Question

The set of all possible values of $\theta$in the interval $(0,\pi )$ for which the points$(1,2)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y=1$is

• A. $\left(0,\frac{\mathrm{\pi }}{4}\right)$
• B. $\left(0,\frac{\pi }{2}\right)$
• C. $\left(0,\frac{3\mathrm{\pi }}{4}\right)$
• D. $\left(\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4}\right)$

Answer 1

The correct option is B

$\left(0,\frac{\pi }{2}\right)$

The explanation for the correct answer.

Compute the required values:

(Put $(1,2)$ in the given line)(Put$(\sin \theta ,\cos \theta )$ in the given line)$>0$

$$\begin{gathered} \Rightarrow (1+2-1)(\sin \theta +\cos \theta -1)>0 \\ \Rightarrow \sin \theta +\cos \theta >1 \\ \Rightarrow \frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta >\frac{1}{\sqrt{2}} \\ \Rightarrow \sin \left(\theta +\frac{\mathrm{\pi }}{4}\right)>\frac{1}{\sqrt{2}} \\ \Rightarrow \frac{\mathrm{\pi }}{4}<\theta +\frac{\mathrm{\pi }}{4}<\frac{\mathrm{\pi }}{2} \\ \Rightarrow 0<\theta <\frac{\mathrm{\pi }}{2} \end{gathered}$$

Hence, option $B$ is the correct answer.