Suppose that $x$ and $y$ are positive number with $xy = \frac{1}{9};\,x\left( {y + 1} \right) = \frac{7}{9};\,y\left( {x + 1} \right) = \frac{5}{{18}}$ . The value of $(x + 1) (y + 1)$ equals

A real root of the equation ${\log _4}\{ {\log _2}(\sqrt {x + 8} - \sqrt x )\} = 0$ is

For what value of $\lambda$ the sum of the squares of the roots of ${x^2} + (2 + \lambda )\,x - \frac{1}{2}(1 + \lambda ) = 0$ is minimum

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

If $x$ is real, the function $\frac{{(x - a)(x - b)}}{{(x - c)}}$ will assume all real values, provided

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: