Platinum crystallizes in a face-centered cubic $(FCC)$ crystal with a unit cell edge length $a$. The distance between nearest neighbours is

If $A(2,-1)$ and $B(6,5)$ are two points,the ratio in which the foot of the perpendicular from $P(4,1)$ to $AB$ divides $AB$ is:

The maximum possible number of real roots of the equation $x^5 - 6x^2 - 4x + 5 = 0$ is

$A$ proton, a deuteron, and an $\alpha$-particle having the same momentum enter a region of uniform electric field between the parallel plates of a capacitor. The electric field is perpendicular to the initial path of the particles. Then the ratio of deflections suffered by them is

If $I_n = \int x^n \cdot e^{cx} \, dx$ for $n \geq 1$,then $c \cdot I_n + n \cdot I_{n-1}$ is equal to

Let $y = y(x)$ be the solution of the differential equation $\sin x \frac{dy}{dx} + y \cos x = 4x$,where $x \in (0, \pi)$. If $y(\frac{\pi}{2}) = 0$,then find the value of $y(\frac{\pi}{6})$.