Question : Find the mean proportion of $\frac{a^3+b^3}{a-b}$ and $\frac{a^2-b^2}{a^2-a b+b^2}$.

Option 1: $1$

Option 2: $a+b$

Option 3: $\frac{a+b}{a-b}$

Option 4: $\sqrt{a+b}$

Question : The value of $\frac{1}{4-\sqrt{15}}-\frac{1}{\sqrt{15}-\sqrt{14}}+\frac{1}{\sqrt{14}-\sqrt{13}}-\frac{1}{\sqrt{13}-\sqrt{12}}+\frac{1}{\sqrt{12}-\sqrt{11}}-\frac{1}{\sqrt{11}-\sqrt{10}}+\frac{1}{\sqrt{10}-3}-\frac{1}{3-\sqrt{8}}$ is:

Option 1: $2-2 \sqrt{2}$

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

Option 3: $4-2 \sqrt{2}$

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

Question : Using $\cos (A+B)=\cos A \cos B-\sin A \sin B$, find the value of $\cos 75^\circ$.

Option 1: $\frac{\sqrt{5}-1}{4}$

Option 2: $\frac{\sqrt{5}+1}{4}$

Option 3: $\frac{\sqrt{6}-\sqrt{2}}{4}$

Option 4: $\frac{\sqrt{6}+\sqrt{2}}{4}$

Question : Find the mean proportion between $(6+\sqrt{8})$ and $(3-\sqrt{2})$.

Option 1: $2 \sqrt{12}$

Option 2: $\sqrt{14}$

Option 3: $(6-\sqrt{8})$

Option 4: $\sqrt{15}-7$

Question : Find the value of $\frac{\cos^2 15^{\circ}-\sin^2 15^{\circ}}{\cos^2 145^{\circ}+\sin^2 145^{\circ}}$.

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

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

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

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

Question : If $\tan A=\frac{2}{3}$, then find $\sin A$.

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

Option 2: $\frac{2}{\sqrt{13}}$

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

Option 4: $\frac{3}{\sqrt{13}}$