If for a posiive integer $n$ , the quadratic equation, $x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \right) + .\;.\;.\; + \left( {x + \overline {n - 1} } \right)\left( {x + n} \right) = 10n$ has two consecutive integral solutions, then $n$ is equal to:

If $\alpha ,\beta ,\gamma $are the roots of the equation ${x^3} + x + 1 = 0$, then the value of ${\alpha ^3}{\beta ^3}{\gamma ^3}$

Let $[t]$ denote the greatest integer $\leq t .$ Then the equation in $x ,[ x ]^{2}+2[ x +2]-7=0$ has

If the roots of ${x^2} + x + a = 0$exceed $a$, then

The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is

How many positive real numbers $x$ satisfy the equation $x^3-3|x|+2=0$ ?