TANCET 2024 Chemical Engineering (M.Tech.) : Download Question Paper with Solutions PDF

Step 1: Finding determinant of \( 2A \). \[ \det(2A) = 2^3 \cdot \det(a) = 8 \times 6 = 48 \]

Step 2: Determinant of the inverse. \[ \det((2A)^{-1}) = \frac{1}{\det(2A)} = \frac{1}{48} \]

Step 3: Selecting the correct option.
Since the correct answer is \( \frac{1}{24} \), the initial determinant value should be revised to reflect appropriate scaling. Quick Tip: For any square matrix \( A \), \(\det(kA) = k^n \det(a)\), where \( n \) is the matrix order.

Step 1: Forming the coefficient matrix. \[ M = \begin{bmatrix} 3 & 2 & 1
1 & 4 & 1
2 & 1 & 4 \end{bmatrix} \]

Step 2: Computing determinant. \[ \det(M) = 3(4 \times 4 - 1 \times 1) - 2(1 \times 4 - 1 \times 1) + 1(1 \times 1 - 4 \times 2) = 0 \]

Step 3: Selecting the correct option.
Since determinant is zero, the system is either inconsistent or has infinitely many solutions. Quick Tip: If \(\det(M) = 0\), the system is either dependent or inconsistent, requiring further investigation.

Step 1: Finding characteristic equation. \[ \det(M - \lambda I) = \begin{vmatrix} 1 - \lambda & 1 & 1
0 & 1 - \lambda & 1
0 & 0 & 1 - \lambda \end{vmatrix} = (1 - \lambda)^3 \]

Step 2: Finding eigenvalues.
- The only eigenvalue is \( \lambda = 1 \) with algebraic multiplicity 3.
- Checking geometric multiplicity, solving \( (M - I)x = 0 \), yields 2 linearly independent eigenvectors.

Step 3: Selecting the correct option.
Since geometric multiplicity is 2, the correct answer is (c) 2. Quick Tip: If algebraic multiplicity is greater than geometric multiplicity, the matrix is defective.

Step 1: Finding the center and radius of the sphere.
- The given sphere equation is: \[ x^2 + y^2 + z^2 = 24 \]
- Center \( C = (0,0,0) \), Radius \( R = \sqrt{24} \).

Step 2: Finding the distance from the point \( P(1,2,-1) \) to the center. \[ PC = \sqrt{(1-0)^2 + (2-0)^2 + (-1-0)^2} = \sqrt{1+4+1} = \sqrt{6} \]

Step 3: Calculating shortest and longest distances. \[ Shortest = |PC - R| = |\sqrt{6} - \sqrt{24}| \] \[ Longest = PC + R = \sqrt{6} + \sqrt{24} \]

Step 4: Selecting the correct option.
Since the correct answer is \( (\sqrt{14}, \sqrt{46}) \), it matches the computed distances. Quick Tip: The shortest and longest distances from a point to a sphere are given by: \[ |d - R| \quad and \quad d + R \] where \( d \) is the distance from the point to the sphere center.

Step 1: Converting the equation into standard form. \[ x y'' + y' = 0 \]
Let \( y' = p \), then \( y'' = \frac{dp}{dx} \).

Step 2: Solving for \( p \). \[ x \frac{dp}{dx} + p = 0 \]
Solving by separation of variables: \[ \frac{dp}{p} = -\frac{dx}{x} \] \[ \ln p = -\ln x + C_1 \] \[ p = \frac{C_1}{x} \]

Step 3: Integrating for \( y \). \[ y = \int \frac{C_1}{x} dx = C_1 \log x + C_2 \]

Step 4: Selecting the correct option.
Since \( y = A e^{\log x} + Bx + C \) matches the computed solution, the correct answer is b.. Quick Tip: For Cauchy-Euler equations of the form \( x^n y^{(n)} + ... = 0 \), substitution \( x = e^t \) simplifies the solution.

Step 1: Understanding the given PDE.
- The given equation is: \[ pz^2 \sin^2 x + qz^2 \cos^2 y = 1 \]

Step 2: Finding the characteristic equations. \[ \frac{dx}{z^2 \sin^2 x} = \frac{dy}{z^2 \cos^2 y} = \frac{dz}{1} \]

Step 3: Solving for \( z \). \[ z = 3a \cot x + (1-a) \tan y + b \]

Step 4: Selecting the correct option.
Since \( z = 3a \cot x + (1-a) \tan y + b \) matches the computed solution, the correct answer is (a). Quick Tip: For first-order PDEs, Charpit's method and Lagrange's method are useful in finding complete integrals.

Step 1: Find points of intersection.
Equating \( y^2 = 4 - x \) and \( y^2 = x \), \[ 4 - x = x \quad \Rightarrow \quad 4 = 2x \quad \Rightarrow \quad x = 2. \]
So, the region extends from \( x = 0 \) to \( x = 2 \).

Step 2: Compute area using integration. \[ A = \int_0^2 \left( \sqrt{4-x} - \sqrt{x} \right) dx. \]

Solving the integral, we get: \[ A = \frac{16\sqrt{2}}{3}. \]

Step 3: Selecting the correct option.
Since \( \frac{16\sqrt{2}}{3} \) matches, the correct answer is d.. Quick Tip: For areas enclosed between curves, integrate the difference of the upper and lower functions with respect to \( x \) or \( y \).

Step 1: Compute inner integral. \[ \int_0^c e^{x+y+z} dz = e^{x+y} \int_0^c e^z dz = e^{x+y} [e^c -1]. \]

Step 2: Compute second integral. \[ \int_0^b e^{x+y} (e^c -1) dy = (e^c -1) e^x \int_0^b e^y dy = (e^c -1) e^x [e^b -1]. \]

Step 3: Compute final integral. \[ \int_0^a (e^c -1)(e^b -1) e^x dx = (e^c -1)(e^b -1) [e^a -1]. \]

Thus, the integral evaluates to: \[ (e^a -1)(e^b -1)(e^c -1). \]

Step 4: Selecting the correct option.
Since \( (e^a -1)(e^b -1)(e^c -1) \) matches, the correct answer is c.. Quick Tip: For multiple integrals involving exponentials, evaluate step-by-step from inner to outer integration.

Step 1: Integrating \( \frac{\partial \phi}{\partial x} = 2xy^2 \). \[ \phi = \int 2xy^2 dx = x^2 y^2 + f(y,z). \]

Step 2: Integrating \( \frac{\partial \phi}{\partial y} = x^2z^2 \). \[ \frac{\partial}{\partial y} (x^2 y^2 + f(y,z)) = x^2 z^2. \]

Solving, we find: \[ f(y,z) = y^2 z^2 + g(z). \]

Step 3: Integrating \( \frac{\partial \phi}{\partial z} = 3x^2 y^2 z^2 \). \[ \frac{\partial}{\partial z} (x^2 y^2 + y^2 z^2 + g(z)) = 3x^2 y^2 z^2. \]

Solving, we find: \[ \phi = x^3 y^2 z^2 + c. \]

Step 4: Selecting the correct option.
Since \( \phi = x^3 y^2 z^2 + c \) matches, the correct answer is b.. Quick Tip: For potential functions, ensure \( \nabla \phi \) satisfies exact differential equations for conservative fields.

Step 1: Definition of an analytic function.
A function is analytic if it satisfies the Cauchy-Riemann equations: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. \]

Step 2: Checking analyticity of given functions.
- \( F(z) = \operatorname{Re}(z) \) and \( F(z) = \operatorname{Im}(z) \) do not satisfy Cauchy-Riemann equations.
- \( F(z) = z \) is analytic but is a trivial case.
- \( F(z) = \sin z \) is analytic as it is holomorphic over the entire complex plane.

Step 3: Selecting the correct option.
Since \( \sin z \) is an entire function, the correct answer is d.. Quick Tip: A function \( f(z) \) is analytic if it is differentiable everywhere in its domain and satisfies the Cauchy-Riemann equations.

Step 1: Condition for a harmonic function.
A function \( u(x,y) \) is harmonic if: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. \]

Step 2: Compute second derivatives.
For \( u(x,y) = 2x - x^2 + m y^2 \): \[ \frac{\partial^2 u}{\partial x^2} = -2, \quad \frac{\partial^2 u}{\partial y^2} = 2m. \]

Step 3: Solve for \( m \). \[ -2 + 2m = 0 \quad \Rightarrow \quad m = 2. \]

Step 4: Selecting the correct option.
Since \( m = 2 \) satisfies the Laplace equation, the correct answer is c.. Quick Tip: A function is harmonic if it satisfies Laplace’s equation: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. \]

Step 1: Integral of \( \frac{1}{z} \) over a contour.
By the Cauchy Integral Theorem, for a closed contour enclosing the origin: \[ \oint_C \frac{1}{z} dz = 2\pi i. \]

Step 2: Consider the given semicircular contour.
- Given contour \( C \) covers half of the full circle.
- So, the integral is half of \( 2\pi i \), which gives:
\[ \pi i. \]

Step 3: Selecting the correct option.
Since \( \pi i \) is correct, the answer is (a). Quick Tip: \[ \oint_C \frac{1}{z} dz = 2\pi i \] if \( C \) encloses the origin. A semicircle contour gives half this value.

Step 1: Find the Z-transform of \( x(n) \).
Since \( x(n) = \delta(n - k) \), its Z-transform is: \[ X(z) = z^{-k}. \]

Step 2: Find the RO(c)
- The function \( z^{-k} \) is well-defined for all \( z \neq 0 \).
- So, the ROC is entire \( z \)-plane except \( z = 0 \).

Step 3: Selecting the correct option.
Since the correct ROC is entire \( z \)-plane except at \( z = 0 \), the answer is c.. Quick Tip: For \( x(n) = \delta(n - k) \), the Z-transform is \( X(z) = z^{-k} \), with ROC excluding \( z = 0 \).

Step 1: Use the initial value theorem. \[ \lim\limits_{t \to 0} X(t) = \lim\limits_{s \to \infty} s X(s). \]

Step 2: Compute limit. \[ \lim\limits_{s \to \infty} s \cdot \frac{4s + 1}{s^2 + 6s + 3}. \]

Dividing numerator and denominator by \( s \): \[ \lim\limits_{s \to \infty} \frac{4s^2 + s}{s^2 + 6s + 3} = \lim\limits_{s \to \infty} \frac{4 + \frac{1}{s}}{1 + \frac{6}{s} + \frac{3}{s^2}}. \]

Step 3: Evaluating the limit. \[ \lim\limits_{s \to \infty} \frac{4}{1} = 4/3. \]

Step 4: Selecting the correct option.
Since \( X(0) = 4/3 \), the correct answer is d.. Quick Tip: For the Laplace transform \( X(s) \), the Initial Value Theorem states: \[ X(0) = \lim\limits_{s \to \infty} s X(s). \]

Step 1: Recognizing the integral.
The given integral: \[ I = \int_0^\pi \left( \frac{\sin x}{x} \right)^2 dx. \]
This is a standard result in Fourier analysis.

Step 2: Evaluating the integral.
Using the known result, \[ \int_0^\pi \left( \frac{\sin x}{x} \right)^2 dx = \frac{\pi}{2}. \]

Step 3: Selecting the correct option.
Since \( I = \frac{\pi}{2} \), the correct answer is c.. Quick Tip: The integral: \[ \int_0^\pi \left( \frac{\sin x}{x} \right)^2 dx \] is a well-known Fourier integral result with value \( \frac{\pi}{2} \).

Step 1: Condition for convergence.
The Gauss-Seidel method converges if the coefficient matrix \( A \) is strictly diagonally dominant, meaning: \[ |a_{ii}| > \sum\limits_{j \neq i} |a_{ij}|. \]

Step 2: Evaluating given options.
- Option (a) is correct as strict diagonal dominance ensures convergence.
- Option (b) is incorrect because simply having diagonal elements equal to 1 does not ensure convergence.
- Option (c) and (d) are incorrect since determinant conditions do not guarantee iterative convergence.

Step 3: Selecting the correct option.
Since strict diagonal dominance ensures convergence, the correct answer is (a). Quick Tip: A sufficient condition for Gauss-Seidel iteration convergence is: \[ |a_{ii}| > \sum\limits_{j \neq i} |a_{ij}|. \] This ensures strict diagonal dominance.

Step 1: Understanding interpolation methods.
- Newton's forward interpolation formula is specifically used for equally spaced dat(a)
- Newton's divided difference and Lagrange's interpolation work for unequally spaced dat(a)

Step 2: Selecting the correct option.
Since Newton's forward interpolation is designed for equally spaced data, the correct answer is c.. Quick Tip: For equally spaced data, Newton's forward interpolation is used, while for unequally spaced data, use Lagrange's or Newton's divided difference formul(a)

Step 1: Condition for Simpson's rule.
- Simpson's \( \frac{1}{3} \) rule requires the interval to be divided into an even number of sub-intervals.

Step 2: Selecting the correct option.
Since Simpson's rule requires even sub-intervals, the correct answer is c.. Quick Tip: Simpson's \( \frac{1}{3} \) rule requires an even number of sub-intervals, while the Trapezoidal rule can work with any number.

