Engineering Mathematics

Given \(f(t)=e^{-at}\). The Laplace transform \(\mathcal{L}[f(t)]=F(s)\). Which is correct?

Correct pair of eigenvectors for \(\begin{bmatrix}1&2\\2&4\end{bmatrix}\)?

Given \(w=f(ax+by)\), value of \(\left(b\frac{\partial w}{\partial x}-a\frac{\partial w}{\partial y}\right)\) is:

Value of scalar triple product \(\vec{a}\cdot(\vec{b}\times\vec{c})\) (answer in integer).
\(\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}\), \(\vec{b}=\hat{i}+2\hat{j}-3\hat{k}\), \(\vec{c}=3\hat{i}+4\hat{j}-\hat{k}\)

20 thermometers, 3 defective. Draw 2 without replacement. Probability (%) that none is defective (round to 2 decimal places) is _____ %.

Steady-state laminar flow: \(\eta\frac{1}{r}\frac{d}{dr}\!\left(r\frac{dv_z}{dr}\right)-\frac{dP}{dz}=0\)
Which statement is NOT correct?

PDF: \(f(x)=0.5\) for \(0

NAT2M

Which one of the following matrices has eigenvalues 1 and 6?

Which of the following functions is/are expandable using Maclaurin series?
(A) \(\ln(1+z)\)   (B) \(\ln z\)   (C) \(1/z^2\)   (D) \(\exp(z)\)

The hydrostatic stress for the stress tensor below is ______ MPa (integer).
\[\boldsymbol{\sigma}=\begin{pmatrix}150&0&0\\0&-100&100\\0&100&250\end{pmatrix}\text{ MPa}\]

For two continuous functions \(M(x,y)\) and \(N(x,y)\), the relation \(M\,dx+N\,dy=0\) describes an exact differential equation if

Which one of the following matrices is orthogonal?

For \(T(x,y)=\frac{1}{3}xy(x+y)\), find the magnitude of its gradient \(|\nabla T|\) at point (1, 1) (2 decimal places).

Molten steel at 1900 K to be vacuum degassed. What equilibrium \(P_{\text{H}_2}\) (in Torr) achieves 1 ppm dissolved H? \(\log_{10}K_{eq}=-1900/T+2.4\) (K\(_{eq}\) in ppm/\(\sqrt{\text{atm}}\)); 1 atm = 760 Torr. (2 decimal places)

Find the value of \(\displaystyle\lim_{x\to0}\frac{6(x-\sin x)}{x^3}\) (integer).

Given (in J): \(\text{Fe}(s)+\frac{1}{2}\text{O}_2\rightleftharpoons\text{FeO}(s)\), \(\Delta G^\circ=-264900+65T\); \(2\text{H}_2+\text{O}_2\rightleftharpoons 2\text{H}_2\text{O}(g)\), \(\Delta G^\circ=-492900+109T\). Find \(P_{\text{H}_2\text{O}}/P_{\text{H}_2}\) to reduce FeO at \(T=1000\) K (2 decimal places). \(R=8.314\) J mol\(^{-1}\)K\(^{-1}\).

Diameter of spherical galena particles having same Stokes settling velocity as spherical quartz particles of diameter 25 µm (both in water) is ______ µm (1 decimal place).
\(\rho_\text{galena}=7400\), \(\rho_\text{quartz}=2600\), \(\rho_\text{water}=1000\) kg m\(^{-3}\).

Two consecutive Newton–Raphson estimates: \(x_i=8.5\) and \(x_{i+1}=13.5\). If \(f(x_i)=15\), the numerical value of \(f'(x_i)\) is ______ (integer).

If \(X_1\) and \(X_2\) are independent normally distributed random variables with means \(\mu_1, \mu_2\) and variances \(\rho_1, \rho_2\), then \(X=X_1+X_2\) has mean \(\mu\) and variance \(\rho\) such that

Which one of the following is the Taylor-series expansion of \(\ln\!\left(\dfrac{1+x}{1-x}\right)\) about the origin for \(|x|<1\)?

Consider the normal (Gaussian) distributions a, b, c shown in the figure. \(\sigma_p\) and \(\mu_p\) are the standard deviation and mean of distribution p (means are positive). Which deduction is correct?

Two rods of different metals of equal lengths L/2, diameter d (d<<L), with thermal conductivities \(k_1\) and \(k_2\) (\(k_1>k_2\)) are connected in series. Left end at \(T_1\), right end at \(T_2\) (\(T_1>T_2\)). Which graph represents the steady-state temperature distribution?

