The ratio of the longest wavelengths corresponding to the Lyman and Balmer series in the hydrogen spectrum is

If the roots of $x^3-42 x^2+336 x-512=0$ are in increasing geometric progression,then its common ratio is (in $: 1$)

If $y(x)$ is the solution of the differential equation $(x + 2)\frac{dy}{dx} = x^2 + 4x - 9, x \ne -2$ and $y(0) = 0$,then $y(-4)$ is equal to

Let $Z$ denote the set of integers. Define $f: Z \rightarrow Z$ by $f(x) = \begin{cases} \frac{x}{2}, & x \text{ is even} \\ 0, & x \text{ is odd} \end{cases}$. Then $f$ is:

If $\alpha$ is a non-real root of the equation $x^6-1=0$,then $\frac{\alpha^2+\alpha^3+\alpha^4+\alpha^5}{\alpha+1}$ is equal to

The interval in which the function $f(x) = 2x^2 - \log x$,for $x > 0$ decreases,is