If $\alpha, \beta, \gamma$ are roots of $x^3 + x^2 - 5x - 1 = 0$,then the value of $[\alpha] + [\beta] + [\gamma]$ is (where $[.]$ denotes the greatest integer function):

Let $\alpha \neq \beta$ satisfy $\alpha^2+1=6 \alpha$ and $\beta^2+1=6 \beta$. Then,the quadratic equation whose roots are $\frac{\alpha}{\alpha+1}$ and $\frac{\beta}{\beta+1}$ is

The number of real values of $m$ such that the equation $x^2+(2m+1)x+m=0$ has equal roots is

If $f(x) = x^2 + bx + c$ and $f(1+k) = f(1-k)$ for all $k \in R$,for two real numbers $b$ and $c$,then:

Let $S = \{ \alpha : \log_2(9^{2\alpha-4} + 13) - \log_2(\frac{5}{2} \cdot 3^{2\alpha-4} + 1) = 2 \}$. Then the maximum value of $\beta$ for which the equation $x^2 - 2(\sum_{\alpha \in S} \alpha)^2 x + \sum_{\alpha \in S} (\alpha+1)^2 \beta = 0$ has real roots,is $...........$

How many roots does the equation $x - \frac{2}{x - 1} = 1 - \frac{2}{x - 1}$ have?