Q 11 :

The value of $\tan 9°-\tan 27°-\tan 63°+\tan 81°$ is ________ .               [2023]

Q 12 :

Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function 

$f\left(\theta \right)=4\left({\sin }^4\right(\frac{{\displaystyle 7\pi }}{{\displaystyle 2}}-\theta )+{\sin }^4(11\pi +\theta \left)\right)-2\left({\sin }^6\right(\frac{{\displaystyle 3\pi }}{{\displaystyle 2}}-\theta )+{\sin }^6(9\pi -\theta \left)\right), \theta \in \mathit{R}.$

Then $\alpha +2\beta$ is equal to:                                                               [2026]

• 5

   

• 3

   

• 4

   

• 6

   

Q 13 :

The value of cosec ${10}^{\mathrm{o}}-\sqrt{3}$ sec 10° is equal to             [2026]

• 2

   

• 4

   

• 6

   

• 8

   

Q 14 :

If $\cot x=\frac{5}{12}$ for some $x\in (\pi ,\frac{3\pi }{2}),$ then $\sin 7x(\cos \frac{{\displaystyle 13x}}{{\displaystyle 2}}+\sin \frac{{\displaystyle 13x}}{{\displaystyle 2}})+\cos 7x(\cos \frac{{\displaystyle 13x}}{{\displaystyle 2}}-\sin \frac{{\displaystyle 13x}}{{\displaystyle 2}})$ is equal to            [2026]

 

• $\frac{1}{\sqrt{13}}$

   

• $\frac{6}{\sqrt{26}}$

   

• $\frac{5}{\sqrt{13}}$

   

• $\frac{4}{\sqrt{26}}$

   

Q 15 :

The value of $\frac{\sqrt{3}cosec20°-\sec 20°}{\cos 20°\cos 40°\cos 60°\cos 80°}$ is equal to:            [2026]

• 64

   

• 16

   

• 12

   

• 32

   

Q 16 :

If $\frac{{\cos }^{2}48°-{\sin }^{2}12°}{{\sin }^{2}24°-{\sin }^{2}6°}=\frac{\alpha +\beta \sqrt{5}}{2},$ where $\alpha ,\beta \in \mathrm{\mathbb{N}}$, then $\alpha +\beta$ is equal to __________ .                 [2026]

Q 17 :

The number of elements in the set $\{x\in [0,180°]:\tan (x+100°)=\tan (x+50°\left)\tan x\tan \right(x-50°\left)\right\}$ is __________ .           [2026]

Q 18 :

Let $\frac{\pi }{2}<\theta <\pi$ and  $\cot \theta =-\frac{1}{2\sqrt{2}}$. Then the value of $\sin \left(\frac{15\theta }{2}\right)(\cos 8\theta +\sin 8\theta )+\cos \left(\frac{15\theta }{2}\right)(\cos 8\theta -\sin 8\theta )$ is equal to [2026]

• $\frac{\sqrt{2}-1}{\sqrt{3}}$

   

• $\frac{1-\sqrt{2}}{\sqrt{3}}$

   

• $-\frac{\sqrt{2}}{\sqrt{3}}$

   

• $\frac{\sqrt{2}}{\sqrt{3}}$

   

Q 19 :

Let $\cos (\alpha +\beta )=-\frac{1}{10}$ and $\sin (\alpha -\beta )=\frac{3}{8}$, where $0<\alpha <\frac{\pi }{3}$ and $0<\beta <\frac{\pi }{4}$, If $\tan 2\alpha =\frac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}, r,s\in \mathrm{\mathbb{N}},$ then r + s is equal to______. [2026]

Q 20 :

If $\frac{\tan (A-B)}{\tan A}+\frac{{\sin }^{2}C}{{\sin }^{2}A}=1, A,B,C\in \left((0,\frac{\pi }{2})\right)$, then      [2026]

• tanA,tanC,tanB are in G.P.

   

• tanA,tanB,tanC are in A.P.

   

• tanA,tanB,tanC are in G.P.

   

• tanA,tanC,tanB are in A.P.