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 $x: y: z=3: 4: 5$, then what will be the ratio of $\left(\frac{x}{y}\right):\left(\frac{y}{z}\right):\left(\frac{z}{x}\right)$?

Option 1: 49 : 37 : 100

Option 2: 45 : 48 : 100

Option 3: 41 : 37 : 100

Option 4: 37 : 47 : 100

Question : If $x$ = $y$ = $z$, then $\frac{\left (x+y+z \right )^{2}}{x^{2}+y^{2}+z^{2}}$ is equal to:

Option 1: 4

Option 2: 2

Option 3: 3

Option 4: 1

Question : If $x+y+z=13$ and $x^2+y^2+z^2=69$, then $xy+z(x+y)$ is equal to:

Option 1: 70

Option 2: 40

Option 3: 50

Option 4: 60

Question : If ${x^2+y^2+z^2=2(x+z-1)}$, then the value of $x^3+y^3+z^3$ is equal to:

Option 1: 6

Option 2: 1

Option 3: 2

Option 4: 8