Question : If $x^3=270+y^3$ and $x=(6+y)$, then what is the value of $(x+y)? $(given that $x>0$ and $y>0$)

Option 1: $2 \sqrt{3}$

Option 2: $\sqrt{3}$

Option 3: $4 \sqrt{3}$

Option 4: $4 \sqrt{2}$

Question : If $x=\sqrt{3}-\frac{1}{\sqrt{3}}, y=\sqrt{3}+\frac{1}{\sqrt{3}}$, then the value of $\frac{x^2}{y}+\frac{y^2}{x}$ is:

Option 1: $\sqrt{3}$

Option 2: $3\sqrt{3}$

Option 3: $16\sqrt{3}$

Option 4: $2\sqrt{3}$

Question : If $x=\frac{1}{x-5}(x>0)$, then the value of $x+\frac{1}{x}$ is:

Option 1: $\sqrt{41}$

Option 2: $\sqrt{29}$

Option 3: $\sqrt{23}$

Option 4: $\sqrt{43}$

Question : If $\left(y^2+\frac{1}{y^2}\right)=74$ and $y>1$, then find the value of $\left(y-\frac{1}{y}\right)$.

Option 1: $6 \sqrt{2}$

Option 2: $-2 \sqrt{19}$

Option 3: $2 \sqrt{19}$

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

Question : If $\sin (x - y) = \frac{1}2$ and $\cos (x + y) = \frac{1}2$, then what is the value of $\sin x \cos x + 2\sin^2x + cos^3x \sec x$?

Option 1: $2$

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

Option 3: $1$

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