mathrm{\theta } \\ =2{\sin }^2\mathrm{\theta }-\left(1-2{\sin }^2\mathrm{\theta }\right) \\ =4{\sin }^2\mathrm{\theta }-1 \\ \because 0\le {\sin }^2\mathrm{\theta }\le 1 \\ \Rightarrow 4\times 0\le 4\times {\sin }^2\mathrm{\theta }\le 4\times 1 \\ \Rightarrow 0\le 4{\sin }^2\mathrm{\theta }\le 4 \\ \Rightarrow 0-1\le 4{\sin }^2\mathrm{\theta }-1\le 4-1 \\ \Rightarrow -1\le 4{\sin }^2\mathrm{\theta }-1\le 3 \\ \Rightarrow -1\le A\le 3 \\ \Rightarrow A\in \left[-1,3\right] \end{gathered}$$