Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?
If $a^2-2 b < 0$, then the equation has one real and two imaginary roots
If $a^2-2 b \geq 0$, then the equation has all real roots.
If $a^2-2 b > 0$, then the equation has all real and distinct roots.
If $4 a^3-27 b^2 > 0$, then the equation has real and distinct roots.
The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x+3^y=5^{x y}$ is
If the equation $\frac{{{x^2} + 5}}{2} = x - 2\cos \left( {ax + b} \right)$ has atleast one solution, then $(b + a)$ can be equal to
The number of real solution of equation $(\frac{3}{2})^x = -x^2 + 5x-10$ :-
If graph of $y = ax^2 -bx + c$ is following, then sign of $a$, $b$, $c$ are
If $a, b, c, d$ are four distinct numbers chosen from the set $\{1,2,3, \ldots, 9\}$, then the minimum value of $\frac{a}{b}+\frac{c}{d}$ is