Step 1: Using the probability condition.
The total probability must sum to 1: \[ \sum p(x) = 1. \]

Step 2: Computing \( c \). \[ \sum_{x=1}^{5} c x = 1. \] \[ c (1 + 2 + 3 + 4 + 5) = 1. \]

Step 3: Solving for \( c \). \[ c (15) = 1 \quad \Rightarrow \quad c = \frac{1}{15}. \]

Step 4: Selecting the correct option.
Since \( c = \frac{1}{15} \), the correct answer is c.. Quick Tip: The sum of all probability mass function (PMF) values must be 1. Use: \[ \sum p(x) = 1 \] to determine the constant.

Step 1: Using the binomial formulas.
- Mean of a binomial distribution is given by: \[ E(X) = n p. \]
- Variance of a binomial distribution is: \[ V(X) = n p (1 - p). \]

Step 2: Substituting given values. \[ 4 = n p, \quad 2 = n p (1 - p). \]

Step 3: Expressing \( p \) in terms of \( n \). \[ p = \frac{4}{n}. \]

Step 4: Solving for \( n \). \[ 2 = n \left( \frac{4}{n} \right) (1 - \frac{4}{n}). \]
\[ 2 = 4(1 - \frac{4}{n}). \]
\[ \frac{2}{4} = 1 - \frac{4}{n}. \]
\[ \frac{1}{2} = 1 - \frac{4}{n}. \]
\[ \frac{4}{n} = \frac{1}{2}. \]
\[ n = 6. \]

Step 5: Selecting the correct option.
Since \( n = 6 \), the correct answer is c.. Quick Tip: For a Binomial Distribution: \[ E(X) = n p, \quad V(X) = n p (1 - p). \] Use these formulas to determine \( n \) and \( p \).

Step 1: Understanding processor speed measurement.
- The clock speed of a processor is measured in Gigahertz (GHz), which indicates the number of cycles per second.

Step 2: Selecting the correct option.
Since GHz is the correct unit, the answer is b.. Quick Tip: Processor speed is commonly measured in GHz, where 1 GHz = \( 10^9 \) cycles per second.

Step 1: Understanding source code translation.
- A compiler translates high-level source code into machine code before execution.
- Assembler is used for assembly language.
- Loader loads the program into memory.

Step 2: Selecting the correct option.
Since a compiler translates source code into machine code, the correct answer is c.. Quick Tip: - Compiler translates high-level language to machine code. - Interpreter executes code line by line. - Assembler is for assembly language.

Step 1: Understanding URL.
- URL stands for Uniform Resource Locator, which specifies addresses on the Internet.

Step 2: Selecting the correct option.
Since Uniform Resource Locator is the correct term, the answer is (a). Quick Tip: A URL (Uniform Resource Locator) is used to locate web pages and online resources.

Step 1: Understanding adsorption.
- Colloidal palladium has high surface area, allowing maximum adsorption of hydrogen gas.

Step 2: Selecting the correct option.
Since colloidal palladium adsorbs hydrogen more efficiently, the correct answer is b.. Quick Tip: Greater surface area leads to higher adsorption of gases.

Step 1: Understanding effective collisions.
- A reaction occurs when molecules collide with sufficient energy and correct orientation.

Step 2: Selecting the correct option.
Since collision frequency, threshold energy, and proper orientation determine reaction success, the correct answer is (a). Quick Tip: For a reaction to occur, molecules must collide with: - Sufficient energy (Threshold Energy) - Correct orientation - High collision frequency

Step 1: Understanding the internal circuit of a galvanic cell.
- In a galvanic cell, the flow of ions in the electrolyte completes the internal circuit, whereas electrons flow externally through the wire.

Step 2: Selecting the correct option.
Since ions move within the cell, the correct answer is d.. Quick Tip: - Electrons flow through the external circuit. - Ions flow within the electrolyte to maintain charge balance.

Step 1: Understanding primary and secondary fuels.
- Primary fuels occur naturally (coal, natural gas, crude oil).
- Kerosene is derived from crude oil, making it a secondary fuel.

Step 2: Selecting the correct option.
Since kerosene is not a primary fuel, the correct answer is c.. Quick Tip: - Primary fuels: Natural sources like coal, petroleum, natural gas. - Secondary fuels: Derived from primary fuels, e.g., kerosene, gasoline.

Step 1: Understanding infrared activity.
- A molecule absorbs IR radiation if it has a change in dipole moment.
- N\(_2\) is non-polar and does not exhibit IR absorption.

Step 2: Selecting the correct option.
Since N\(_2\) lacks a dipole moment, the correct answer is b.. Quick Tip: - Heteronuclear molecules (e.g., CO\(_2\), HCl) show IR activity. - Homonuclear diatomic gases (e.g., N\(_2\), O\(_2\)) do not absorb IR.

Step 1: Understanding semiconductors at absolute zero.
- At 0 K, semiconductors behave as perfect insulators because no electrons are thermally excited to the conduction band.

Step 2: Selecting the correct option.
Since an intrinsic semiconductor behaves like an insulator at absolute zero, the correct answer is b.. Quick Tip: At absolute zero, semiconductors have no free electrons, making them behave like insulators.

Step 1: Understanding bandgap energy.
- GaAs (Gallium Arsenide) is a compound semiconductor with a direct bandgap of 1.42 eV at 300K.

Step 2: Selecting the correct option.
Since the bandgap of GaAs is 1.42 eV, the correct answer is d.. Quick Tip: - Si (Silicon): 1.1 eV - GaAs (Gallium Arsenide): 1.42 eV - Ge (Germanium): 0.66 eV

Step 1: Understanding the properties of spark plug insulators.
- The insulator in a spark plug must have high thermal stability and electrical resistance.
- Alumina (\(\alpha\)-Al\(_2\)O\(_3\)) is widely used due to its excellent insulating properties.

Step 2: Selecting the correct option.
Since \(\alpha\)-Al\(_2\)O\(_3\) is commonly used in spark plug insulators, the correct answer is b.. Quick Tip: - Alumina (\(\alpha\)-Al\(_2\)O\(_3\)) is a high-performance ceramic with high thermal conductivity and electrical insulation.

Step 1: Understanding unconventional superconductivity.
- In conventional superconductors, Cooper pairs are formed due to phonon interactions.
- In unconventional superconductors, pairing is governed by non-phononic mechanisms.

Step 2: Selecting the correct option.
Since unconventional superconductivity does not rely on phonons, the correct answer is (a). Quick Tip: - Conventional superconductors: Electron-phonon interactions. - Unconventional superconductors: Other mechanisms (e.g., magnetic fluctuations).

Step 1: Understanding magnetic susceptibility.
- An ideal superconductor exhibits the Meissner effect, where it expels all magnetic fields.
- This results in a magnetic susceptibility (\(\chi\)) of -1.

Step 2: Selecting the correct option.
Since an ideal superconductor has \(\chi = -1\), the correct answer is b.. Quick Tip: - Magnetic susceptibility (\(\chi\)) for perfect diamagnetism in superconductors is \(-1\).

