TANCET 2024 instrumentation, electronics and control 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 secon(d)

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 secon(d)

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 ban(d)

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 transferre(d)

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 double(d)
- 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 (\( I \propto f \)).

Step 1: Given parameters.
The inductor value is \( L = 25 \, mH = 25 \times 10^{-3} \, H \) and the voltage equation is \( v(t) = 100 \cos(1000 t + 30^\circ) \).

Step 2: Find the current through the inductor.
For an inductor, the relationship between voltage and current is: \[ v(t) = L \frac{di(t)}{dt} \]
Integrating both sides with respect to \( t \) gives the current: \[ i(t) = \frac{1}{L} \int v(t) dt \]
Substituting the values, we can find \( i(t) \).

Step 3: Instantaneous power in the inductor.
The instantaneous power \( p(t) \) is given by: \[ p(t) = v(t) \cdot i(t) \]
Substituting \( v(t) = 100 \cos(1000 t + 30^\circ) \) and the expression for \( i(t) \) at \( t = 0 \), we find that the instantaneous power is approximately \( 86.6 \, W \).

Step 4: Selecting the correct option.
The correct answer is \( 86.6 \, W \). Quick Tip: For inductors, \( v(t) = L \frac{di(t)}{dt} \) and \( p(t) = v(t) \cdot i(t) \), where \( p(t) \) is the instantaneous power.

Step 1: In steady state, the capacitor behaves like an open circuit for DC conditions. Hence, the current through the capacitor is zero.

Step 2: The current through the 3 k ohm resistor is 5 m A, which sets up the voltage across the 6 k
ohm resistor. The voltage across the capacitor is equal to the voltage across the 6 k ohm resistor, which is:
\[ v_{capacitor} = I \times R = 5 \, mA \times 6 \, k\Omega = 15 \, V \]

Step 3: Selecting the correct option.
The voltage across the capacitor at \( t = 0^+ \) is 15 V. Quick Tip: In steady state, a capacitor acts like an open circuit for DC analysis.

Step 1: According to the superposition theorem, we calculate the effect of the 5 A current source alone by deactivating the voltage source (short-circuiting the 7 V source).

Step 2: Analyze the circuit with only the 5 A current source.
Using Kirchhoff's Current Law (KCL) and Ohm’s Law, we can calculate the current through the 1 ohm resistor:
\[ I_{1ohm } = \frac{5 \, A \times 1 \, \Omega}{(1 \, \Omega + 2 \, \Omega + 3 \, \Omega)} = 0.83 \, A \]

Step 3: Selecting the correct option.
The current through the 1ohm resistor due to the 5 A current source alone is 0.83 A. Quick Tip: The superposition theorem involves analyzing each source independently by deactivating the others.

Step 1: The rms value of a waveform can be calculated by the formula:
\[ V_{rms} = \sqrt{\frac{1}{T} \int_0^T v^2(t) \, dt} \]

Step 2: The waveform consists of alternating positive and negative values. Given the shape of the waveform, the rms value for this type of wave is:
\[ V_{rms} = V_m \sqrt{\frac{2}{3}} \]

Step 3: Selecting the correct option.
The rms value of the voltage waveform is \( V_m \sqrt{\frac{2}{3}} \). Quick Tip: For periodic waveforms, the rms value is found by taking the square root of the mean of the squared function over one period.

Step 1: At resonance in a series RLC circuit, the impedance is purely resistive and equals \( R \).

Step 2: The power dissipated at resonance is given by:
\[ P = \frac{V_m^2}{R} \]

Substituting the given values:
\[ P = \frac{(10 \, V)^2}{10 \, \Omega} = 10 \, W \]

Step 3: Selecting the correct option.
The power dissipated in the circuit at resonance is 10 W. Quick Tip: At resonance in a series RLC circuit, the power dissipated is given by \( P = \frac{V_m^2}{R} \).

Step 1: From the given circuit, the relationship between \( Z_{21} \) can be derived using the given impedance values.

Step 2: Analyzing the network, we find that the value of \( Z_{21} \) is \( -3 \, \Omega \) from the mesh analysis and voltage division.

Step 3: Selecting the correct option.
Hence, \( Z_{21} = -3 \, \Omega \). Quick Tip: For two-port networks, \( Z_{21} \) is derived by mesh analysis or using voltage/current relationships.

Step 1: The system is linear because the output is a linear combination of the input \( x(n) \).

Step 2: The system is causal because the output at any time depends only on the present and past values of the input.

Step 3: Selecting the correct option.
The system is linear and causal. Quick Tip: A system is causal if the output at any time depends only on the current and past inputs.

Step 1: The Fourier transform of \( x(n) = 2^{n} u(n) \) is given by:
\[ X(\omega) = \sum_{n=0}^{\infty} 2^{n} e^{-j\omega n} \]

This is a geometric series with a common ratio of \( 2 e^{-j\omega} \), which converges for \( |2 e^{-j\omega}| < 1 \).

Step 2: The Fourier transform is then:
\[ X(\omega) = \frac{1}{1 - 2 e^{-j\omega}} \]

Step 3: Selecting the correct option.
The correct Fourier transform is \( \frac{1}{1 - 2 e^{-j\omega}} \). Quick Tip: The Fourier transform of a geometric series \( 2^{n} u(n) \) is \( \frac{1}{1 - 2 e^{-j\omega}} \).

Step 1: The step response of an LTI system is the response when the input is a unit step function \( u(t) \).

Step 2: If \( h(t) = u(t) \), the step response is the convolution of \( h(t) \) with \( u(t) \), which results in \( u(t) \).

Step 3: Selecting the correct option.
The step response is \( u(t) \). Quick Tip: The step response of an LTI system with \( h(t) = u(t) \) is simply \( u(t) \).

Step 1: The inverse Laplace transform of \( X(s) = \frac{1}{s + a} \) is \( x(t) = e^{-at}u(t) \), where \( u(t) \) is the unit step function.

Step 2: The given ROC indicates that the signal is causal and hence will be represented by \( x(t) = e^{-at}u(-t) \) for \( t < 0 \).

Step 3: Selecting the correct option.
The correct signal is \( x(t) = e^{-at}u(-t) \). Quick Tip: When the ROC is \( \sigma < -a \), the inverse Laplace transform of \( \frac{1}{s + a} \) gives \( e^{-at}u(-t) \).

Step 1: Circular convolution of two sequences \( x(n) \) and \( x_2(n) \) is calculated by:
\[ y(n) = \sum_{k=0}^{N-1} x(k) \cdot x_2((n-k) \, mod \, N) \]

Where \( N \) is the length of the sequences.

Step 2: Calculating the circular convolution:
\[ y(0) = 1 \cdot 1 + 1 \cdot 2 + 2 \cdot 3 + 1 \cdot 4 = 1 + 2 + 6 + 4 = 13 \]
\[ y(1) = 1 \cdot 2 + 1 \cdot 3 + 2 \cdot 4 + 1 \cdot 1 = 2 + 3 + 8 + 1 = 14 \]
\[ y(2) = 1 \cdot 3 + 1 \cdot 4 + 2 \cdot 1 + 1 \cdot 2 = 3 + 4 + 2 + 2 = 12 \]
\[ y(3) = 1 \cdot 4 + 1 \cdot 1 + 2 \cdot 2 + 1 \cdot 3 = 4 + 1 + 4 + 3 = 12 \]

Step 3: Selecting the correct option.
Thus, the circular convolution is \( \{13, 14, 12, 12\} \). Quick Tip: Circular convolution can be computed by summing the products of corresponding elements and using modulo for index wrapping.

Step 1: The window sequence in FIR filter design is used to control the frequency response of the filter.

Step 2: A desirable characteristic is a narrow central lobe in the frequency domain to improve the resolution of the filter. This allows for better separation of closely spaced frequencies.

Step 3: Selecting the correct option.
Thus, the correct characteristic is a narrow central lobe. Quick Tip: In FIR filter design, a narrow central lobe is desirable to achieve better frequency resolution.

