If in $6$ trials, X is a binomial random variable which follows the relation $9P(x = 4) = P(x = 2)$, then the probability of failure is...

Given the probability density function (p.d.f.) of the random variable X, $f(x) = \dfrac{1}{2a}$, $0 < x < 2a$, $a > 0$
$= 0$, otherwise, then which of the following is correct ?

Consider a game of tossing a six sided fair die. If the face that comes up is $6$, the player wins Rs. $36$ and he loses Rs. $k^2$, where $k$ is the face that comes up $k = \{1, 2, 3, 4, 5\}$, then the expected winning amount in this game in Rs. is...

For two vectors $\vec{P}$ and $\vec{Q}$, $\vec{P} \cdot \vec{Q} = |\vec{P} \times \vec{Q}|$
The magnitude of $\vec{R} = \vec{P} + \vec{Q}$ is ($\cos 45^\circ = \dfrac{1}{\sqrt{2}}$) ?