If $\alpha$ satisfies the equation $\sqrt{\frac{x}{2x+1}} + \sqrt{\frac{2x+1}{x}} = 2$,then the roots of the equation $\alpha^2 x^2 + 4\alpha x + 3 = 0$ are

The number of distinct quadratic equations $ax^2 + bx + c = 0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c$ such that $a \neq b \neq c$ from the set $\{0, 1, 2, 4\}$ is:

Let $\theta$ be an acute angle such that the equation $x^3+4 x^2 \cos \theta+x \cot \theta=0$ has multiple roots. Then the value of $\theta$ (in radians) is

The product of all the rational roots of the equation $(x^2-9x+11)^2-(x-4)(x-5)=3$ is equal to:

Let $a, b$ and $c$ be the sides of a scalene triangle. If $\lambda$ is a real number such that the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real,then the interval in which $\lambda$ lies is

If the equation $x^4+ax^3+bx^2+cx+d=0$ has three equal roots,then that root is