If $x$ is a real number,for what values of $x$ does $3x^2 + 14x + 11 > 0$ hold?

If the inequality $kx^2 - 2x + k \geq 0$ holds good for at least one real $x$,then the complete set of values of $k$ is

For the inequality $2^{\log_{\sqrt{2}}(x - 1)} > x + 5$,the set of real values of $x$ is:

Assertion $(A)$: $3x^2 - 16x + 4 > -16$ is satisfied for some values of real $x$ in $(0, \frac{10}{3})$.
Reason $(R)$: $ax^2 + bx + c$ and $a$ will have the same sign for some values of $x \in \mathbb{R}$ when $b^2 - 4ac > 0$.
The correct option among the following is

Number of integral values of $m$ for which $\{x\}^2 + 5m\{x\} - 3m + 1 < 0$ for all $x \in \mathbb{R}$ is (where $\{.\}$ denotes the fractional part function).

The integral value of $k$ for which $x^2 - 2(4k - 1)x + 15k^2 - 2k - 7 > 0$ for all $x \in R$ is