Sum of the series 5, 10, 15, ………., 500 is ______.

The sample standard deviation for the following set of observations is ______.

40, 45, 50 and 55

For the given function $f(x, y) = (3 + x)(4 + y)$, the value of $\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}$ at $x = 2$ and $y = 1$ is ______.

Given $y = x^2 + x + 6$, the value of $\frac{d}{dx}(\ln y)$ at $x = 2$ is ______.

A portion of a ventilation system has two splits as shown. Split ‘A’ has a resistance of 0.2 $\text{Ns}^2/\text{m}^8$ and a regulator of size 2.0 $\text{m}^2$. The resistance of split ‘B’ in $\text{Ns}^2/\text{m}^8$ is ______.

A force diagram is shown below. Considering clockwise moments to be positive, the resultant moment about A in Nm is ______.

The experiment to determine permeability of a soil sample is illustrated below. The cross-sectional area of the sample is 20.0 cm$^2$. The permeability of the soil sample in cm/s is ______.

The following figure shows the designed blast pattern of a bench. The explosive column is charged at 18 kg/m. If the unit weight of the blasted material is 2.5 tonne/m$^3$, the powder factor for the blast in tonne/kg is ______.

From an elevated point A, when a stone is thrown vertically, it attains an upward velocity of $v$ at a height of $h$ from the point A. While falling, its downward velocity becomes $2v$ at a distance $h$ below the point A. The maximum height attained by the stone from point A is

Out of the support categories given for an underground coal mine, identify the ‘active support’.