Quadratic Equations — NDA Mathematics PYQs

If the highest degree coefficient is equal to $1$, then what is the total number of quadratic equations which are unchanged on squaring their roots ?

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2-2bx+c^2=0$ where $b$, $c$ are positive real numbers. Let $A$ be the arithmetic mean of $\alpha$ and $\beta$; and $G$ be the geometric mean of $\alpha$ and $\beta$. What are the roots of the quadratic equation $x^2-(b+c)x+bc=0$ ?

For a given $k$, what is the minimum value of $x^2+kx+k^2$ ?

Consider the following statements :
I. $\sqrt{x}+x+1=0$ has two irrational roots.
II. $5\sqrt{x}-x-4=0$ has two rational roots.
Which of the statements given above is/are correct ?

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?

Directions : For the following two (02) items :
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, ${\beta }^2=\alpha ({\log }_{a^2}(0.5))$.

What is $\beta$ equal to?

Directions : For the following two (02) items :
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, ${\beta }^2=\alpha ({\log }_{a^2}(0.5))$.

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

Directions : For the following two (02) items :
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?

Directions : For the following two (02) items :
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