Which of the following statements is/are correct for a square matrix A with real number entries? (\(A^T\) = transpose, \(A^{-1}\) = inverse)

The divergence of the vector field \(\vec{V}=x^2y\,\hat{i}+y^3z\,\hat{j}+z^4\hat{k}\) at the point (1,1,1) is __________ . (Round off to the nearest integer)

If \(\begin{bmatrix}1&2\\8&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\lambda\begin{bmatrix}x\\y\end{bmatrix}\), where x, y are not identically zero, the values of \(\lambda\) are

An isotropic metallic cuboid (\(\alpha\), E, \(\nu\); dimensions a, b, c in X, Y, Z) is rigidly constrained in X but free in Y and Z. Initially stress-free, temperature increases by \(\Delta T\). The CHANGE in dimension in the Y direction is

The following data is obtained from an experiment: X: 1, 2, 3; Y: 8, 15, 19. If the data is fit using the straight line \(y=mx+c\) using the least-squares method, the value of m is __________. (Round off to one decimal place)

The integral \(\displaystyle\int_0^1 xe^{-x}\,dx\) evaluates to __________. (Round off to two decimal places)

𝑑y 3 𝑑2y
Order (O) and degree (D) of the differential equation ( ) = √ + 10 are
𝑑x 𝑑x2

The sum of eigen values of the matrix
4 3 2
[0 -1 2 ] is__________ (in integer).
0 0 -3

1 1 1
The probability of setting an easy exam paper by three setters are , , and .
2 3 4
If all three are setting one paper each, then the probability that at least one of the
papers will be easy is __________ (round off to 2 decimal places).

Given, phi̅ = xy i-hat + y𝑧 j-hat + x𝑧 k-hat . 𝑆 is a surface bounded by the planes x = 0 ,
y = 0 , 𝑧 = 0 , x = 3 , y = 2 , and 𝑧 = 1 . If 𝑛̂ is the unit vector normal to 𝑆, then
∬ phi̅ . 𝑛̂ 𝑑𝑆 is
𝑆

For the equation
x + 3 3x + 4 4x + 5
| -2 -3 -4 | = 0
-3 -4 -5
the value of x is____________(in integer).

The Taylor series expansion around \(x = 0\) of the function \(f(x) = \dfrac{x+1}{e^x+1}\) truncated to first two terms is __________

According to Sieverts' law, the equilibrium solubility of N (gas) in molten steel
2
is proportional to _________
Given: Equilibrium partial pressure of N (gas) is 𝑝
2

Given that V is a closed volume in space bounded by the surface S with unit
  
normal n . If f is any non-zero vector and is the gradient operator, then the
       
volume integral  . f dV is equal to the surface integral  n . f dS by
V S
virtue of ___________

A set of observations with normal distribution of error as ±1.96 (where  is
standard deviation) is equivalent to the confidence interval of ________% (round
off to the nearest integer).

The general solution to the following differential equation is __________ , where
A and B are constants
\[\dfrac{d^2y}{dt^2} - 4\dfrac{dy}{dt} + 4y = 0\]

Which one of the following equations will fail to converge to a root with an
initial guess value of x = 0.5, using the Newton-Raphson method?

For a 3×3 matrix, the value of the determinant is \(-48\) and the trace is 8. If one of the eigenvalues is 4, the other two are ________

The integral of the function \(f(x) = 0.2 + 10x^2\) estimated by the trapezoidal rule
with a single segment from x = 0 to x =1is _________ (round off to 1 decimal
place).

Given the stress tensor \(\begin{bmatrix}130 & 30 & 0\\ 30 & 50 & 0\\ 0 & 0 & 0\end{bmatrix}\) MPa,
the maximum shear stress is _________ MPa (round off to the nearest integer).

A set of 11 (x, y) data points is least-squares fitted to a quadratic polynomial. If
the sum of squares of error is 2.4, the variance of error is ________ (round off to 1
decimal place).

For the matrix given below, the eigenvalues are:
1 0 -1
0 1 0
-1 0 1

The divergence of a vector field V(x, y, z), where its three components (V, ,
V,, V,) are functions of x,y,z, is:
0) Ee
Ox Ody dz
| (a%_ 2), (M%_MMy,, (O_O
Oy Oz Oz Ox Ox dy
Ox oa oy Upp dz i
D)|a%% a | oY,
Ox4- dys dz?

The grain size (X) of annealed specimens follows a symmetric distribution
percentage of specimens with grain size in the range 5 to 6 ym is expected
to be: (round off to nearest integer).
Given: For the symmetric distribution: Probability P(X < 1: + 2c) = 0.98

Consider the function f(x) = x - cos x. Using Newton-Raphson method,
the estimated root of f(x) after the first iteration is: (round off to 3
decimal places).
Assume: Initial guess of the root = 0.5 radians.

The general solution to the following homogeneous ODE,
\(\dfrac{d^2y}{dt^2} + 4\dfrac{dy}{dt} + 3y = 0\),
is \(y(t) = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t}\).
The values of \(\lambda_1\) and \(\lambda_2\) are:

For the function \(y = a^x\), the derivative \(\dfrac{dy}{dx}\) at \(x = 1\) is:

Which one of the following dislocation reactions is NOT feasible in a FCC crystal?

The divergence of the vector field \((x^3 + y^3)\mathbf{i} + 3xy^2\mathbf{j} + 3zy^2\mathbf{k}\) is:

The production process of cylindrical pipes results in a statistical scatter in their diameter which is modelled by a normal distribution with a mean value of 10 mm. If the area under the normal curve between 9 mm and 10 mm is 0.35, then the probability of producing pipes of diameter greater than 11 mm is __________ (round off to two decimal places).

The solution (using trapezoidal rule) of the integral \(\displaystyle\int_0^1 e^{-x^2}\,dx\) by dividing the range 0 to 1 into two equal intervals is __________ (round off to two decimal places).

One of the eigenvalues for the following matrix is _____________.
\[\begin{bmatrix} a & 2 \\ 8 & a \end{bmatrix}\]

The curl of vector fields shown below is not zero for _____________.

The smallest period of function \(f(x) = \sin\!\left(\dfrac{nx}{k}\right)\) is ____________.

The directional derivative of \(\phi = x^2 + y\) along the unit vector \(\hat{u} = \tfrac{1}{5}(3\hat{i} + 4\hat{j})\) at \((1,1)\) is _____________.

The standard deviation (rounded off to one decimal place) of the following set of five numbers is _________.
6, 8, 8, 9, 9

A plate of width 100 cm and thickness 5 cm is rolled to a thickness of 3 cm. If the entry velocity is 10 cm·s⁻¹, the exit velocity of the plate (in cm·s⁻¹, rounded off to one decimal place) is __________.
Assume no change in the width of the plate.

A 50 cm long rod is placed against a vertical wall such that the bottom of the rod is 30 cm away from the wall. If the bottom of the rod is pulled horizontally away from the wall at 4 cm·s⁻¹, the top of the rod starts sliding down the wall with an instantaneous velocity (in cm·s⁻¹, rounded off to two decimal places) of magnitude _______________.

The probability of solving a problem by Student A is 1/3, and the probability of solving the same problem by Student B is 2/5. The probability (rounded off to two decimal places) that at least one of the students solves the problem is _______________.

Numerical value of work done (rounded off to the nearest integer) by a position-dependent force \(\vec{F} = x\hat{i} + 5xy\hat{j}\) along the path \(y = \dfrac{x^2}{2}\), from (0,0) to (2,2) in the xy plane is _______________.

The estimated value of the cube root of 37 (rounded off to two decimal places) obtained from the Newton-Raphson method after two iterations (\(x_2\)) is ________________.
[Start with an initial guess value of \(x_0 = 1\)].

Equilibrium concentration of dissolved nitrogen (in wt.%, rounded off to three decimal places) in pure liquid iron exposed to atmospheric air at 1873 K is __________.
Given: Sieverts' law constant: \(\log_{10} K_{[N]} = \left[-\dfrac{518}{T} - 1.063\right]\) where \(K_{[N]}\) has dimensions of atm\(^{-1/2}\). Assume \([h_N] = [\text{wt.\% N}]\).

Cold working of iron increases dislocation density from \(10^{10}\) to \(10^{15}\) m⁻². The associated stored energy (in MJ·m⁻³, rounded off to one decimal place) is ________________.
Given: Shear modulus of iron = 82 GPa; Burger's vector \(\vec{b} = \frac{a_0}{2}[111]\); \(a_0 = 0.2856\) nm.

