Question : Three circles of equal radius '$a$' cm touch each other. The area of the shaded region is:

Option 1: $\left ( \frac{\sqrt{3}+\pi}{2} \right )a^{2}$ sq. cm

Option 2: $\left ( \frac{6\sqrt{3}-\pi }{2} \right )a^{2}$ sq. cm

Option 3: $\left ( \sqrt{3}-\pi \right )a^{2}$ sq. cm

Option 4: $\left ( \frac{2\sqrt{3}- \pi}{2} \right )a^{2}$ sq. cm

Question : If $k\left(\tan 45^{\circ} \sin 60^{\circ}\right)=\cos 60^{\circ} \cot 30^{\circ}$, then the value of k is:

Option 1: $1$

Option 2: $2$

Option 3: $\frac{1}{\sqrt{3}}$

Option 4: $\sqrt{3}$

Question : Find the value of $\frac{\sin ^2 39^{\circ}+\sin ^2\left(90^{\circ}–39^{\circ}\right)}{\cos ^2 35^{\circ}+\cos ^2\left(90^{\circ}–35^{\circ}\right)}+3 \tan 25^{\circ} \tan 75^{\circ}$:

Option 1: 2

Option 2: 4

Option 3: 3

Option 4: 1

Question : If $3+\cos ^2 \theta=3\left(\cot ^2 \theta+\sin ^2 \theta\right), 0^{\circ}<\theta<90^{\circ}$, then what is the value of $(\cos \theta+2 \sin \theta)$ ?

Option 1: $\frac{2 \sqrt{3}+1}{2}$

Option 2: $3 \sqrt{2}$

Option 3: $\frac{3 \sqrt{3}+1}{2}$

Option 4: $\frac{\sqrt{3}+2}{2}$

Question : The value of $\tan ^2 48^{\circ}-\operatorname{cosec}^2 42^{\circ}+\operatorname{cosec}\left(67^{\circ}+\theta\right)-\sec \left(23^{\circ}-\theta\right)$ is:

Option 1: $-1$

Option 2: $0$

Option 3: $1$

Option 4: $-2$