CDS Mathematics Complex Numbers and Quadratic Equations Medium Questions

If $p+q+r=0$, then what is $z^{\frac{p^2}{qr}}\times z^{\frac{q^2}{rp}}\times z^{\frac{r^2}{pq}}$ equal to ?

What is the value of $k$ for which $(k^2-5k+4)x^2+(k^2-3k-4)x+(k^2-4k)=0$ is an identity?

If $(x-5)$ is the HCF of $x^2-x-p$ and $x^2-qx-10$, then what is the value of $(p+q)$ ?

Let $Q=x^2+bx+c$. If the sum of the roots is equal to product of the roots of the equation $Q=0$, then which of the following statements is/are correct? I. $Q$ can be a perfect square. II. $Q$ is positive for all real values of $x$. Select the answer using the code given below: (A) I only (B) II only (C) Both I and II (D) Neither I nor II

For the next two items that follow: $16\left(x^4+\frac{1}{x^4}\right)-257=0$. What is $\left(x^2+\frac{1}{x^2}\right)$ equal to ?

Let $x=\frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}-\sqrt{5}}$ and $y=\frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}+\sqrt{5}}$. What is $\left(x^2-y^2\right)$ equal to ?

Let $x=\frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}-\sqrt{5}}$ and $y=\frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}+\sqrt{5}}$. What is $\left(x^2-xy+y^2\right)$ equal to ?

If $x^4+p{x}^3+q{x}^2+x+6$ is divisible by $x^2-x-6$. What is the value of p?

If $x^4+p{x}^3+q{x}^2+x+6$ is divisible by $x^2-x-6$. What is the value of q?

If $x^4=x^2+1$, where $x>0$, then what is $2x^4$ equal to?

If $\left(\frac{a-b}{2}\right)x^2-\left(\frac{a+b}{2}\right)x+b=0$, then what are the roots of this equation?

If one root of the equation $2x^2-5px+2p^2=0$ exceeds the other by 4, then what is the value of p ?