If a line in space makes angles $\alpha, \beta,$ and $\gamma$ with the coordinate axes, then the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ is equal to:

If the coefficient of $x$ in the expansion of $(x^2 + \frac{k}{x})^5$ is $270$,then $k =$

If $x^2 y - x^3 \frac{dy}{dx} = y^4 \cos x$, then $x^3 y^{-3}$ is equal to

Let electronegativity,ionisation energy and electron affinity be represented as $EN$,$IP$ and $EA$ respectively. Which one of the following equations is correct according to Mulliken?

$A$ narrow tube is bent in the form of a circle of radius $R,$ as shown in the figure. Two small holes $S$ and $D$ are made in the tube at positions right-angled to each other. $A$ source placed at $S$ generates a wave of intensity $I_0$ which is equally divided into two parts: one part travels along the longer path,while the other travels along the shorter path. Both the waves meet at the point $D$ where a detector is placed. The maximum value of $\lambda$ to produce a maxima at $D$ is given by

Radius of gyration of a body depends on