Question : If $\sqrt{\frac{a}{b}}=\frac{8}{3}+\sqrt{\frac{b}{a}}$ and $(a+b)=30$, then what is the value of $ab$?

Option 1: 64

Option 2: 28

Option 3: 81

Option 4: 36

Question : If $a^2+\frac{1}{a^2}=\frac{7}{3}$, then what is the value of $\left(a^3-\frac{1}{a^3}\right)?$

Option 1: $\frac{5}{3 \sqrt{3}}$

Option 2: $\frac{10}{3 \sqrt{3}}$

Option 3: $\frac{7}{3 \sqrt{3}}$

Option 4: $\frac{8}{3 \sqrt{3}}$

Question : If $A=30^{\circ}$, then find the value of $\frac{(2 \tan A)}{\left(1-\tan^2 A\right)}$.

Option 1: $4 \sqrt{3}$

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

Option 3: $3$

Option 4: $2 \sqrt{3}$

Question : If $\sqrt{\frac{\mathrm{a}}{\mathrm{b}}}=\frac{8}{3}-\sqrt{\frac{\mathrm{b}}{\mathrm{a}}}$ and $a-b=10$, then the value of $ab$ is:

Option 1: $32 \frac{1}{7}$

Option 2: $32 \frac{3}{7}$

Option 3: $32 \frac{4}{7}$

Option 4: $32 \frac{2}{7}$

Question : If $\sin \theta = \frac{3}{11}$, then what is the value of $\cot \theta$?

Option 1: $\frac{3 \sqrt{7}}{11}$

Option 2: $\frac{4 \sqrt{7}}{11}$

Option 3: $\frac{4 \sqrt{7}}{3}$

Option 4: $\frac{3 \sqrt{7}}{28}$

Question : If $\sin A-\cos A=\frac{\sqrt{3}-1}{2}$, then the value of $\sin A\cdot \cos A$ is:

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

Option 2: $\frac{3}{2}$

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

Option 4: $\frac{1}{\sqrt{3}}$