If $\alpha, \beta$ are the roots of the equation $x^{2}+kx+12=0$ such that $\alpha-\beta=1$,the value of $k$ is

The roots of the equation ${2^{x + 2}} \cdot {27^{x/(x - 1)}} = 9$ are given by

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.
$I.$ $\frac{15}{\sqrt{x}} - \frac{9}{\sqrt{x}} = (x)^{1/2}$
$II.$ $y^{10} - (36)^{5} = 0$

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$ and $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in $G.P.$,where $\Delta = b^2 - 4ac$,then:

If $x^{2}+y^{2}+1=2x,$ then the value of $x^{3}+y^{5}$ is