Let $\mathrm{S}$ be the set of positive integral values of $a$ for which $\frac{\mathrm{ax}^2+2(\mathrm{a}+1) \mathrm{x}+9 \mathrm{a}+4}{\mathrm{x}^2-8 \mathrm{x}+32}<0, \forall \mathrm{x} \in \mathbb{R}$. Then, the number of elements in $\mathrm{S}$ is :
$1$
$0$
$\infty$
$3$
If $a < 0$ then the inequality $a{x^2} - 2x + 4 > 0$ has the solution represented by
Number of integers satisfying inequality, $\sqrt {{{\log }_3}(x) - 1} + \frac{{\frac{1}{2}{{\log }_3}\,{x^3}}}{{{{\log }_3}\,\frac{1}{3}}} + 2 > 0$ is
If the inequality $kx^2 -2x + k \geq 0$ holds good for atleast one real $'x'$ , then the complete set of values of $'k'$ is
Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{x}^{2}-\mathrm{x}-1=0 .$ If $\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,$ then which one of the following statements is not true?
The set of all $a \in R$ for which the equation $x | x -1|+| x +2|+a=0$ has exactly one real root is: