Question : If $a + b = 10$ and $\sqrt{\frac{a}{b}}-13=-\sqrt{\frac{b}{a}}-11,$ then what is the value of $3ab+4a^{2}+5b^{2}?$

Option 1: $450$

Option 2: $300$

Option 3: $600$

Option 4: $750$

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 : If $y+\frac{1}{y}=11$, then the value of $y^3-\frac{1}{y^3}$ is:

Option 1: $345 \sqrt{13}$

Option 2: $360 \sqrt{13}$

Option 3: $352 \sqrt{13}$

Option 4: $368 \sqrt{13}$

Question : Let $x=\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$ and $y=\frac{1}{x}$, then the value of $3x^{2}-5xy+3y^{2}$ is:

Option 1: 1717

Option 2: 1177

Option 3: 1771

Option 4: 1171

Question : If $2 \cot \theta = 3$, find the value of $\frac{\sqrt{13} \sin \theta – 3 \tan \theta}{3 \tan \theta + \sqrt{13} \cos \theta}$

Option 1: $\frac{1}{\sqrt{13}}$

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

Option 3: 0

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

Question : If $(2+\sqrt{3})a=(2-\sqrt{3})b=1$, then the value of $\frac{1}{a}+\frac{1}{b}$ is:

Option 1: $1$

Option 2: $2$

Option 3: $2\sqrt{3}$

Option 4: $4$