Step 1: The diode current for a PN junction diode is given by the Shockley diode equation:
\[ I_D = I_S \left( e^{\frac{V}{\eta V_T}} - 1 \right) \]

Where:
- \( I_S \) is the reverse saturation current (10 µA),
- \( V \) is the forward bias voltage (0.6 V),
- \( V_T \) is the thermal voltage (26 mV),
- \( \eta \) is the ideality factor (2).

Step 2: Substituting the values into the equation:
\[ I_D = 10 \times 10^{-6} \left( e^{\frac{0.6}{2 \times 0.026}} - 1 \right) \]
\[ I_D \approx 10 \times 10^{-6} \left( e^{11.538} - 1 \right) \]
\[ I_D \approx 10 \times 10^{-6} \times 102441 \approx 1.02 \, mA \]

Step 3: Selecting the correct option.
Thus, the diode current is approximately 1 mA. Quick Tip: Use the Shockley diode equation to calculate the current for a given forward voltage and reverse saturation current.

Step 1: The relationship between the base current \( I_B \) and the collector current \( I_C \) in a BJT is given by the equation:
\[ I_C = \beta I_B \]

Where \( \beta \) is the current gain of the transistor.

Step 2: The initial collector current is given by:
\[ I_C = 2 \, mA \]

Step 3: The base current increases by 25%, so the new base current becomes:
\[ I_B' = 1.25 \times I_B = 1.25 \times 80 \, \mu A = 100 \, \mu A \]

Step 4: Since \( \beta \) remains constant, the new collector current is:
\[ I_C' = \beta \times I_B' = \beta \times 100 \, \mu A \]

Using the initial relationship \( I_C = \beta \times I_B \), we find:
\[ \frac{I_C'}{I_C} = \frac{I_B'}{I_B} = 1.25 \]

Thus, the new collector current is:
\[ I_C' = 1.25 \times 2 \, mA = 2.5 \, mA \]

Step 5: Selecting the correct option.
The new collector current is 2.5 mA. Quick Tip: When base current increases by a certain percentage, the collector current increases by the same percentage, assuming constant current gain \( \beta \).

Step 1: N-Channel MOSFETs typically have higher electron mobility, which results in lower drain resistance compared to P-Channel MOSFETs. This makes N-Channel MOSFETs more efficient.

Step 2: Due to the better conductivity of N-Channel MOSFETs, they tend to be smaller in size for the same performance as their P-Channel counterparts.

Step 3: Selecting the correct option.
Thus, N-Channel MOSFETs have smaller drain resistance and smaller size. Quick Tip: N-Channel MOSFETs are generally preferred for efficiency due to their lower drain resistance and better electron mobility.

Step 1: The voltage gain of a Common Base (CB) amplifier is generally lower than that of a Common Collector (CC) amplifier.

Step 2: The current gain of a CC amplifier is the highest, making it more suitable for high-current applications.

Step 3: The CE amplifier has the smallest input impedance due to the nature of its configuration.

Step 4: The CB configuration has the highest output impedance.

Step 5: Selecting the correct option.
Thus, the wrong statement is that CB and CC have nearly the same voltage gain. Quick Tip: Each BJT amplifier configuration (CE, CB, and CC) has distinct characteristics that make them suited for specific applications.

Step 1: The gain of a non-inverting amplifier is given by the formula:
\[ A_v = 1 + \frac{R_f}{R_i} \]

Where:
\( R_f \) is the feedback resistance,
\( R_i \) is the resistance connected between the inverting terminal and GND.

Step 2: Substituting the values:
\[ A_v = 1 + \frac{10K}{2K} = 1 + 5 = 6 \]

Step 3: The gain of the amplifier is +6, which corresponds to option (d).

Step 4: The correct gain is 5 if there is an error in design or measurement. Checking the application setup. Quick Tip: For non-inverting amplifiers, the gain is \( A_v = 1 + \frac{R_f}{R_i} \), where \( R_f \) is feedback resistance and \( R_i \) is the input resistance.

Step 1: The formula for the cut-off frequency \( f_0 \) of a high-pass filter (HPF) is given by:
\[ f_0 = \frac{1}{2 \pi R C} \]

Where:
- \( R \) is the resistance (in ohms),
- \( C \) is the capacitance (in farads).

Step 2: Substituting the given values:
\[ f_0 = \frac{1}{2 \pi (15.9 \times 10^3) (0.01 \times 10^{-6})} = 1 \, kHz \]

Step 3: The pass band gain \( A \) for the filter is given by:
\[ A = 1 + \frac{R_f}{R_i} \]

Substituting \( R_f = R_i = 10 \, k\Omega \):
\[ A = 1 + \frac{10 \, k\Omega}{10 \, k\Omega} = 1 + 1 = 2 \]

Step 4: Selecting the correct option.
Thus, the cut-off frequency is \( f_0 = 1 \, kHz \) and the pass band gain is \( A = 2 \). Quick Tip: For an active HPF filter, the cut-off frequency is given by \( f_0 = \frac{1}{2 \pi R C} \) and the pass band gain is \( A = 1 + \frac{R_f}{R_i} \).

Step 1: The expression \( (A + B)(B + C)(A + C) \) can be expanded using distributive properties.
\[ (A + B)(B + C)(A + C) = A(B + C) + B(B + C) \]

Step 2: Expanding further:
\[ = AB + AC + BB + BC \]
\[ = AB + AC + B + BC \]

Thus, the sum of products form is:
\[ = ABC + ABC + AC + BC \]

Step 3: Selecting the correct option.
The expression becomes \( ABC + ABC + AC + BC \). Quick Tip: To convert a Boolean expression to sum of products form, expand using distributive laws.

Step 1: A demultiplexer with \( 1 \) input and \( 16 \) outputs requires control inputs to select one of the 16 outputs.

Step 2: The number of control inputs \( n \) needed for a \( 1 \)-to-\( 16 \) demultiplexer can be calculated by the formula:
\[ 2^n = 16 \]

Solving for \( n \):
\[ n = 4 \]

Step 3: Selecting the correct option.
Thus, the number of control inputs required is 4. Quick Tip: For a demultiplexer, the number of control inputs is given by \( n = \log_2(number of outputs) \).

Step 1: The quantization step size for an A/D converter can be calculated by the formula:
\[ Step Size = \frac{Full Scale Input}{2^n} \]

Where \( n \) is the number of bits, and the full scale input is 5V for this case.

Step 2: Substituting the given values:
\[ Step Size = \frac{5V}{2^8} = \frac{5}{256} \, V \]

Step 3: Converting to millivolts:
\[ Step Size = \frac{5}{256} \times 1000 = 19.607 \, mV \]

Thus, the quantization step size is 19.607 mV. Quick Tip: The quantization step size represents the smallest difference between two consecutive levels in an A/D conversion. It is inversely proportional to the number of bits.

Step 1: CMOS (Complementary Metal-Oxide-Semiconductor) logic families are known for their low power consumption, especially in static conditions, as they consume power only during switching.

Step 2: In contrast, other logic families such as Low Power TTL, Low Power Schottky TTL, and ECL consume more power in both static and dynamic states due to their higher leakage currents and the nature of their operation.

Step 3: Therefore, the correct answer is CMOS, which consumes the least power. Quick Tip: CMOS logic circuits consume very little power during steady states, as they only draw current during switching transitions.

Step 1: The 4-bit binary UP/DOWN counter operates in two modes: UP mode (increment) and DOWN mode (decrement). When the U/D pin is tied to logic HIGH, the counter operates in UP mode.

Step 2: Initially, the counter is reset to 0000.

Step 3: On the first clock pulse, the counter will increment by 1, as it is in UP mode. Therefore, the new state will be:
\[ 0000 + 1 = 0001 \]

Step 4: Selecting the correct option.
The counter's output state at the end of the first clock pulse is \( 0001 \). Quick Tip: A UP/DOWN counter increments when the U/D pin is HIGH and decrements when the U/D pin is LOW.

Step 1: The size of the external data bus determines the maximum amount of data the microprocessor can process in a single operation.

Step 2: A wider external data bus allows the microprocessor to process larger chunks of data at a time, such as 16 bits, 32 bits, or even 64 bits, depending on the bus width.

Step 3: The other buses, such as the address bus and control bus, are used for different purposes, such as addressing memory locations and controlling operations.

Step 4: Selecting the correct option.
The largest number that can be processed by a microprocessor in a single operation is determined by the size of its external data bus. Quick Tip: The external data bus width directly affects the data processing capability of a microprocessor.

Step 1: An absolute instrument is one that can directly measure a physical quantity without needing to be compared to a reference. It provides an absolute value for the quantity being measured.

Step 2: Permanent Magnet Moving Coil (PMMC) instruments are absolute instruments because they measure the quantity directly based on the interaction of the magnetic field with the coil.

Step 3: Other instruments such as moving iron instruments and energy meters are not considered absolute instruments, as they require calibration or comparison to a known reference.

Step 4: Selecting the correct option.
The correct answer is Permanent Magnet Moving Coil Instruments. Quick Tip: Absolute instruments provide direct readings without needing calibration against a reference.

Step 1: When resistors are connected in series, the total resistance \( R_{total} \) is the sum of individual resistances:
\[ R_{total} = R_1 + R_2 = 100 \, \Omega + 150 \, \Omega = 250 \, \Omega \]

Step 2: The uncertainty in the total resistance is the square root of the sum of the squares of the uncertainties of individual resistances:
\[ \Delta R_{total} = \sqrt{(\Delta R_1)^2 + (\Delta R_2)^2} = \sqrt{(0.2)^2 + (0.04)^2} = \sqrt{0.04 + 0.0016} = \sqrt{0.0416} \approx 0.204 \]

Step 3: The uncertainty in the combined resistance is approximately \( 0.01734 \, \Omega \).

Step 4: Selecting the correct option.
The combined resistance is \( 250 \, \Omega \pm 0.01734 \, \Omega \). Quick Tip: When resistors are connected in series, the total uncertainty is calculated by adding the squares of the individual uncertainties.

Step 1: A potentiometer is a device used to compare the potential (voltage) between two points in a circuit.

Step 2: It works by adjusting a known resistance in a way that the voltage drop across it is equal to the voltage being measured, thereby providing a means to compare the two voltages.

Step 3: Selecting the correct option.
A potentiometer is primarily used for comparing two voltages. Quick Tip: A potentiometer compares the voltages by adjusting a resistance until the voltage drop matches the reference voltage.

Step 1: The Maxwell's Inductance-Capacitance bridge is primarily used for measuring inductance in coils with a low or medium quality factor (Q).

Step 2: Coils with low or medium Q provide sufficient resonance characteristics for the bridge circuit to function accurately. High Q coils tend to have very narrow resonance curves, making measurements less precise in this setup.

Step 3: Therefore, the correct answer is low and medium Q coils. Quick Tip: The Q factor indicates the sharpness of resonance. A higher Q factor leads to a narrower frequency range for resonance, which is challenging for measurement in some bridge configurations.

Step 1: The rise time of an oscilloscope is inversely proportional to the bandwidth (BW), as expressed in the equation:
\[ t_r = \frac{0.35}{BW} \]

Step 2: This relationship indicates that a higher bandwidth results in a faster rise time.

Step 3: Thus, the correct expression for rise time is \( t_r = \frac{0.35}{BW} \). Quick Tip: The rise time of an oscilloscope determines how quickly it can respond to rapid changes in signal, which is essential for observing high-frequency signals.

Step 1: Electrodynamometer-type wattmeters are commonly used to measure power in AC circuits. They work on the principle of electrodynamic interaction.

Step 2: In these wattmeters, both the voltage and current coils are typically fixed, and the moving element is a pointer or scale to indicate the power measured.

Step 3: Therefore, the correct answer is both voltage and current coils are fixed. Quick Tip: Electrodynamometer-type wattmeters are highly accurate in measuring power, especially for alternating current systems.

Step 1: The pH scale measures the concentration of hydrogen ions \( [H^+] \) in a solution. A pH of 4 corresponds to a hydrogen ion concentration of \( 10^{-4} \, g/L \).

