The number of solutions of $\frac{\log 5 + \log (x^2 + 1)}{\log (x - 2)} = 2$ is

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

The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation $6x^2 - 11x + \alpha = 0$ are rational numbers is

Babu starts from his house at a certain time with a certain speed to pick up his girlfriend from the office at $5:00 \, PM$. One day,his girlfriend left the office at $3:00 \, PM$ and started walking home at a speed of $40 \, km/h$. She met Babu on the way,who had left his home at his usual time. They reached home $40 \, min$ earlier than their usual time. If Quantity $I$ is the speed of Babu and Quantity $II$ is $120 \, km/h$,compare the two quantities.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2} = 49$
$II.$ $y^{2} + 15y + 56 = 0$

Number of integral values of $a$ for which the smaller root of the quadratic equation $x^2 - 2ax - 4 + a^2 = 0$ is smaller than $1$ and the larger root is greater than $6$ is