Algebra — CDS Elementary Mathematics PYQs

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 $\frac{a^2}{b^2+c^2}=\frac{b^2}{c^2+a^2}=\frac{c^2}{a^2+b^2}$, then what is the value of $a^4+b^4+c^4$ equal to?

If $\frac{1}{x}=\frac{1}{p}+\frac{1}{q}$, then what is $\frac{pq}{p^2-q^2}\left(\frac{x+p}{x-p}-\frac{x+q}{x-q}\right)$ equal to?

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

The LCM and HCF of two polynomials $p(x)$ and $q(x)$ are $(x+a)(x^3-a^3)$ and $(x^2-ax+a^2)$, respectively. If $p(x)=x^4+a^2{x}^2+a^4$, then what is $q(x)$ equal to?

Which of the following expressions can divide both the polynomials $x^3+2x^2-5x+2$ and $x^3+4x^2+x-6$ exactly?
I. $x-1$
II. $x+1$
III. $x+2$

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For what integral value of $k$, the system of equations $kx-5y+6=0$ and $4(k-1)y-12x+3=0$ has no solution?

Let $Q=x^2+bx+c$. If the sum of the roots is equal to the 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 $x$.

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If ${\log }_{10}(80)=p$, ${\log }_{10}(45)=q$ and ${\log }_{10}(216)=r$, then what is ${\log }_{10}(384)$ equal to?