During decarburization in a steel bath at 1550 oC, the compositions of dissolved C (wt.%C)
and dissolved O (wt.%O) follow the relation:
(wt.%C)(wt.%O) = K
When the partial pressure of CO (𝑝𝐶𝑂) is 1 atm, K = 0.002.
If 𝑝𝐶𝑂 = 0.1 atm, what is the value of K, at the same temperature?
Note: Assume Henry’s law is applicable.

Two equal and opposite point charges \(+Q\) and \(-Q\) are located as shown in the figure below. A surface integral, \(F_i\) is defined on surface \(S_i\) of a sphere of radius \(r_i\) as follows:

\[F_i = \oint_{S_i} (\vec{E}\cdot\hat{n})\,dS\]

where \(\vec{E}\) is the electric field, and \(\hat{n}\) is the unit normal to the surface of integration.
If \(r_1:r_2:r_3\) are in the ratio 1:2:5, use the Gauss divergence theorem to determine the ratio \(F_1:F_2:F_3\).

Consider the following Ordinary Differential Equation:

\[\frac{d}{dx}\!\left(c\,\frac{dc}{dx}\right) = 0\]

In a domain \(0 \le x \le t\), with boundary conditions \(c(0)=0.5\) and \(c(t)=1.0\), pick the appropriate choice for \(c(x)\) from the following options:

A classroom of 20 students can be categorized on the basis of blood-types: 5 students each
with “A”, “B”, “AB”, and “O” blood-types. If four students are selected at random from
this class, what is the probability that each student has a different blood-type?

Consider the following stress state imposed on a material:

\[\sigma = \begin{bmatrix} 90 & 50 & 0 \\ 50 & -20 & 0 \\ 0 & 0 & 140 \end{bmatrix}\,\mathrm{MPa}\]

If the material responds elastically with a volumetric strain \(\Delta = 3.5\times10^{-4}\), what is its bulk modulus?

Using the trapezoidal rule with two equal intervals (\(n = 2\), \(\Delta x = 1\)), the definite integral \[\int_2^4 \ln(x)\,dx = \text{______________ (to two decimal places).}\]

For the matrix \(A = \begin{bmatrix}1&1&2\\2&1&1\\1&1&2\end{bmatrix}\), \(AA^T\) is

The mean of a numerical data-set is \(\bar{X}\) and the standard deviation is \(S\). If a number \(K\) is added to each term in the data-set then the mean and standard deviation become:

If \(f(x) = e^{|x|}\) then at \(x = 0\), the function \(f(x)\) is

A rod is elastically deformed by a uniaxial stress resulting in a strain of 0.02. If the Poisson’s ratio is 0.3, the volumetric strain is ___

Assume that the probability of South Africa winning against India is 1/3. If South Africa plays a 3 match cricket series against India, the probability that South Africa wins only one match is ___

The definite integral \(\int_0^4 e^{-x^2}\,dx\) is to be evaluated numerically. Divide the integration interval into exactly 2 subintervals of equal length. Applying the trapezoidal rule, the approximate value of the integral is ___

For the second order linear ordinary differential equation, \(\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = 0\), the following function is a solution: \(y = e^{\lambda x}\). Which one of the following statements is NOT TRUE?

For the linear transformation, if one of the eigenvalues is 0, the other eigenvalue is ___

The general solution of the ordinary differential equation d²y/dx² + dy/dx = 0 is:

If V = x²yz î + xy²z ĵ + xyz² k̂, the divergence of V is:

A hot body cools according to dT/dt = −cT, where T is instantaneous temperature and c = 0.05 s⁻¹. Using forward difference, the maximum time step Δt (in seconds) for numerical stability is ___

Solve x = e^−x using Newton-Raphson method. Starting with x₀ = 0, the value of x after the first iteration is ___

A coin is tossed three times. It is known that out of the three tosses, one is a HEAD. The probability of the other two tosses also being HEADs is ___

The value of the integral ∫₀^π/2 x sin x dx = ___

For a hypothetical binary liquid system A-B at 1073 K. Given the partial pressures of A at various compositions, when the atom fraction of A is 0.4, the activity of A in the liquid is ___

Consider the following five readings: 19, 17, 15, 13, 11. The standard deviation is ___

\(\frac{f(x+h)-f(x)}{h}\) is a numerical approximation for:

One of the eigenvalues of the matrix shown below is \(-3\). The other eigenvalue is ___

Given \(f = xyz\), the magnitude of the gradient \(|\nabla f|\) at the point (0, 2, 2) is ___

The determinant of the matrix \(\begin{bmatrix}\cos\theta & \sin\theta & 0\\-\sin\theta & \cos\theta & 0\\0 & 0 & 1\end{bmatrix}\) is ___

The solution of \(\frac{dy}{dx} = 5x\) with \(y(0) = 0\) is:

In FCC, the strain energy of a \(\frac{1}{2}[110]\) dislocation is ___ times that of a \(\frac{1}{2}[112]\) dislocation.

