Question : Simplify the given expression. $\frac{x^3+y^3+z^3-3 x y z}{(x-y)^2+(y-z)^2+(z-x)^2}$

Option 1: $\frac{1}{3}(x+y+z)$

Option 2: $(x+y+z)$

Option 3: $\frac{1}{4}(x+y+z)$

Option 4: $\frac{1}{2}(x+y+z)$

Question : If $x\cos \theta -y\sin \theta =\sqrt{x^{2}+y^{2}}$ and $\frac{\cos ^2{\theta }}{a^{2}}+\frac{\sin ^{2}\theta}{b^{2}}=\frac{1}{x^{2}+y^{2}},$ then the correct relation is:

Option 1: $\frac{x^{2}}{b^{2}}-\frac{y^{2}}{a^{2}}=1$

Option 2: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

Option 3: $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$

Option 4: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

Question : What is the value of $\frac{4x^2+9y^2+12xy}{144}$?

Option 1: $(\frac{x}{3} + \frac{y}{4})^2$

Option 2: $(\frac{x}{3} + y)^2$

Option 3: $(\frac{x}{4} + \frac{y}{6})^2$

Option 4: $(\frac{x}{6} + \frac{y}{4})^2$

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

Option 1: $\frac{4a}{a^{2}+2}$

Option 2: $\frac{2a}{a^{2}+2}$

Option 3: $\frac{4a}{a^{2}+4}$

Option 4: $\frac{2a}{a^{2}+4}$

Question : If $\frac{x+1}{x-1}=\frac{a}{b}$ and $\frac{1-y}{1+y}=\frac{b}{a}$, then the value of $\frac{x-y}{1+xy}$ is:

Option 1: $\frac{2ab}{a^{2}-b^{2}}$

Option 2: $\frac{a^{2}-b^{2}}{2ab}$

Option 3: $\frac{a^{2}+b^{2}}{2ab}$

Option 4: $\frac{a^{2}-b^{2}}{ab}$