Question

If 2 cos $\theta$ + sin $\theta$ = 1 $\left(\theta \ne \frac{\pi}{2}\right),$ then 7 cos $\theta$ + 6 sin $\theta$ is equal to :

• A. $\frac{1}{2}$
• B. 2
• C. $\frac{11}{2}$
• D. $\frac{46}{5}$

Answer 1

Answer: (D)

$ 2 \cos \Theta=1-\sin \Theta $
on squaring both the sides
$ \begin{array}{l} \Rightarrow 4 \cos ^{2} \Theta=\sin ^{2} \Theta-2 \sin \theta+1 \\ \Rightarrow 4\left(1-\sin ^{2} \Theta\right)=\sin ^{2} \Theta-2 \sin \Theta+1 \\ \Rightarrow 5 \sin ^{2} \Theta-2 \sin \Theta-3=0 \\ \Rightarrow \sin \Theta=1 ; \sin \Theta=-\frac{3}{5} \end{array} $
As per the question $\theta=\frac{\pi}{2}$
hence here $\sin \theta=-\frac{3}{5}$ so the $\cos \Theta=\frac{4}{5}$
required answer $=7\left(\frac{4}{5}\right)+6\left(-\frac{3}{5}\right)=2$