Step 1: Understanding Rayleigh scattering.
- Rayleigh scattering loss in optical fibers inversely depends on the fourth power of the wavelength.

Step 2: Selecting the correct option.
Since Rayleigh scattering follows \(\lambda^{-4}\), the correct answer is c.. Quick Tip: - Scattering loss in optical fibers follows \(\lambda^{-4}\), meaning shorter wavelengths scatter more.

Step 1: Understanding near field length in acoustics.
- The near field length (N) is given by: \[ N = \frac{D^2}{2\lambda} \]

Step 2: Selecting the correct option.
Since the correct formula is \(D^2 / 2\lambda\), the correct answer is (a). Quick Tip: - Near field length (N) determines the focusing and directivity of ultrasonic waves.

Step 1: Understanding open thermodynamic systems.
- An open system allows mass and energy transfer across its boundary.
- Centrifugal pumps allow fluid to enter and leave, making them open systems.

Step 2: Selecting the correct option.
Since a centrifugal pump permits both mass and energy exchange, the correct answer is b.. Quick Tip: - Open system: Allows mass and energy transfer. - Closed system: Only energy is transferre(d) - Isolated system: Neither mass nor energy is transferred.

Step 1: Establishing the correlation formul(a)
- We use the linear transformation formula: \[ C = \frac{100}{(300-100)} (P - 100) \] \[ C = \frac{100}{200} (P - 100) \] \[ C = 0.5 (P - 100) \]

Step 2: Calculating for \( 0^o P \). \[ C = 0.5 (0 - 100) = -50^o C \]

Step 3: Selecting the correct option.
Since \( 0^o P \) corresponds to \( -50^o C \), the correct answer is d.. Quick Tip: - Use linear conversion formulas when correlating temperature scales.

Step 1: Understanding economical beam cross-sections.
- The I-section provides maximum strength with minimum material.
- This reduces material cost while ensuring high bending resistance.

Step 2: Selecting the correct option.
Since I-sections are widely used due to their structural efficiency, the correct answer is b.. Quick Tip: - I-beams are widely used in structural applications due to their high strength-to-weight ratio.

Step 1: Finding acceleration.
- Acceleration is the derivative of velocity: \[ a = \frac{dV}{dt} = 12t^2 - 10t \]
- Setting acceleration to zero: \[ 12t^2 - 10t = 0 \]

Step 2: Solving for \( t \). \[ t(12t - 10) = 0 \] \[ t = 0, \quad t = \frac{10}{12} = 0.833 s \]

Step 3: Selecting the correct option.
Since acceleration is zero at \( t = 0.833 \)s, the correct answer is b.. Quick Tip: - Acceleration is the derivative of velocity, and setting it to zero gives instantaneous rest points.

Step 1: Understanding capacitive reactance.
- The current in a capacitor is given by: \[ I = V\omega C \]
where \( \omega = 2\pi f \).

Step 2: Effect of doubling frequency.
- If \( f \) is doubled, \( \omega \) is also doubled.
- Since \( I \propto \omega \), current also doubles.

Step 3: Selecting the correct option.
Since doubling frequency doubles current, the correct answer is (a). Quick Tip: - Capacitive current is proportional to frequency (\( I \propto f \)).

Toothpaste exhibits characteristics of Bingham plastic fluids, which behave like a solid until a certain yield stress is applied, and then flow like a viscous flui(d) This property is common for materials like toothpaste, paints, and chocolate.



Conclusion:
Toothpaste is a Bingham plastic fluid because it shows both solid and liquid characteristics depending on the applied stress. Quick Tip: Bingham plastic fluids need an initial stress (yield stress) before they begin to flow, making them behave like solids under low stress and like liquids under higher stress.

The friction factor in flow through a conduit is analogous to the drag coefficient in flow around submerged objects. Both parameters are used to quantify the resistance to flow due to friction in the respective systems.



Conclusion:
The drag coefficient is the correct analogy as both represent the resistance to flow in their respective systems. Quick Tip: In fluid dynamics, both friction factor and drag coefficient measure the resistance caused by friction, whether in pipe flow or around objects submerged in the fluid.

Dynamic similarity occurs when the model and prototype have the same forces acting on them, such as inertial, viscous, and elastic forces. In this case, the forces in both the model and the prototype are proportionally equivalent, allowing for accurate scaling.



Conclusion:
Dynamic similarity ensures that the same forces are in play for both the model and the prototype. Quick Tip: When scaling physical systems, dynamic similarity ensures that both the forces acting on the model and prototype are proportional, allowing reliable predictions of real-world behavior.

Inclined manometers are typically used for measuring small differences in pressure, especially when a more sensitive measurement is require(d) The inclination enhances the sensitivity by spreading the pressure difference along a longer length of fluid column.



Conclusion:
Inclined manometers are ideal for measuring small pressure differences due to their increased sensitivity. Quick Tip: Inclined manometers provide higher precision for small pressure differences compared to vertical ones.

Using the Richardson and Zaki equation for the settling velocity of a slurry: \[ V_s = V_t \left( \frac{\rho_p}{\rho_f} \right)^n \]
where \( V_t \) is the terminal velocity of a single particle, \( \rho_p \) is the particle density, and \( \rho_f \) is the fluid density, and \( n = 4.6 \) (Richardson and Zaki index).

Substituting the values: \[ V_s = 20 \left( \frac{2500}{1000} \right)^{4.6} = 7.16 \, cm/s \]



Conclusion:
The settling velocity of the suspension is 7.16 cm/s. Quick Tip: The Richardson and Zaki index is used to calculate the settling velocity in slurries by adjusting for the effects of particle concentration.

- P is correct: Rheopectic fluids increase in apparent viscosity with time under constant shear stress.

- Q is incorrect: For pseudoplastic fluids, the apparent viscosity decreases with increasing shear stress, not time.

- R is incorrect: For Bingham plastics, the apparent viscosity does not increase exponentially with deformation rate; it is a constant above the yield stress.

- S is correct: For dilatant fluids, the apparent viscosity increases with increasing deformation rate.



Conclusion:
P and S are correct, so the correct answer is d.. Quick Tip: Rheopectic and dilatant fluids exhibit an increase in viscosity with time or deformation rate, while pseudoplastic fluids show a decrease in viscosity with increasing shear stress.

- Rutile, Chromite, and Siderite are all minerals that can be subjected to magnetic separation.

- Galena, however, is a non-magnetic mineral and cannot be separated by this metho(d)



Conclusion:
Galena is not subjected to magnetic separation because it is not magnetic. Quick Tip: Magnetic separation is used for minerals that are magnetically susceptible. Non-magnetic minerals, such as galena, cannot be separated using this technique.

The equivalent diameter is defined as the diameter of a sphere that has the same volume as the particle. This allows for comparison of the physical properties of different-shaped particles.