Identify the wave equation from the following:

If one row of a 3×3 matrix is multiplied by 3, the determinant changes by a factor of ___

The trapezoidal rule approximates the function in each interval as:

The curl of the gradient of a scalar field, \(\nabla \times \nabla\phi\), is:

The divergence of the position vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) in 3D space, \(\nabla \cdot \vec{r}\), is:

For \(K_{IC} = 45\) MPa√m and applied stress = 400 MPa, the critical half-crack length (in mm) is ___

Numerical calculation problem (see question paper for details). The answer is ___

Numerical calculation (see question paper for details). The answer is ___

Numerical answer type (see question paper). The answer is ___

The degree and order of the differential equation \(\frac{d^2y}{dx^2} + x^2\left(\frac{dy}{dx}\right)^3 - 6y = 0\) are:

In a binary A–B system, a miscibility gap occurs if:

The isothermal compressibility of a material is:

\(\nabla \cdot (\nabla \varphi \times \nabla \psi) = \) ___

The atomic packing fraction of the diamond cubic structure is ___

Which of the following series is divergent?

The Taylor series expansion of \(e^x\) around \(x = 0\) is:

Which of the following attributes is NOT correct for the rotation matrix \(\begin{bmatrix}\cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1\end{bmatrix}\) at \(\theta = 60°\)?

A box contains 150 gumballs: 112 within tolerance, 23 below, 15 above. If two gumballs are picked without replacement, the probability of picking one below tolerance and then one above tolerance is:

The critical initial crack length for Mode-I fracture: \(K_{IC} = 45\) MPa\(\sqrt{m}\), applied stress \(\sigma = 400\) MPa. The value of \(a_c\) (in mm) is ___

A is a 2×2 matrix with det(A) = 2. The det(2A) is:

The eigenvalues of the matrix shown in the figure are:

Which of the following methods is NOT used for numerical integration?

The polynomial function has the following behaviour at x = 0.5:

Match equations in Group I with physical meaning in Group II.

Temperature field of a slab is given by T = 400 − 50 exp(−t − x² − y²). The temperature gradient in the y-direction is:

The solution of the given differential equation represents:

A thin layer of material B is plated on the end faces of two long rods of material A. These are then joined together on the plated side and heated. Assuming diffusion coefficient of B in A is D, the composition profile along the rod axis after time t is described by:

The terminal velocity of a fine spherical steel particle (diameter d_m in mm) falling in quiescent liquid aluminium is:

Which one of the following methods is NOT used for numerically solving an ordinary differential equation?

The inflection point of a nonlinear function U(r) is at

A box contains 5 white balls and 3 red balls. Two balls are withdrawn from the box randomly, one after another (without replacement). The probability that the two balls withdrawn are of different colour is

A large set of data for a given measurement has been found to be normally distributed around a mean μ, with standard deviation σ. Which of the following limits would have about 95% of the data points around the mean and rest outside?

For the thermodynamic relation G = U + PV − TS, which one of the following partial derivative relations is CORRECT?

Which of the following is NOT a property of a 4 × 4 singular matrix?

Which of the following is an iterative technique to solve a linear system of equations?

Which of the following partial derivative is equal to (dT/dP) at constant S?

Two fluids of densities ρ₁ and ρ₂ are flowing at velocities v₁ and v₂, respectively, through smooth pipes of identical diameter and pressure per unit length. When the friction factor is same, the ratio v₁/v₂ is equal to

Given the polynomial: x³ − 3x² + 4x − 2.5 = 0. Starting from a guess value x = 0 what will be the value of x after iterating twice using the Newton-Raphson method?

