In an acute angled triangle,$\cot B \cot C + \cot A \cot C + \cot A \cot B$ is equal to

The numbers $a_n = 6^n - 5n$ for $n = 1, 2, 3, \ldots$ when divided by $25$ leave the remainder:

$A$ convex lens of focal length $0.15 \ m$ is made of a material of refractive index $\frac{3}{2}$. When it is placed in a liquid,its focal length increases by $0.225 \ m$. The refractive index of the liquid is

The population of an insect species shows an explosive increase in numbers during the rainy season,followed by its disappearance at the end of the season. What does this show?

The values of $a$ and $b$,so that the function $f(x) = \begin{cases} x + a \sqrt{2} \sin x, & 0 \le x \le \frac{\pi}{4} \\ 2x \cot x + b, & \frac{\pi}{4} < x \le \frac{\pi}{2} \\ a \cos 2x - b \sin x, & \frac{\pi}{2} < x \le \pi \end{cases}$ is continuous for $0 \le x \le \pi$,are respectively:

$\sum_{k=1}^5 \frac{1^3+2^3+\ldots+k^3}{1+3+5+\ldots+(2k-1)}$ is equal to (in $.5$)