Quadratic Equations — NDA Mathematics PYQs

If one root of the equation $x^2-kx+k=0$ exceeds the other by $2\sqrt{3}$, then which one of the following is a value of $k$?

If $x+\frac{5}{y}=4$ and $y+\frac{5}{x}=-4$, then what is $(x+y)$ equal to?

Consider the following for the two (02) items that follow :
Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2+({\log }_{0.5}{a}^2)x+({\log }_{0.5}{a}^2{)}^4=0$ where $a^2\ne 1$ and ${\log }_{0.5}{a}^2>0$. Further, ${\beta }^2=\alpha \cdot {\log }_{a^2}0.5$.

What is $\beta$ equal to?

Consider the following for the two (02) items that follow :
Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2+({\log }_{0.5}{a}^2)x+({\log }_{0.5}{a}^2{)}^4=0$ where $a^2\ne 1$ and ${\log }_{0.5}{a}^2>0$. Further, ${\beta }^2=\alpha \cdot {\log }_{a^2}0.5$.

What is the relation between $\alpha$ and $\beta$?

Consider the following for the two (02) items that follow :
Consider the equation $ab{x}^2+bcx+ca=ca{x}^2+abx+bc$.

If the roots of the equation are equal, then which one of the following is correct?

Consider the following for the two (02) items that follow :
Consider the equation $ab{x}^2+bcx+ca=ca{x}^2+abx+bc$.

If the roots of the equation are equal, then $a,b,c$ are in

Consider the following for the two (02) items that follow :
Let $f(x)=a{x}^2+bx+c$ be a quadratic polynomial such that $f(1)=f(4)=2$. Further, $2$ is a root of $f(x)=0$.

What is the other root of $f(x)=0$?

Consider the following for the two (02) items that follow :
Let $f(x)=a{x}^2+bx+c$ be a quadratic polynomial such that $f(1)=f(4)=2$. Further, $2$ is a root of $f(x)=0$.

What is $(a+b+c)$ equal to?