If $\alpha$ and $\beta$ are the roots of $6x^2 - 6x + 1 = 0$,then the value of $\frac{1}{2}[a + b\alpha + c\alpha^2 + d\alpha^3] + \frac{1}{2}[a + b\beta + c\beta^2 + d\beta^3]$ 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.

If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then the quadratic equation whose roots are $(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$ and $\alpha^3\beta^2 + \alpha^2\beta^3$ is (where $S = p[p^4 - 5p^2q + 5q^2]$ and $P = p^2q^2(p^4 - 5p^2q + 4q^2)$).

Solve the given two equations and choose the correct option.
$I.$ $\frac{3x}{3x+7} - \frac{3x+7}{3x} = 14$
$II.$ $\frac{y}{18y-5} - \frac{18y-5}{y} = 2$

The range of $a$ for which the roots of $x^2 - 2x - a^2 + 1 = 0$ lie between the roots (exclusive) of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ is:

If the equations $x^2 + px + 2q = 0$ and $x^2 + qx + 2p = 0$ $(p \ne q)$ have a common root,then the value of $p + q$ is