$A$ coin and a six-faced die, both unbiased, are thrown simultaneously. The probability of getting a head on the coin and an odd number on the die is:

If the points with position vectors $60 \hat{i}+3 \hat{j}$, $40 \hat{i}-8 \hat{j}$, and $a \hat{i}-52 \hat{j}$ are collinear, then $a$ is equal to

Suppose $X$ follows a binomial distribution with parameters $n$ and $p$, where $0 < p < 1$. If $\frac{P(X=r)}{P(X=n-r)}$ is independent of $n$ for every $r$, then $p$ is equal to

$\int_0^{\pi / 2} \sin ^8 x \cos ^2 x \, dx$ is equal to

The uncertainties in the velocities of two particles $A$ and $B$ are $0.05 \ m \ s^{-1}$ and $0.02 \ m \ s^{-1}$ respectively. The mass of $B$ is five times that of mass $A$. What is the ratio of uncertainties $\left(\frac{\Delta x_A}{\Delta x_B}\right)$ in their positions?