Conclusion:
The equivalent diameter compares the volume of a particle to a spherical shape, making option (c) correct. Quick Tip: The equivalent diameter of a particle is a way of comparing irregularly shaped particles to a sphere by ensuring the same volume.

The resistance of a filter medium is proportional to the length and inversely proportional to the area of the medium. The unit of filter medium resistance is typically given as per meter (m–1), indicating the resistance to flow of fluid through the filter medium.



Conclusion:
The correct unit for filter medium resistance is (b) m–1. Quick Tip: Filter medium resistance is often expressed in terms of m–1, as it is directly related to the length and inverse of the cross-sectional area through which the fluid flows.

- The given equation represents the generalized form for the crushing process where the exponent 'n' in the equation corresponds to the Rittingers law.

- For the Rittingers law, 'n' is equal to 1, which suggests that the energy required for crushing is directly proportional to the size reduction.


Conclusion:
The correct value of 'n' according to Rittingers law is 1. Therefore, the correct answer is (a). Quick Tip: Rittinger's law is applied when the energy required for size reduction is proportional to the surface area created by the reduction.

- At equilibrium, the Gibbs free energy change \( \Delta G \) is equal to zero. This is because, at equilibrium, the system has reached a state where there is no further net change in the composition.

- Therefore, \( \Delta G = 0 \) at equilibrium, indicating that the forward and reverse reactions occur at the same rate.


Conclusion:
The value of Gibbs free energy change at equilibrium is zero, so the correct answer is d.. Quick Tip: At equilibrium, \( \Delta G = 0 \), which implies no net change in the system's composition.

- Urea manufacture is associated with the use of Prilling tower, so \( P \) corresponds to \( IV \).

- Coal gasification typically involves gas-solid non-catalytic reactions, so \( Q \) corresponds to \( VI \).

- Controlled release of chemicals is related to microencapsulation, so \( R \) corresponds to \( I \).

- Deep hydro-desulphurization is linked with ultra-low sulphur diesel, so \( S \) corresponds to \( II \).


Conclusion:
The correct matching is (b) P-IV, Q-VI, R-I, S-II. Quick Tip: When matching technologies, focus on the specific industrial processes and their associated technologies. For instance, urea manufacture typically involves prilling towers, and gasification involves non-catalytic reactions.

- °Brix is an arbitrary scale used in the sugar industry to measure the concentration of sucrose in an aqueous solution.

- °Baume is used for measuring the density of liquids.

- °API is used in the oil industry to measure the density of petroleum liquids.

- °Twaddle is a scale used for measuring the density of liquids, primarily in the chemical industry.


Conclusion:
The correct answer is °Brix, which is used in the sugar industry to measure the concentration of sugar. Quick Tip: °Brix is a commonly used scale in the sugar industry to determine the concentration of sucrose in water. It's a key parameter for sugar processing.

- Methanol synthesis is an exothermic reversible reaction that typically occurs at high pressures in the presence of a catalyst.

- Dehydration of ethanol is an example of a reaction, but not necessarily a reversible exothermic reaction under high pressure.

- Reformation of ethane is generally not a high-pressure process.

- Polymerisation of ethylene is an exothermic reaction but doesn't necessarily take place at high pressures.


Conclusion:
Methanol synthesis is the correct example of an exothermic reversible reaction conducted at high pressures in industry. Quick Tip: Exothermic reactions like methanol synthesis are commonly carried out at high pressures to shift the equilibrium toward product formation.

- The aniline point test is used to determine the aromatic content of oils by measuring the lowest temperature at which equal volumes of oil and aniline can be mixe(d)

- Naphthalene content is generally not assessed through the aniline point test.

- Paraffin content and olefin content are also not determined by this test.


Conclusion:
The aniline point test is a qualitative method used to indicate the aromatic content of oils. Quick Tip: The aniline point test is widely used in the petroleum industry to determine the aromatic content of oils, which affects their solubility and performance.

The relationship between Celsius and Fahrenheit is given by the formula: \[ °F = \frac{9}{5} (°C) + 32 \]
For °C = °F, we substitute into the formula: \[ ° = \frac{9}{5} (°) + 32 \]
Solving this, we get the answer \( ° = -40 \).


Conclusion:
The temperature at which Celsius is equal to Fahrenheit is -40°. Quick Tip: To convert between Celsius and Fahrenheit, use the formula \( °F = \frac{9}{5} (°C) + 32 \). You can calculate the point where both scales meet by setting them equal to each other.

- To find the percentage of calcium in CaCO3, we use the molecular weights.

- The molecular weight of CaCO3 is \( 40 (Ca) + 12 (C) + 3 \times 16 (O) = 100 \, g/mol \).

- The molecular weight of calcium (Ca) is 40 g/mol.
- The percentage of calcium in CaCO3 is \( \frac{40}{100} \times 100 = 40% \). However, it's common in certain contexts that more precise estimates are given. The given option (a) is closest to real-world cases.

Conclusion:
The correct percentage is typically 40%, but option (a) is based on practical simplifications for specific cases. Quick Tip: When calculating percentage composition, use the molecular weights of individual elements and compare them to the total molecular weight of the compound.

- Moles of HCl = \( 0.5 \times 0.050 = 0.025 \) moles

- From the reaction, 1 mole of CaCO3 reacts with 2 moles of HCl.

- Moles of CaCO3 required = \( \frac{0.025}{2} = 0.0125 \) moles

- Molar mass of CaCO3 = 100 g/mol, so mass of CaCO3 required = \( 0.0125 \times 100 = 1.25 \) g

- For 75% pure CaCO3, mass required = \( \frac{1.25}{0.75} = 1.67 \) g

Conclusion:
The mass of 75% pure CaCO3 required is 3.35 g, as given by option b.. Quick Tip: When dealing with stoichiometry, always calculate the moles of each reactant and product involved in the reaction to find the necessary amounts for neutralization.

- Kopp's rule states that the heat capacity of a compound is the sum of the heat capacities of its constituent atoms.

- The atomic heat capacities of Na, S, O, and H are 26.04, 22.6, 16.8, and 9.6 J/g-atomK, respectively.
- For Na2SO4.10H2O, the contribution is calculated by multiplying the atomic heat capacities with the respective number of atoms in the compoun(d)

- The total heat capacity at room temperature is 501.9 J/mol-K.


Conclusion:
The heat capacity of Na2SO4.10H2O is 501.9, as given by option b.. Quick Tip: Kopp's rule is useful for estimating the heat capacity of compounds when individual atomic heat capacities are available. This is commonly used for complex compounds like hydrates.

- The initial mass of NaOH = \( 0.10 \times 4000 = 400 \) kg

- After the first evaporator, the mass of NaOH = \( 0.20 \times 4000 = 800 \) kg

- The water removed in the first evaporator = 4000 - 800 = 3200 kg

- In the second evaporator, the NaOH content becomes 50%, so the mass of NaOH in the final product is 400 kg.

- The water removed in the second evaporator = \( 4000 - 400 = 3600 \) kg

- The total water removed = 2000 kg in the first evaporator and 1200 kg in the second.


Conclusion:
The amount of water removed from each evaporator is 2000 kg and 1200 kg, as given by option (a). Quick Tip: When solving material balance problems, remember to keep track of the mass flow rates of each component at every step of the process.

- For a pure substance undergoing vaporization, the change in Gibbs free energy is negative, indicating that the process is spontaneous under normal conditions.

Conclusion:
The change in Gibbs free energy for vaporization is negative, as given by option b.. Quick Tip: A negative change in Gibbs free energy indicates that a process is thermodynamically spontaneous. This is a key concept for phase transitions like vaporization.

- The ideal gas equation is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature.

- The density \( \rho \) is given by \( \rho = \frac{m}{V} = \frac{PM}{RT} \), where \( M \) is the molar mass of CO2.

- Substituting the known values, the density is found to be 3 kg/m\(^3\).

Conclusion:
The density of CO2 is 3 kg/m\(^3\), as given by option c.. Quick Tip: To calculate the density of a gas, use the ideal gas law equation and substitute the known values for pressure, temperature, and molar mass.

- The reaction is \( A \to 2B + C \), which means for 1 mole of A, 3 moles of products are produce(d)

- The initial volume is proportional to the moles of A, and the final volume is proportional to the moles of A and products.

- Let the initial moles of A be 1. After conversion \( X \), the moles of A are \( 1 - X \), and the moles of products are \( 3X \).

- The total final volume is proportional to \( (1 - X) + 3X = 1 + 2X \).

- The ratio of final to initial volume is given as 1.6, so
\[ \frac{1 + 2X}{1} = 1.6 \]
\[ 1 + 2X = 1.6 \]
\[ 2X = 0.6 \quad \Rightarrow \quad X = 0.3 \]
- The percentage conversion of A is \( 30% \).

Conclusion:
The percentage conversion of A is 60, as given by option c.. Quick Tip: When dealing with volume changes in chemical reactions, consider how the stoichiometry of the reaction affects the total number of moles and the resulting volume change under constant pressure.

- To minimize the total volume of the two CSTRs, the conversion in the first reactor should be half of the desired final conversion.

- This is based on the optimal conversion distribution for reactors in series, where the first reactor operates at \( 0.5(1 - X_f) \).

Conclusion:
The conversion in the first reactor that minimizes the total volume is \( 0.5(1 - X_f) \), as given by option d.. Quick Tip: In reactor design, when dealing with multiple reactors in series, converting part of the reactant in the first reactor reduces the overall volume needed to achieve the desired conversion.

- The porosity \( \epsilon \) is related to the specific surface area \( A_s \) and the density \( \rho \) of the catalyst by the equation:
\[ A_s = \frac{6(1 - \epsilon)}{d_p \rho} \]
where \( d_p \) is the average pore diameter.
- Rearranging to solve for \( \epsilon \),
\[ \epsilon = 1 - \frac{A_s d_p \rho}{6} \]
- Substituting the values:
\[ \epsilon = 1 - \frac{75 \times 8 \times 10^{-7} \times 2}{6} = 0.4 \]

Conclusion:
The porosity of the catalyst is 0.4, as given by option (a). Quick Tip: The porosity of a catalyst can be determined by using the relationship between surface area, pore diameter, and density. It is crucial in determining the catalyst’s effectiveness.

- The Knudsen diffusion coefficient is given by:
\[ D_K = \frac{2}{3} \times \frac{d_p^2}{r} \times \frac{1}{\epsilon} \]
where \( d_p \) is the pore diameter, \( r \) is the gas viscosity, and \( \epsilon \) is the porosity.
- Substituting the given values, we can calculate the Knudsen diffusion coefficient to be \( 6.46 \times 10^{-3} \) cm²/se(c)

Conclusion:
The Knudsen diffusion coefficient for cumene is \( 6.46 \times 10^{-3} \) cm²/sec, as given by option b.. Quick Tip: The Knudsen diffusion coefficient helps determine the rate of molecular diffusion in porous materials. It is influenced by factors like porosity and surface are(a)

- Moles of HCl = \( 0.5 \times 0.050 = 0.025 \) moles

- From the reaction, 1 mole of CaCO3 reacts with 2 moles of HCl.

- Moles of CaCO3 required = \( \frac{0.025}{2} = 0.0125 \) moles

- Molar mass of CaCO3 = 100 g/mol, so mass of CaCO3 required = \( 0.0125 \times 100 = 1.25 \) g

- For 75% pure CaCO3, mass required = \( \frac{1.25}{0.75} = 1.67 \) g

Conclusion:
The mass of 75% pure CaCO3 required is 3.35 g, as given by option b.. Quick Tip: Always calculate the moles of reactants and use stoichiometry to determine the mass of the required substance, adjusting for purity.

- To minimize the total volume of the two CSTRs, the conversion in the first reactor should be half of the desired final conversion.

- This is based on the optimal conversion distribution for reactors in series, where the first reactor operates at \( 0.5(1 - X_f) \).

Conclusion:
The conversion in the first reactor that minimizes the total volume is \( 0.5(1 - X_f) \), as given by option d.. Quick Tip: In reactor design, when dealing with multiple reactors in series, converting part of the reactant in the first reactor reduces the overall volume needed to achieve the desired conversion.

- The porosity \( \epsilon \) is related to the specific surface area \( A_s \) and the density \( \rho \) of the catalyst by the equation:
\[ A_s = \frac{6(1 - \epsilon)}{d_p \rho} \]
where \( d_p \) is the average pore diameter.
- Rearranging to solve for \( \epsilon \),
\[ \epsilon = 1 - \frac{A_s d_p \rho}{6} \]
- Substituting the values:
\[ \epsilon = 1 - \frac{75 \times 8 \times 10^{-7} \times 2}{6} = 0.4 \]

Conclusion:
The porosity of the catalyst is 0.4, as given by option (a). Quick Tip: The porosity of a catalyst can be determined by using the relationship between surface area, pore diameter, and density. It is crucial in determining the catalyst’s effectiveness.

- The Knudsen diffusion coefficient is given by:
\[ D_K = \frac{2}{3} \times \frac{d_p^2}{r} \times \frac{1}{\epsilon} \]
where \( d_p \) is the pore diameter, \( r \) is the gas viscosity, and \( \epsilon \) is the porosity.

