The value of $k$ for which the quadratic equation $kx^2 + 1 = kx + 3x - 11x^2$ has real and equal roots is:

The value of $2 + \frac{1}{2 + \frac{1}{2 + \dots \infty}}$ is

Let $\alpha, \beta, \gamma$ be the three roots of the equation $x^3+bx+c=0$. If $\beta \gamma=1=-\alpha$,then $b^3+2c^3-3\alpha^3-6\beta^3-8\gamma^3$ is equal to $......$.

The product of the real roots of the equation $x^2 - |x| - 6 = 0$ is .......

If $f(x) \in \mathbb{Q}[x]$ is a non-zero polynomial such that all its roots are irrational,then the degree of $f(x)$ is

Let $S = \{ \sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0 \text{ has real roots} \}$. If $\alpha$ and $\beta$ are the smallest and largest elements of the set $S$,respectively,then $3((\alpha - 2)^2 + (\beta - 1)^2)$ equals: