Question

If $2\cos \theta +\sin \theta =1$ then $(1\cos \theta +6\sin \theta )$ has on of the possible value is

• A. $3$
• B. $6$
• C. $-2$
• D. $7$

Answer 1

The correct option is B $6$

$\begin{array}{c}2\cos x+\sin x=1\\ \left(2\cos x+S_{m}x\right)=1\\ 4{\cos }^{2}x+4\cos x{S}_{m}x+{S_m}^{2}x={S_m}^{2}x+{\cos }^{2}x\\ 4{\cos }^{2}x-{\cos }^{2}x+4\cos x{S}_m=0\\ 3{\cos }^{2}x+4\cos x{S}_{m}x=0\\ \cos x\left(3\cos x+4S_{m}x\right)=0\\ \cos x=0x=\frac{\pi }{2}\\ 3\cos x+4S_{n}x=0\tan x=\frac{-3}{4}\\ \sin x=\frac{-3}{5}\cos x=\frac{4}{5}\\ now,\left(\cos \theta +6S_m\theta \right)\\ \cos \frac{\pi }{2}+6S_m\frac{\pi }{2}=0+6\times 1=6\end{array}$