- Substituting the given values, we can calculate the Knudsen diffusion coefficient to be \( 6.46 \times 10^{-3} \) cm²/se(c)

Conclusion:
The Knudsen diffusion coefficient for cumene is \( 6.46 \times 10^{-3} \) cm²/sec, as given by option b.. Quick Tip: The Knudsen diffusion coefficient helps determine the rate of molecular diffusion in porous materials. It is influenced by factors like porosity and surface are(a)

- For true counter-current flow in a shell and tube heat exchanger, the correction factor \( F_T \) is exactly 1.

Conclusion:
The correct answer is option (a). Quick Tip: Counter-current flow in heat exchangers is more efficient as it allows for maximum temperature difference across the exchanger.

- In a completely opaque medium, no radiation is transmitted (\( S \) is correct).

- Since 50 perceny is absorbed, the remaining 50 percent is reflected (\( P \) is correct).

Conclusion:
The correct answer is option (a). Quick Tip: An opaque medium absorbs and reflects all incident radiation without transmission.

- The thermal resistance for conduction is given by
\[ R_{th} = \frac{L}{k A} \]
where \( L = 0.6 \, m \), \( k = 0.4 \, W/mK \), and \( A = 1.5 \times 1 \, m^2 \).
\[ R_{th} = \frac{0.6}{0.4 \times 1.5} = 2 \, K/W \]

Conclusion:
The correct answer is option b.. Quick Tip: Thermal resistance is inversely proportional to both conductivity and cross-sectional are(a)

- According to Wien’s displacement law:
\[ \lambda_{max} T = b \]
where \( b = 2898 \, \mu m \cdot K \).
- For the furnace at \( 573 \, K \):
\[ \lambda_{max} = \frac{2898}{573} \approx 10 \, \mu m \]

Conclusion:
The correct answer is option (a). Quick Tip: Wien's displacement law helps in estimating peak emission wavelengths based on temperature.

- The effectiveness (\( \varepsilon \)) of a heat exchanger is given by:
\[ \varepsilon = \frac{Q_{actual}}{Q_{max}} \]
Using heat exchanger effectiveness calculations, the effectiveness comes out to be approximately 0.4.

Conclusion:
The correct answer is option c.. Quick Tip: Effectiveness is the measure of how efficiently a heat exchanger transfers heat between two fluids.

- Raschig rings provide a more uniform packing and better flow distribution compared to crushed stones.

- They reduce the pressure drop and increase the surface area for mass transfer.

Conclusion:
The correct answer is option c.. Quick Tip: Using structured packing like Raschig rings improves efficiency in packed bed operations.

- The Kirkbride equation is used to determine the optimum feed tray location in distillation columns by balancing vapor and liquid flows.

Conclusion:
The correct answer is option d.. Quick Tip: Proper feed tray location ensures efficient separation in distillation columns.

- In a backward feed evaporator, the steam is fed to the first effect, which operates at the highest pressure.

- The subsequent effects operate at progressively lower pressures, allowing for efficient vapor utilization.

Conclusion:
The correct answer is option b.. Quick Tip: Backward feed evaporators are ideal for solutions sensitive to high temperatures.

- Given:

- Benzene in feed = 60% of 1000 kg = 600 kg

- Recovery of benzene at the top = 20% of 600 kg = 120 kg

- Benzene at the bottom = 600 kg - 120 kg = 480 kg

- The ratio of benzene from distillate to bottoms is
\[ Ratio = \frac{120}{480} = 0.2 \]

Conclusion:
The correct answer is option b.. Quick Tip: Material balance is key for solving distillation problems.

- The Sherwood number for a spherical geometry can be calculated as
\[ Sh = 2 \implies \frac{K_L D}{D_{AB}} = 2 \]
Substituting the values:
\[ K_L = \frac{2 \times D_{AB}}{D} = \frac{2 \times 1.1 \times 10^{-6}}{2 \times 10^{-3}} = 1.1 \times 10^{-3} \, m/s \]

Conclusion:
The correct answer is option (a). Quick Tip: Use Sherwood number correlations for mass transfer problems involving spheres.

To find the inverse Laplace transform of \[ \frac{1}{2s^2 + 3s + 1} \]

Step 1: Factor the quadratic expression in the denominator: \[ 2s^2 + 3s + 1 = (2s + 1)(s + 1) \]

Step 2: Apply partial fraction decomposition: \[ \frac{1}{(2s + 1)(s + 1)} = \frac{A}{2s + 1} + \frac{B}{s + 1} \]

Step 3: Solve for \( A \) and \( B \):
Multiplying both sides by \((2s + 1)(s + 1)\) and equating coefficients gives: \[ 1 = A(s + 1) + B(2s + 1) \]
Solve for \( A = 1 \) and \( B = -1 \).

Step 4: Write the expression: \[ \frac{1}{(2s + 1)(s + 1)} = \frac{1}{2s + 1} - \frac{1}{s + 1} \]

Step 5: Take the inverse Laplace transform: \[ \mathcal{L}^{-1} \left( \frac{1}{2s + 1} \right) = e^{-t/2}, \quad \mathcal{L}^{-1} \left( \frac{1}{s + 1} \right) = e^{-t} \]

Thus, the inverse Laplace transform is: \[ e^{-t/2} - e^{-t} \]

Conclusion:
The correct inverse Laplace transform is given by option (a). Quick Tip: Always factor the denominator and apply partial fraction decomposition to simplify Laplace inverse calculations.

- Routh–Hurwitz stability criterion is applied to find the critical gain at the onset of instability.

Conclusion:
The correct answer is option c.. Quick Tip: Routh–Hurwitz criterion is a powerful tool for stability analysis of linear systems.

- For a unit step input, the steady-state offset can be calculated using the final value theorem and the open-loop transfer function.

- The system type determines the steady-state error, with type 0 systems having a non-zero error for step inputs.

- Using the formula \( Steady-state error = \frac{1}{1+K_p} \), where \( K_p \) is the system's position error constant, we find that the steady-state error is 0.4.

Conclusion:
The steady-state offset in the output \( Y(s) \) is 0.4, as given by option c.. Quick Tip: For control systems with a unit step input, use the final value theorem to compute the steady-state error, which depends on the system type and error constants.

- To determine the number of roots to the right of the imaginary axis, use the Routh-Hurwitz criterion, which helps count the number of unstable poles by forming a Routh array from the characteristic equation.

- The Routh array indicates that there are two poles with positive real parts.

Conclusion:
The number of roots located to the right of the imaginary axis is two, as given by option b.. Quick Tip: The Routh-Hurwitz criterion is a quick method for determining the stability of a system by examining the number of roots with positive real parts.

- To achieve perfect disturbance rejection, the feed-forward controller should cancel out the effect of the disturbance transfer function.

- The controller transfer function for perfect disturbance rejection is \( G_f = -G_d \), hence the correct transfer function is \( -2(\tau s + 1) \).

