SSC CGL 2021 PYQ — Question 55

The correct answer is option 1 i.e. $\frac{\sqrt{6}}{2}$

 $\begin{aligned} &5 \sin ^{2} \theta-4 \cos \theta-4=0 \\ &\Rightarrow 5\left(1-\cos ^{2} \theta\right)-4 \cos \theta-4=0 \\ &\Rightarrow 5-5 \cos ^{2} \theta-4 \cos \theta-4=0 \\ &\Rightarrow 5 \cos ^{2} \theta+4 \cos \theta-1=0 \\ &\Rightarrow 5 \cos ^{2} \theta+5 \cos \theta-\cos \theta-1=0 \\ &\Rightarrow 5 \cos \theta(\cos \theta+1)-1(\cos \theta+1)=0 \\ &\Rightarrow(5 \cos \theta-1)(\cos \theta+1)=0 \\ &\Rightarrow \cos \theta=1 / 5 \text { or }-1 \end{aligned}$

cos $\theta$ = 1/5 = B/H.

P² = H² - B²

P² = 25 - 1 = 24

P = $2\sqrt6$

cot $\theta$ = B/P = 1/$2\sqrt6$ 

cosec $\theta$ = 5/$2\sqrt6$.

(cot $\theta$ + cosec $\theta$) = 6/$2\sqrt6$ = $\frac{\sqrt{6}}{2}$.