The probability of obtaining “head” n times, on tossing an unbiased coin N times is given by

The function \(f(x) = ax^2 + bx + c\) has a maxima only if

A furnace wall consists of four layers of different materials, M1, M2, M3 and M4. If the layers are of equal thickness and the steady state temperature profile is as shown below, then the material with the lowest thermal conductivity is

\(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}\) is equal to

Fick’s first law relates

A unit dislocation with a Burgers vector \(\vec{b}_1\) will dissociate into two partial dislocations with Burgers vectors \(\vec{b}_2\) and \(\vec{b}_3\), if and only if
P. \(b_1^2 > b_2^2 + b_3^2\)
Q. \(b_1^2 < b_2^2 + b_3^2\)
R. \(\vec{b}_1 = \vec{b}_2 + \vec{b}_3\)
S. \(\vec{b}_1 = \vec{b}_2 \times \vec{b}_3\)

The solution function \(y = f(x)\) for the ordinary differential equation \(dy/dx = 3x^2 - 2x\), passes through (1,1). The magnitude of \(y\) at \(x = 3\) is

What is the magnitude of the following integral using single step application of trapezoidal rule?
\(\displaystyle\int_0^4 (3x^2 + 4x - 2)\,dx\)

The stress applied on a material is
\(\sigma_{ij} = \begin{bmatrix} 21 & 0 & 0 \\ 0 & 21 & 0 \\ 0 & 0 & 21 \end{bmatrix}\) MPa.
The maximum shear stress experienced by it is

Taylor series can be used to approximate the value of \(f(x) = \cos x\) by expanding around \(x = 0\). If only the first three terms of the series are considered, the magnitude of deviation from the actual value of \(\cos(\pi/3)\) will be

Determinant \(\begin{vmatrix} 3 & 1 & 2 \\ 1 & 3 & 4 \\ 2 & 1 & 3 \end{vmatrix}\) is

\(\int \frac{dx}{a+bx}\) is

Which of the following is a solution for \(\frac{\partial z}{\partial t} = \frac{\partial^2 z}{\partial x^2}\)

Deoxidation of liquid steel with ferrosilicon produces spherical silica particles. The particles of 5 \(\mu\)m diameter take 3000 minutes to float up through a 2 m height of liquid steel. For particles of 50 \(\mu\)m diameter to float up through the same height, the time required in minutes is

For Ni + 0.5O\(_2\) = NiO, \(\Delta G^0 = -250{,}000 + 100T\) Joules. At 1000 K, the \(p_{O_2}\) in equilibrium with Ni/NiO in atm is

In a gaseous mixture, CO, CO\(_2\), and O\(_2\) are in equilibrium at temperature T. For the reaction CO + \(\frac{1}{2}\)O\(_2\) = CO\(_2\), \(\Delta G^0 = -281{,}000 + 87.6T\) Joules. The correct statements among the following are:
(P) The reaction will shift to left on increasing T
(Q) The reaction will shift to right on increasing T
(R) The reaction will shift to left on increasing pressure
(S) The reaction will shift to right on increasing pressure

The number of boundary conditions required to solve a steady-state two-dimensional diffusion equation (\(\nabla^2 C = 0\)) is

The determinant of the matrix \(\begin{bmatrix} 1 & 3 & 2 \\ 2 & 6 & 4 \\ -5 & 3 & 1 \end{bmatrix}\) is

The configurational entropy \(S_c\) of an ideal solid solution is given by \(S_c = -R[x \ln x + (1-x) \ln(1-x)]\), where \(x\) is the mole fraction of solute. The value of \(S_c\) as \(x\) tends to zero (\(\lim_{x\to 0} S_c\)) is

In a BOF steelmaking, the hydrogen mass balance is governed by the following equation: \(-W\frac{dC_H}{dt} = R(C_H - C_{H,eq})\), where \(W\) is the capacity of the degasser in tons, \(C_H\) is the hydrogen concentration at any time \(t\), and \(R\) is the recirculation rate in tons per minute. If hydrogen concentration in liquid steel, if it drops from 5 ppm to 1 ppm in 20 minutes, \(R\) is

If \(\mathbf{V} = (4xy - 3z^2)\mathbf{i} + 2x^2\mathbf{j} - 9xz^2\mathbf{k}\), the divergence of \(\mathbf{V}\) is