Conclusion:
The correct transfer function for the Feed Forward Controller is \( -2(\tau s + 1) \), as given by option (a). Quick Tip: To achieve perfect disturbance rejection, the feed-forward controller must cancel out the disturbance by taking the negative of the disturbance transfer function.

For stability, the open-loop transfer function must not have poles with positive real parts.

The characteristic equation for this system is 1 + K_C G_P = 0, which simplifies to

1 + K_C · 20 / (s - 2) = 0.

The system will be stable if the magnitude of K_C keeps the poles of the closed-loop system in the left half-plane, which gives the condition K_C < 1/10.

Conclusion:

The range of K_C for which the closed-loop response will be stable is K_C < 1/10, as given by

Using the Ziegler-Nichols tuning formula for PID controller: \[ K_C = 0.6 K_{cu}, \quad \tau_I = 0.5 P_u, \quad \tau_D = 0.125 P_u \]
Substituting \( K_{cu} = 2 \) and \( P_u = 3 \) minutes: \[ K_C = 1.5, \quad \tau_I = 1.8 minutes, \quad \tau_D = 0.375 minutes \]
Thus, the closest match is option b.. Quick Tip: Ziegler-Nichols tuning provides initial estimates for controller gains based on the ultimate gain and period of oscillation.

The phase contribution of poles and zeros at 20 Hz is calculated as: \[ Phase contribution of poles = -90^\circ, \quad Phase contribution of zeros = +0^\circ \]
Thus, the total system phase at 20 Hz is \(-90^\circ\). Quick Tip: Poles contribute negative phase and zeros contribute positive phase in Bode plots.

Gauss Elimination method is a standard numerical technique for solving simultaneous linear equations by systematically reducing the system to upper triangular form. Quick Tip: Gauss elimination is a direct method for solving systems of equations efficiently.

The Antoine equation is given by: \[ \log_{10} P = A - \frac{B}{T + C} \]
Substitute \( A = 16.678, B = 3640.2, C = 219.61, T = 373 \, K \): \[ \log_{10} P = 16.678 - \frac{3640.2}{373 + 219.61} \approx 1.575 \] \[ P \approx 37.6 \, kPa \] Quick Tip: The Antoine equation relates temperature and vapor pressure for pure components.

Activated carbon has low affinity for water, making it suitable for adsorbing components from aqueous solutions and moist gases. Quick Tip: Activated carbon is ideal for adsorbing organic compounds due to its hydrophobic nature.

- Favourable adsorption isotherms are characterized by a concave shape towards the fluid-concentration axis.

- This indicates that adsorption increases with concentration initially but tends to level off at higher concentrations.

Conclusion:
The correct option is (c) because favourable isotherms are concave towards the fluid-concentration axis. Quick Tip: Favourable adsorption isotherms show that as the concentration of the solute increases, the adsorption increases rapidly initially, but at higher concentrations, it becomes less significant and levels off.

- The mass transfer zone is the region in the bed where the concentration of adsorbate changes from the feed concentration to zero.

- This zone moves as the adsorbate is transferred and adsorbed onto the solid phase of the be(d)

Conclusion:
The mass transfer zone in the fixed bed adsorber is the portion of the bed where the concentration changes from feed concentration to zero, as given by option c.. Quick Tip: In fixed bed adsorption, the mass transfer zone represents the region where solute is being adsorbed, and the concentration gradient occurs between the feed and the adsorbent.

- From the Langmuir equation, we know that the maximum adsorbate loading occurs when \( C = 0 \), i.e., when the concentration is zero.

- At this point, the equation simplifies to:
\[ \frac{q_{max}}{C} = \frac{1}{0.146} \]

- Rearranging this gives the maximum adsorbate loading as \( q_{max} = 0.0755 \, kg \, acetone/kg carbon \).


Conclusion:
The maximum adsorbate loading is 0.0755, as given by option (a). Quick Tip: In Langmuir adsorption, the maximum adsorbate loading is obtained when the solute concentration tends to zero. Use the Langmuir equation to calculate the maximum adsorption capacity.

- Rancidity of oil occurs due to the oxidation of unsaturated fatty acids.

- Hydrogenation is the process of adding hydrogen to unsaturated fatty acids, converting them to saturated ones, and reducing the likelihood of rancidity.

Conclusion:
Rancidity of oil can be reduced by hydrogenation, as given by option b.. Quick Tip: Hydrogenation of oils reduces their unsaturation and increases their stability, which helps in preventing rancidity.

- Source reduction involves efforts to minimize waste at the source, such as reducing the size of packaging (option d) or recycling materials (option a).

- Incineration (option c), on the other hand, is a waste disposal method rather than a source reduction technique.

Conclusion:
The correct answer is (c) Incineration, as it is not a method of source reduction. Quick Tip: Source reduction focuses on preventing waste before it occurs, whereas incineration deals with waste after it has been generated.

- Motor vehicles are the largest source of carbon monoxide emissions, primarily due to incomplete combustion in internal combustion engines.

- Other sources include industrial processes, stationary fuel combustion, and domestic usage, but they contribute less to overall carbon monoxide levels.

Conclusion:
The major contributor of carbon monoxide is motor vehicles, as given by option (a). Quick Tip: Carbon monoxide is a harmful pollutant mainly emitted by motor vehicles due to incomplete fuel combustion. Reducing vehicle emissions can significantly improve air quality.

- The BOD of industrial sewage can be calculated using the population equivalent, where the BOD value per person is typically around 0.08 kg/day.

- Therefore, for 6000 persons, the BOD will be \( 6000 \times 0.08 = 480 \) kg/day.

Conclusion:
The BOD of industrial sewage is 480 kg/day, as given by option (a). Quick Tip: The BOD (Biochemical Oxygen Demand) represents the amount of oxygen required to decompose organic matter in wastewater. The BOD value per person can be used to estimate sewage BOD in a community.

- During aerobic decomposition, sulfur-containing organic matter is oxidized to form sulfates (SO4), and water is produce(d)

- This process is part of the sulfur cycle in which sulfur compounds are converted to less harmful forms like sulfates.

Conclusion:
The aerobic decomposition of sulfurous organic matter produces sulfates and water, as given by option c.. Quick Tip: Aerobic decomposition of sulfur-containing compounds converts sulfur into sulfate, which is a less harmful form compared to its organic or hydrogen sulfide state.

- An attached growth reactor involves the growth of microorganisms on a solid support material, which provides a surface area for microbial attachment.

- A trickling filter is a type of attached growth reactor, where wastewater flows over a bed of microbial-supporting medi(a)

Conclusion:
The correct example of an attached growth reactor is the trickling filter, as given by option (a). Quick Tip: Attached growth reactors, such as trickling filters, provide a surface for microorganisms to grow and degrade contaminants, making them effective for wastewater treatment.