Question : If $\left(x^3+\frac{1}{x^3}-k\right)^2+\left(x+\frac{1}{x}-p\right)^2=0$, where k and p are real numbers and $x \neq 0$, then $\frac{k}{p}$ is equal to:

Option 1: ${p}^2+1$

Option 2: ${p}^2+3$

Option 3: ${p}^2-1$

Option 4: $p^2-3$

Question : The value of $\frac{(x-y)^3+(y-z)^3+(z-x)^3}{\left(x^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3}$, where $x \neq y \neq z$, is:

Option 1: $0$

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

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

Option 4: $1$

Question : If $x^2-3 x+1=0$, then the value of $\left(x^4+\frac{1}{x^2}\right) \div\left(x^2+1\right)$ is:

Option 1: 5

Option 2: 6

Option 3: 9

Option 4: 7

Question : If $x^2-5 x+1=0$, then the value of $\left(x^4+\frac{1}{x^2}\right) \div\left(x^2+1\right)$ is:

Option 1: 21

Option 2: 22

Option 3: 25

Option 4: 24

Question : If $\sqrt{x}+\frac{1}{\sqrt{x}}=3, x>0$, then $x^2\left(x^2-47\right)=?$

Option 1: $0$

Option 2: $2$

Option 3: $-2$

Option 4: $-1$

Question : If $\frac{x^{2}-x+1}{x^{2}+x+1}=\frac{2}{3}$, then the value of $\left (x+\frac{1}{x} \right)$ is:

Option 1: 4

Option 2: 5

Option 3: 6

Option 4: 8