SSC CGL 2024 PYQ — Question 55

The correct answer is Option 2 i.e. $-{2121\over 769}$.

${7sinθ\ +\ 4 cosθ \over 9sinθ\ -\ 2cosθ} = {5\over 4}$

Dividing both sides by cosθ

⇒ ${7{sinθ\over cos \theta}\ +\ 4 \over 9{sinθ\over cos\theta} -\ 2} = {5\over 4}$

⇒ ${7{tan \theta}\ +\ 4 \over 9{tan\theta} -\ 2} = {5\over 4}$    [sinθ/cosθ = tanθ]

⇒ 4(7 tanθ + 4) = 5(9 tanθ -2)

⇒ 28 tanθ + 16 = 45 tanθ - 10

⇒ 17 tanθ = 26

⇒ tanθ = 26/17

Now,

The value of $tan^2θ\ +\ 5 \over tan^2θ\ -\ 5$ 

⇒ $({26\over17})^2\ +\ 5 \over ({26\over17})^2 -\ 5$

⇒ $({676\over289})\ +\ 5 \over ({676\over289}) -\ 5$

⇒ $676\ +\ 1445 \over {676} -\ 1445$ = $-{2121\over 769}$