List of practice Questions

Given, $y = f(x)$; $\dfrac{d^2 y}{dx^2} + 4y = 0$; $y(0) = 0$; $\dfrac{dy}{dx}(0) = 1$. The problem is a/an
Average values of the re-aeration rate constant for river X and Y are 0.92 day$^{-1}$ and 1.12 day$^{-1}$ respectively. Average de-oxygenation rate constants are 0.23 day$^{-1}$ and 0.35 day$^{-1}$ for the same rivers. Which of the following statement(s) is/are correct?
The area of the region (rounded off to one decimal place) enclosed between the curves $y = x$ and $y = 3\sqrt{x}$ and between the lines $x = 0$ and $x = 1$ is ____________ units
The Henry’s law constant of CO$_2$ is $3.4 \times 10^{-2}$ M/atm at 25$^\circ$C.
Dissolved CO$_2$ undergoes:
CO$_2 \cdot$H$_2$O $\rightleftharpoons$ H$^+$ + HCO$_3^-$ ($K_1 = 4.3 \times 10^{-7}$ M)
HCO$_3^- \rightleftharpoons$ H$^+$ + CO$_3^{2-}$ ($K_2 = 4.7 \times 10^{-11}$ M)
If ambient CO$_2$ concentration is 300 ppm, the total dissolved CO$_2$ (in $\mu$M, rounded off to one decimal place) is __________.
An intake shaft has resistance of 0.05 Ns$^2$/m$^8$ up to depth 400 m. Airflow rate = 100 m$^3$/s, air density = 1.2 kg/m$^3$. Barometer reads 99.375 kPa at surface. Find the barometer reading at 400 m depth (kPa), rounded to two decimals.
The NPVs of project proposals A and B are: \[ NPV_A = -0.01i^2 - 0.02i + 4.44, \qquad NPV_B = -0.03i^2 - 0.01i + 6.55, \] where $i$ is the discount rate. The discount rate for which both proposals have equal possibility of acceptance or rejection (rounded off to two decimals) is __________.
The value of \[ \int_0^1 x \log(1+x)\, dx \] (rounded off to two decimals) is __________.
A coal seam of uniform thickness 12 m dips at 30$^\circ$. The ultimate pit limit is based on an instantaneous stripping ratio of 10 m$^3$/tonne and a safe slope angle of 45$^\circ$. Coal density = 1.41 tonne/m$^3$. Find length $L$ in m (rounded off to two decimals).
A mine has a reserve of 150 million tonne (Mt) and a maximum production capacity of 5 Mt/year. First-year production is 2 Mt and increases by 20% each year. Find the reserve remaining at the end of 15 years (rounded off to two decimal places).
Information on Activity-Time duration of a project is provided below. The expected project duration in weeks, is ____________. (in integer)
A pump lifts mine water (density 1020 kg/m$^3$) at 250 m$^3$/hr from 150 m depth. Head loss = 15 m, efficiency = 68%. Find motor input power (in kW), rounded to two decimals.
In a surface mine bench, overburden is removed by shovel–dumper combination. Data for the dumper is given. Minimum number of dumpers required to avoid shovel idle time (in integer) is __________.
In a bord and pillar panel, with headings of 4.4 m × 2.5 m advanced by solid blasting, the average pull per round = 1.2 m and 12 faces are blasted per day. Density of coal = 1500 kg/m$^3$. Mine works three shifts; daily employment = 330 persons. Labour productivity (OMS) in tonne (rounded off to two decimals) is __________.
In a longwall face, the full seam thickness of 3 m is cut by a shearer with a web depth of 0.7 m. Hauling speed = 12 m/min. AFC trough cross-section = 0.4 m$^2$, loading coefficient = 0.7. Find the AFC speed (in m/s), rounded to one decimal place.
A city of area 20 km × 40 km receives wind perpendicular to the 20 km side at 4 m/s. Inversion layer height = 100 m. PM$_1$ emission rate = 1 kg/s. Find steady-state PM$_1$ concentration (µg/m$^3$), rounded to one decimal place, using a Box model.
The point A(7,17) in the Coward flammability diagram represents the gas composition of a sealed-off area (volume = 10000 m$^3$). Inert gas is injected so that composition moves below the LEL boundary. Minimum required inert gas volume (in m$^3$, rounded off to one decimal place) is __________.
An 8 m × 6 m × 4 m workshop has four identical luminaires at roof corners. Each is 100 W with luminous efficacy 100 lumen/W. Light is transmitted spherically with no reflections. Illumination at the centre of the floor (in lux, rounded off to two decimals) is __________.
An underground AC plant requires 250 US gpm of chilled water. Ice pellets at 0$^\circ$C (latent heat = 334 kJ/kg) are mixed with 20$^\circ$C water (cp = 4.18 kJ/kg$^\circ$C). Mixture arrives underground at 7$^\circ$C. Find ice requirement (tonne/hr), rounded to two decimals.
In a health centre, the probability of ‘full occupancy’ of COVID beds for a day is 0.8. Assuming Binomial probability distribution, the probability of full occupancy exactly for 5 days in a week is __________ (rounded off to three decimal places).
Information for a drilling operation in an overburden mine is given: Volume of rock blasted per round = \(3.2 \times 10^5\ \text{m}^3\)
Number of blast holes = 100
Drill hole diameter \(D = 200\ \text{mm} = 0.2\ \text{m}\)
Length of subgrade drilling = \(8D = 1.6\ \text{m}\)
Stemming length = \(25D = 5\ \text{m}\)
Bench height = 30 m
Powder factor = \(3.2\ \text{m}^3/\text{kg}\) Find the explosive per unit charge length (kg/m), rounded to two decimals).
The shaft-top coordinates of two vertical shafts are given below. The depth of the shaft A and B are 200 m and 149 m, respectively. The downward gradient of the line joining the bottom of the two shafts in degrees, is __________. (round off to two decimal places)
For the oxygen-balanced explosive ANFO reaction \[ 3NH_4NO_3 + CH_2 \rightarrow 7H_2O + CO_2 + 3N_2 \] the amount of ammonium nitrate (in kg, rounded off to two decimals) required for 100 L of fuel oil (density 850 kg/m$^3$) is __________.
Two weightless cables of equal length and equal cross-sectional area hang from a ceiling. They are connected by a horizontal light bar of length 1.0 m. A downward force $F$ is applied at a point located $x$ meters from Cable–1. The moduli of elasticity of Cable–1 and Cable–2 are 50 GPa and 200 GPa respectively. If the deformation in both cables is equal, find $x$ (in m), rounded to one decimal place.
A circular tunnel of radius 3 m is constructed in a hydrostatic stress field of 15 MPa. The rock has modulus of elasticity $E = 5$ GPa and Poisson’s ratio $\nu = 0.25$. A uniform support pressure $p_i$ is applied at the tunnel boundary to restrict the radial deformation to 4 mm. The value of $p_i$ in MPa (rounded off to two decimals) is __________.
A bord and pillar panel initially has an extraction ratio of 0.15. After widening galleries, the extraction ratio changes to 0.25. Using the tributary area method, the percentage change in pillar stress (rounded off to two decimals) is __________.