Question : If $\small x=a\left (b-c \right),\; y=b\left (c-a \right) ,\; z=c\left (a-b \right)$, then the value of $\left (\frac{x}{a} \right)^{3}+\left (\frac{y}{b} \right)^{3}+\left (\frac{z}{c} \right)^{3}$ is:

Option 1: $\frac{2xyz}{abc}$

Option 2: $\frac{xyz}{abc}$

Option 3: $0$

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

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 $\left(x-\frac{1}{x}\right) =4$, then what is the value of $\left(x^6+\frac{1}{x^6}\right)$?

Option 1: 4689

Option 2: 4786

Option 3: 5832

Option 4: 5778

Question : If $3a^{2}= b^{2}\neq0$, then the value of $\frac{\left (a+b \right)^{3}-\left (a-b \right)^{3}}{\left (a+b \right)^{2}+\left (a-b \right)^{2}}$ is:

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

Option 2: $b$

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

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

Question : If $x=(\sqrt{6}-1)^{\frac{1}{3}}$, then the value of $\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right)$ is:

Option 1: $\frac{2 \sqrt{6}-6}{5}$

Option 2: $\frac{4 \sqrt{6}-6}{5}$

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

Option 4: $\frac{4 \sqrt{3}-6}{5}$