The carbon concentration profile \(C(x,t)\) during decarburization is given by: \(C(x,t) = L + M \operatorname{erf}\left(\frac{x}{2\sqrt{Dt}}\right)\). The furnace atmosphere is free of carbon and maintained at 927\(^\circ\)C. For a steel with initial carbon of 1.2%, how long to attain 0.8% C at 0.5 mm below the surface?
[Given: \(D = 1.28 \times 10^{-11}\) m\(^2\)/s at 927\(^\circ\)C; erf(0.65) = 0.64, erf(0.69) = 0.667, erf(0.71) = 0.678]

The probability distribution function, \(p(x)\), for a random variable, \(x\), is given by: \(p(x) = \frac{1}{\sqrt{\pi}}\exp(-x^2)\). The probability that \(x\) lies between \(x_1 = 0.6\) and \(x_2 = 0.8\) is [Use single-step trapezoidal rule]

The settling velocity of a 0.5 \(\mu\)m diameter particle (density = 4900 kg/m\(^3\)) under laminar flow conditions is
[Given: viscosity of water = 1 centipoise]

The recovery of gold in the following operation is
Feed: 8000 tons per day, with 8.6 g of Gold per ton. Concentrate: 100 tons per day. 0.71 g of Gold per ton in tailings.

The atomic packing factor for the diamond cubic structure is

In a cubic crystal with lattice parameter \(a\), the dislocation reaction that is vectorially correct and energetically feasible is

When one mole of copper is quenched from 1000 K to 300 K, the amount of heat released is
[Given: the specific heat capacity of copper in J K\(^{-1}\) mol\(^{-1}\): \(C_p = 22.68 + 6.3 \times 10^{-3} T\), where \(T\) is temperature]

If \(\Delta G_v = 3.0 \times 10^7\) J m\(^{-3}\), and \(\gamma = 3.3 \times 10^{-2}\) J m\(^2\), the number of atoms in the critical cluster is approximately
[Given: the solid is an FCC crystal with a lattice parameter of 0.495 nm]

Euclidean norm of the matrix \(\begin{bmatrix} 5 & 9 \\ -2 & 1 \end{bmatrix}\) is

If F, the gradient of a differentiable function f, is a vector function of x and the function F is continuous in a region \(\Omega\) of \(R^n\), then for any closed curve lying in \(\Omega\), the line integral \(\int F(x)\,dx\) is

The weight of an assembly of N particles of given density determines its average size, \(\bar{x}\). The variance of the size distribution may be estimated from the measurements of size on each particle, \(x_j\), as

The general solution of the first order differential equation \(\frac{dy}{dx} = \cot(ax)\) is

\(\lim_{x\to 0}\frac{1-\cos x}{x^2}\) is

The order, O, of local error (in terms of step size, h) in the Runge-Kutta 4th order method for solving ODE is

The general solution of the integral, \(\int\frac{x\,dx}{4-x^2+\sqrt{4-x^2}}\) is

The general solution of the differential equation, \(\frac{dy}{dx}-\frac{2}{x+1}y=(x+1)^3\) is

The function f(x) is known from the following table: x=2.4, f(x)=0.318; x=2.5, f(x)=0.286; x=2.6, f(x)=0.253. The first derivative f'(x) at x=2.5 is estimated by central difference as

The matrix operator for 60\(^\circ\) anticlockwise rotation of a point (x, y, z) along the z-axis is

The eigenvectors of the matrix \(\begin{pmatrix}4 & 2\\ 3 & 3\end{pmatrix}\) are

Identify the first derivatives of the thermodynamic functions in Group 1 that are equal to the corresponding state variables given in Group 2. Group 1: (P) \((\partial U/\partial S)_V\), (Q) \((\partial G/\partial T)_P\), (R) \((\partial G/\partial P)_T\), (S) \((\partial U/\partial V)_S\). Group 2: (1) \(-p\), (2) \(-S\), (3) V, (4) T

The solubility of nitrogen in liquid iron at 1600\(^\circ\)C under 1 atm pressure of nitrogen gas is 0.046 wt %. Nitrogen solubility in the binary Fe-N system obeys Sievert's law. Interaction parameters for solutes in molten iron at 1600\(^\circ\)C are \(e_N^C=0.11\) and \(e_N^{CC}=0.0067\). Other higher order interaction parameters are assumed to be zero. If the metal contains 4.4 wt % C, the solubility (in wt %) of nitrogen is

