The number of ordered pairs $(a, b)$ of positive integers such that $\frac{2a-1}{b}$ and $\frac{2b-1}{a}$ are both integers is

For any real value of $x$,if $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b]$,then the value of $x$ for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is

For real values of $x$ and $a$,if the expression $\frac{x+a}{2 x^2-3 x+1}$ assumes all real values,then

If the quadratic equation formed by eliminating $x$ from $x^2+\alpha x+\beta=0$ and $xy+l(x+y)+m=0$ has the same roots as that of the given quadratic equation,then the set of values of $\beta$ is

Let $a, b, c, d \in R^+$ such that $256abcd \geq (a+b+c+d)^4$ and $3a + b + 2c + 5d = 11$. Then $a^3 + b + c^2 + 5d$ is equal to:

Consider the quadratic equation $nx^2 + 7\sqrt{n}x + n = 0$ where $n$ is a positive integer. Which of the following statements are necessarily correct?
$I$. For any $n$,the roots are distinct.
$II$. There are infinitely many values of $n$ for which both roots are real.
$III$. The product of the roots is necessarily an integer.