List of top Questions asked in GATE MT

Given a scalar field \(\phi(x,y,z) = x^2 - yz\).
Magnitude of a vector in the direction of the most rapid increase of \(\phi\) at a point P(3,4,1), rounded off to two decimal places, is _______________

\(Al_2O_3\) can be electrolyzed with an inert anode or a carbon anode.
Overall reactions are as follows:
Reaction I: For inert anode,
\[ \frac{2}{3}Al_2O_3(s) \to \frac{4}{3}Al(l) + O_2(g); \quad \Delta G^\circ \text{ (in Joules)} = 1124800 - 218T \]
Reaction II: For carbon anode,
\[ \frac{2}{3}Al_2O_3(s) + C(s) \to \frac{4}{3}Al(l) + CO_2(g); \quad \Delta G^\circ \text{ (in Joules)} = 730700 - 218T \]
T denotes temperature in Kelvin.
Given: Faraday constant = 96500 Coulomb.
If the inert anode is replaced by a carbon anode during electrolysis of \(Al_2O_3\) at temperature 1300 K, the decrease in the magnitude of decomposition potential between the two reactions (rounded off to two decimal places) is _________________ Volts.

For the following reaction between methane and stoichiometric air having composition 20 vol.% \(O_2\) and 80 vol.% \(N_2\), the estimated adiabatic flame temperature (rounded off to one decimal place) is ________ Kelvin.
\[ CH_4(g) + 2O_2(g) \to CO_2(g) + 2H_2O(g) \]
Given: \(\Delta H = -850\) kJ/mol at 298 K. Assume the specific heat at constant pressure (\(C_p\)) for each reactant and product is 50 J/mol-K and is independent of temperature.

A copper ore contains 30 wt.% chalcopyrite (\(CuFeS_2\)) and the remaining gangue material. Assuming no copper is present in the gangue material, the amount of copper in the ore (rounded off to one decimal place) is ______________ wt.%.
Given: Atomic weights of Fe, Cu and S are 56, 63.5, and 32 g/mol, respectively.

A blast furnace produces hot metal with the following composition: 4 wt.% C, 1.5 wt.% Si, and the rest Fe.
The only input of iron is through iron ore containing 85 wt.% \(Fe_2O_3\) and 15 wt.% gangue consisting of \(SiO_2\) and \(Al_2O_3\).
2% of all the iron (by weight) is lost in the slag.
The amount of ore used to produce 1000 kg of hot metal (rounded off to one decimal place) is _________________ kg.
Given: Atomic weights of Fe and O are 56 g/mol and 16 g/mol, respectively.

During carburizing of a steel, the surface concentration is kept constant at 1.4 wt.% carbon. Diffusivity of carbon for the steel at \(950^{\circ}\text{C}\) is \(6.25 \times 10^{-11}\) m\(^2\)/s. At \(950^{\circ}\text{C}\), the time required to carburize the steel with an initial composition of 0.2 wt.% carbon to 0.8859 wt.% carbon at a depth of 0.2 mm is _______ seconds (approximate to the nearest integer).
Use the nearest value of the error function from the table given below for your calculation.
\[ \begin{array}{cc} z & \text{erf}(z) \\ 0.3 & 0.3268 \\ 0.4 & 0.4284 \\ 0.5 & 0.5205 \end{array} \]

The figure shows a cylindrical furnace of inner diameter 0.1 m and height 0.2 m. The curved (lateral) side wall is surface \(A_1\), the bottom circular surface is \(A_2\), and the open top circular surface, facing the atmosphere, is \(A_3\).
A cylindrical furnace has 0.1 m inner diameter and 0.2 m height. Walls \(A_1\) (inner section of the cylindrical surface area) and \(A_2\) (inner section of the bottom surface area) are maintained at 1873 K. The sides and bottom are assumed to be black bodies, well insulated and heated electrically. The top area (\(A_3\)) is open to the atmosphere maintained at 300 K, resulting in loss of heat 'q'.
Given: View factors \(F_{13} = 0.1175\) and \(F_{23} = 0.06\), where \(F_{ij}\) is the fraction of radiation leaving surface 'i' that is intercepted by surface 'j'.
Stefan-Boltzmann constant \( = 5.67 \times 10^{-8} \) W/m\(^2\)-K\(^4\).
The power needed to maintain the furnace at 1873 K, is (approximate to the nearest integer) _______ W.

A given volume of liquid is undercooled just below the melting temperature to form a spherical solid nucleus (consider homogeneous nucleation). The Gibbs free energy of solidification (\(\Delta G_v\)) is \((-0.5 \times 10^{8})\) J/m\(^3\). The solid-liquid interfacial energy (\(\gamma\)) is isotropic and its value is \(0.1\) J/m\(^2\).
The critical nucleus size for a stable nucleus is _______ nm (answer in integer).

With reference to the Ellingham diagram, the standard Gibbs free energy change for the oxidation of solid metal M(s) and liquid metal M(l) is given below.
Reaction I: \( M(s) + O_2(g) \rightarrow MO_2(s) \)
\[ \Delta G^{\circ} = (-338900 - 15.2\,T\ln T + 247T) \text{ Joules} \]
from \(T = 300\) K to the melting point.
Reaction II: \( M(l) + O_2(g) \rightarrow MO_2(s) \)
\[ \Delta G^{\circ} = (-390800 - 15.2\,T\ln T + 285.3T) \text{ Joules} \]
from the melting point to \(T = 1800\) K.
The melting point of the metal M (rounded off to one decimal place) is _______ Kelvin.

An alloy having composition W-20 wt.% Ni is prepared by mixing elemental powders. The resulting powder mixture is liquid phase sintered at \(1550^{\circ}\text{C}\) for 1 hour. The liquid phase sintered microstructure consists of interconnected, spherical tungsten grains dispersed in nickel. The tungsten grain size is 70 \(\mu\text{m}\) and the W-W interparticle neck diameter is 35 \(\mu\text{m}\). If W-Ni interfacial energy is 0.30 J/m\(^2\), the W-W interfacial energy (in J/m\(^2\)), rounded off to two decimal places is ______.
(Given: melting point of W: \(3410^{\circ}\text{C}\) and melting point of Ni: \(1455^{\circ}\text{C}\))

Taking number of intervals n = 3, the value of the integral
\[ \int_0^{0.3} e^{-x^2}\, dx \]
using Trapezoidal method, is ______ (rounded off to two decimal places).

Solution of the differential equation \[ 10x^2\frac{d^2y}{dx^2}-20x\frac{dy}{dx}+22.4y=0 \] is \[ y=c_1x^{m_1}+c_2x^{m_2} \] where \(m_1 \neq m_2\).
The value of \(m_1+m_2\) (answer in integer) is ______.

A cylindrical single crystal of copper having 10 mm diameter is deformed under a uniaxial tensile load of 2200 N. The angle between the normal to the slip plane and the tensile loading axis is \(\alpha\), whereas the angle between slip direction and the tensile axis is \(\beta\). The slip direction is in the plane defined by the stress axis and the normal to the slip plane.
If \(\alpha=\beta\), the critical resolved shear stress (CRSS), rounded off to one decimal place, is ______ MPa.

A metal having body-centered cubic structure is analyzed through X-ray diffraction using monochromatic X-ray of wavelength 0.154 nm. The diffraction angle (\(2\theta\)) corresponding to \(\{2\,0\,0\}\) plane is \(60^{\circ}\) (for first order reflection).
The atomic radius of this element (rounded off to three decimal places) is ______ nm.