For the Mn partitioning reaction (MnO)+[C]↔[Mn]+CO(g) between slag and hot metal in a blast furnace, the equilibrium constant is given by \(\log_{10}K_{Mn}=-\frac{15090}{T}+10.97\). Assume that MnO activity in slag and partial pressure of CO(g) are not affected by the temperature change. When temperature is increased from 1300\(^\circ\)C to 1350\(^\circ\)C, the Mn content of the hot metal changes by a factor

For the Si partitioning reaction (SiO\(_2\))+2[C]↔[Si]+2CO(g), the equilibrium constant is given by \(\log_{10}K_{Si}=-\frac{30935}{T}+20.455\). The expression for equilibrium constant for the combined Mn-Si partitioning, \(\log_{10}K_{Mn-Si}\), is given by

During load versus load-line displacement measurement for the determination of \(J_{IC}\), Compliance is measured in terms of

The operation of the matrix \(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\) on vectors in the 2-D Cartesian space corresponds to

The radius of curvature of a curve f(x) at \(x = x_0\) is

The condition for a function f(x) to exhibit a point of inflection at \(x = x_0\) is

The solution of the equation \(\dfrac{\partial c}{\partial t} = D\dfrac{\partial^2 c}{\partial x^2}\) is of the form

The value of the summation \(\displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n!}\) is

Which of the following statements are true: (P) div curl A = 0, (Q) curl grad A = 0, (R) grad div A = \(\nabla^2\)A, (S) div grad A = \(\nabla^2\)A

The flux balance equation for the reaction \(2[Al] + 3(O) = Al_2O_3\), where J is the flux in mol s\(^{-1}\)m\(^{-2}\), is

In spherical coordinates (r = radius, T = temperature, and t = time), the heat conduction equation in steady state is

Decomposition of calcium carbonate occurs as CaCO\(_3\) = CaO + CO\(_2\); \(\Delta G_T^0 = 17710 - 158\) T J mol\(^{-1}\). What will be the decomposition temperature (in K) of CaCO\(_3\) if the activity of CaO is 0.5 and the pressure is 3 atmospheres.

For ideal gases, the difference between heat capacities per mole at constant pressure and at constant volume, \(c_p - c_v = R\), is true because

A furnace wall is made of three materials (I, II and III) of equal thickness and having thermal conductivities k₁, k₂, and k₃ respectively. The steady state temperature profile inside each material is shown in the figure below.
Thermal conductivity of the materials would vary as

If the drift velocity of holes under a field gradient of 150 V/m is 7.5 m/s, their mobility (in SI units) is

A dislocation line in a FCC crystal dissociates into two partials which have their Burgers vectors as (a/6)[1 1̅ 1] and (a/6)[1̅ 1 2̅]. Indicate the correct statement.

Identify the correct statement for the product of three vectors a, b and c and scalar k

A first order ordinary differential equation is given by
dy/dt = ky

The general solution of the equation is

The Maclaurin series expansion of 1/(1−z) is
1/(1−z) = ∑ zⁿ = 1 + z + z² + ...   (|z| < 1)

If we replace z by −z², then the series would be

Given the matrix [−4.0, 4.0; −1.6, 1.2]. The correct pair of eigenvalues is

Identify the correct statement

The order of the following differential equation is y'''y'' + 2y'³ = (x² + 2)² · x⁸

The magnitude of the following determinant is
| 1   0   0 |
| 6   2   5 |
| 1   3   2 |

The rank of the following matrix is[3   0   2]
[−1   7   4]
[6   0   4]
[2   7   6]

In the AOD process of steelmaking, a mixture of argon and oxygen gas is injected into liquid steel in the volume ratio of 3:1. If all the oxygen is completely utilized for the oxidation of carbon, then the partial pressure of CO in the gas coming out of the metal at an ambient pressure of 1 atmosphere is

Match the following:
Group 1: P. Newton Raphson, Q. Gauss Seidel, R. Gauss Quadrature, S. Runge–Kutta
Group 2: 1. Ordinary differential equations, 2. Roots of equations, 3. System of linear equations, 4. Integration, 5. Interpolation, 6. Extrapolation

Fermi level of an atom refers to

The stress required to move a dislocation of Burgers vector 3Å through a matrix having shear modulus of 80 GPa and containing coherent precipitates separated by an average distance of 0.3 μm is

In a 3 × 3 matrix A_ij, it is known that A₂₃ = 3A₃₂. Its determinant is:

A Frank Sessile dislocation in an fcc lattice has the following Burgers vector:

The solution of y″ − y = 0 is given by: