The $y$ intercept of the tangent of curve $y = x^3 - x^2 + x - 1$ at $x = 1$ is ______

A rectangle has two of its corners on the $x$ axis and the other two on the parabola $y = 12 - x^2$. The largest area of the rectangle is ______.

$F(y)$ and $f(y)$ are the probability distribution function and density function respectively of a continuous variable $Y$ in the interval $(0, \infty)$. Which one of the following is TRUE?

The position vector of a moving particle is given by $\vec{r}(t) = t^3\hat{i} + t\hat{j} + t^2\hat{k}$. The acceleration of the particle in the direction of the motion is

Components of a 2-D stress tensor in Cartesian coordinate are $\sigma_{xx} = 5.0\,\text{MPa}, \sigma_{yy} = -10.0 \, \text{MPa}$, and $\tau_{xy} = 2.0 \,\text{MPa}$. The traction vector ($\vec{T}$) in MPa acting on a plane having outward normal $\hat{n} = \frac{\sqrt{3}}{2} \hat{i} - \frac{1}{2} \hat{j}$ is

The magnetic bearing of a line is $65^\circ 30'$. If the declination is $4^\circ 30' \, W$, the true bearing of the line in degrees is ______.

The results of the crossing point temperature experiments for coal A and B are shown in the figure.

The correct interpretation of the plot is that

A double ended ranging drum shearer is employed in a longwall mine of face length 150 m. The mining height is 3.5 m and depth of the web cut is 0.76 m. The cycle time for unidirectional cutting is 40 min. Considering bulk density of the coal to be $1.4 \, \text{t/m}^3$, hourly production from the face in tonne is ______.

A 20 m thick and 30 m wide confined aquifer has two monitoring wells spaced 500 m apart along the direction of groundwater flow. The difference in water level between the wells is 2 m. The hydraulic conductivity is 50 m/day. The rate of flow in $m^3/ \text{day}$ is

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