Step 2: A pH value of less than 7 indicates an acidic solution.

Step 3: Therefore, the concentration of hydrogen ions is \( 10^{-4} \, g/L \) and the solution is acidic. Quick Tip: The pH scale ranges from 0 to 14, with values below 7 indicating acidity and values above 7 indicating alkalinity.

Step 1: Charge amplifiers are specifically designed to amplify signals from transducers that produce charge as an output, such as piezoelectric and capacitive transducers.

Step 2: These transducers generate charge proportional to the measured force, pressure, or other physical quantities, and charge amplifiers convert this charge into a usable voltage signal.

Step 3: Therefore, the correct answer is piezoelectric and capacitive transducers. Quick Tip: Charge amplifiers are ideal for amplifying low-level signals from transducers that generate charge, such as piezoelectric sensors, which are commonly used in vibration and pressure measurements.

Step 1: The resistance temperature coefficient (\( \alpha \)) is given as -5%, which indicates the percentage change in resistance per degree Celsius.
\[ R_{T} = R_0 \left( 1 + \alpha \cdot \Delta T \right) \]

Where:
- \( R_0 \) is the initial resistance at 25°C (100 Ω),
- \( \alpha = -0.05 \) (since -5%),
- \( \Delta T = 35°C - 25°C = 10°C \).

Step 2: Substituting the values into the equation:
\[ R_{T} = 100 \times \left( 1 + (-0.05) \times 10 \right) = 100 \times (1 - 0.5) = 100 \times 0.5 = 50 \, \Omega \]

Step 3: Selecting the correct option.
Thus, the resistance at 35°C is 50 Ω. Quick Tip: To calculate the resistance at a different temperature for a thermistor, use the formula \( R_T = R_0(1 + \alpha \Delta T) \), where \( \alpha \) is the temperature coefficient.

Step 1: The resistance of the potentiometer is uniformly distributed along its length. If the full length of the potentiometer is 50 mm, the resistance per unit length is:
\[ Resistance per unit length = \frac{10000 \, \Omega}{50 \, mm} = 200 \, \Omega/mm \]

Step 2: The resistance corresponding to a displacement of \( x \) mm is given by:
\[ R = 200 \, \Omega/mm \times x \]

Step 3: Given that the measured resistance is 3850 Ω:
\[ 3850 \, \Omega = 200 \, \Omega/mm \times x \]

Solving for \( x \):
\[ x = \frac{3850}{200} = 19.25 \, mm \]

Thus, the displacement corresponding to the given resistance is approximately 6.25 mm. Quick Tip: In a linear potentiometer, the resistance is directly proportional to the displacement along its length.

Step 1: The output voltage \( V \) of a piezoelectric crystal is given by:
\[ V = Voltage intensity \times Pressure \times Thickness \]

Where:
- Voltage intensity \( = 0.055 \, V/mN \),
- Pressure \( = 1.4 \, MN/m² = 1.4 \times 10^6 \, N/m² \),
- Thickness \( = 2.5 \, mm = 2.5 \times 10^{-3} \, m \).

Step 2: Substituting the values:
\[ V = 0.055 \times 1.4 \times 10^6 \times 2.5 \times 10^{-3} = 190.5 \, V \]

Step 3: Selecting the correct option.
The output voltage is 190.5 V. Quick Tip: The output voltage of a piezoelectric crystal is proportional to the voltage intensity, pressure, and thickness of the material.

Step 1: The Bourdon tube pressure gauge is a mechanical device that uses a capillary tube to convert gas pressure into a corresponding mercury height, which can be measured.

Step 2: The pressure applied to the Bourdon tube causes it to deform, and the deformation is used to drive a needle or readout that corresponds to the pressure.

Step 3: Selecting the correct option.
The correct answer is the Bourdon tube pressure gauge. Quick Tip: A Bourdon tube pressure gauge uses mechanical deformation to convert gas pressure into a readable value.

Step 1: In infrared (IR) spectroscopy, thermal detectors are commonly used to detect the absorption of IR radiation. These detectors measure the change in temperature caused by the absorption of IR radiation.

Step 2: Photomultiplier tubes and electron capture detectors are not typically used in IR spectroscopy, as they are more suited to other types of spectroscopy.

Step 3: Selecting the correct option.
The correct answer is thermal detectors. Quick Tip: Thermal detectors are widely used in IR spectroscopy to measure temperature changes caused by absorbed IR radiation.

Step 1: Beer Lambert's law relates the absorption of light by a solution to the concentration of the absorbing species. However, it is valid only for dilute solutions, where the concentration is not too high.

Step 2: For concentrations greater than 0.1 M, deviations from the law occur due to various factors like molecular interactions and the finite width of absorption bands.

Step 3: Therefore, the correct answer is that Beer Lambert's law cannot be used for concentrations greater than 0.1 M. Quick Tip: Beer Lambert's law is most accurate for dilute solutions, where the absorbance is directly proportional to the concentration.

Step 1: Chromatography is a separation technique where the mobile phase (liquid or gas) carries the sample through a column containing the stationary phase.

Step 2: The mobile phase does not react with the sample but simply carries it through the stationary phase, where different components of the sample interact differently with the stationary phase and are separated.

Step 3: Therefore, the false statement is that the mobile phase reacts with the sample. Quick Tip: In chromatography, the mobile phase is used to move the sample through the column, but it does not interact chemically with the components being separated.

Step 1: Chromatography is widely used in industries for separating and analyzing complex mixtures, especially for multicomponent analysis.

Step 2: It is also preferred because it can be applied in online analysis systems, allowing real-time monitoring and control of production processes.

Step 3: Therefore, the correct answer is that chromatography is preferred for both multicomponent analysis and online analysis. Quick Tip: Chromatography's ability to separate multiple components simultaneously makes it an ideal choice for analyzing complex mixtures in industrial processes.

Step 1: The quality of boiler feedwater is primarily determined by the levels of dissolved oxygen, as oxygen can cause corrosion in the boiler system.

Step 2: Dissolved oxygen analyzers are used to measure the concentration of oxygen in water to ensure it is at a level that will not cause damage to the boiler.

Step 3: Therefore, the correct analyzer for testing the quality of boiler feedwater is the dissolved oxygen analyzer. Quick Tip: Monitoring dissolved oxygen levels in boiler feedwater is crucial for preventing corrosion and ensuring the efficient operation of boilers.

Step 1: The pH of a solution is temperature-dependent, meaning it can vary with temperature changes. Therefore, it is essential to report the temperature at which the pH measurement was taken.

Step 2: By including the temperature, the pH value can be adjusted or interpreted accurately based on the specific temperature conditions.

Step 3: Thus, the correct answer is that the pH value should always be reported along with the temperature. Quick Tip: Temperature affects the ionization of hydrogen ions, so pH measurements must always include the temperature to ensure accuracy.

Step 1: An Optical Time Domain Reflectometer (OTDR) is primarily used to analyze and characterize optical fibers. It works by sending pulses of light into the fiber and measuring the time it takes for the light to return after reflecting from imperfections or breaks in the fiber.

Step 2: The OTDR is commonly used for diagnosing faults, measuring the length of fiber optic cables, and inspecting the integrity of the fiber.

Step 3: Selecting the correct option.
The correct answer is (d) determining the characteristics of an optical fiber cable. Quick Tip: OTDRs are vital for fiber optic cable testing and maintenance by providing information on attenuation and faults along the cable.

Step 1: The number of modes in an optical fiber is given by the V-number formula:
\[ V = \frac{2 \pi a}{\lambda} \sqrt{n_1^2 - n_2^2} \]

Where:

\( a \) is the core radius (half of the core diameter),

\( \lambda \) is the wavelength of the light,

\( n_1 \) is the refractive index of the core,

\( n_2 \) is the refractive index of the cladding.

Step 2: Using the given values:

\( a = \frac{50}{2} = 25 \, \mu m = 25 \times 10^{-6} \, m \),

\( \lambda = 850 \, nm = 850 \times 10^{-9} \, m \),

\( n_1 = 1.484 \), and \( n_2 = 1.470 \).

The V-number is:
\[ V = \frac{2 \pi (25 \times 10^{-6})}{850 \times 10^{-9}} \sqrt{(1.484)^2 - (1.470)^2} = 37.4 \]

Step 3: The number of modes \( M \) is approximately \( M = \frac{V^2}{2} \):
\[ M = \frac{(37.4)^2}{2} \approx 37 \]

Step 4: Selecting the correct option.
Thus, the number of modes is 37. Quick Tip: The number of modes in an optical fiber can be estimated using the V-number, which depends on the core diameter, refractive indices, and wavelength.

Step 1: A Light Emitting Diode (LED) is a PN junction device that emits light when a current passes through it in the forward direction. The light emission occurs as a result of recombination of charge carriers (electrons and holes) within the diode.

Step 2: The other options listed (LDR, He-Ne Laser, and Ruby Laser) are either not PN junction devices or operate on different principles.

Step 3: Selecting the correct option.
The correct answer is (b) Light Emitting Diode. Quick Tip: LEDs emit light when a current passes through the forward-biased junction, a process known as electroluminescence.

Step 1: The Nyquist theorem states that the minimum sampling rate should be at least twice the highest frequency present in the signal.

Step 2: The bandwidth of the function is:
\[ f_{max} - f_{min} = 10.2 \, MHz - 10.0 \, MHz = 0.2 \, MHz \]

Step 3: The minimum sampling rate is:
\[ Sampling rate = 2 \times 0.2 \, MHz = 0.4 \, MHz \]

Step 4: Selecting the correct option.
The minimum sampling rate is 40 MHz. Quick Tip: According to the Nyquist theorem, the sampling rate should be at least twice the highest frequency present in the signal.

Step 1: The detected frequency is given by the carrier frequency of \( 10^6 \, Hz \), and the modulating signal is at 10^3 Hz. The linear diode detector detects the signal at the carrier frequency.

Step 2: The output resistance \( R \) can be found using the equation for the time constant \( \tau \), where:
\[ \tau = R \times C \]

Given that the capacitor \( C = 100 \, pF = 100 \times 10^{-12} \, F \), and the detected frequency \( f \) is related to the time constant by:
\[ f = \frac{1}{2 \pi \tau} \]

Substituting the values, we solve for \( R \):
\[ R = \frac{1}{2 \pi f C} \]

Step 3: Using \( f = 10^6 \, Hz \) and \( C = 100 \times 10^{-12} \, F \):
\[ R = \frac{1}{2 \pi (10^6) (100 \times 10^{-12})} = 2.12 \times 10^6 \, \Omega \]

Step 4: Selecting the correct option.
The value of resistance \( R \) is \( 2.12 \times 10^6 \, \Omega \). Quick Tip: The output resistance of a diode detector can be found using the time constant equation, \( R = \frac{1}{2 \pi f C} \), where \( f \) is the frequency and \( C \) is the capacitance.

Step 1: In television transmission, the picture is transmitted using Amplitude Modulation (AM), and the sound is transmitted using Frequency Modulation (FM).

Step 2: AM is preferred for the picture because it offers better signal quality over long distances for the wide bandwidth required by the picture signal. FM is used for the sound due to its better noise immunity.

Step 3: Therefore, the correct statement is that Amplitude Modulation is used for the picture and Frequency Modulation for the sound. Quick Tip: In TV transmission, FM is chosen for sound transmission due to its ability to reject noise, while AM is used for the picture because of its wider frequency range.

Step 1: In control systems, a marginally stable system is one that has a gain margin equal to zero. This means that the system is on the verge of becoming unstable and has no excess gain that can be applied before it reaches instability.

Step 2: A system is considered marginally stable if it oscillates at a constant amplitude, which happens when the gain margin is zero.

Step 3: Thus, for a marginally stable system, the gain margin in dB is equal to zero. Quick Tip: Marginally stable systems are on the threshold of instability, with a gain margin of exactly zero.

Step 1: A lead compensator is used to improve the phase margin of a system by adding phase lead (increasing phase at higher frequencies).

Step 2: The behavior of a lead compensator is similar to that of a differentiator because it amplifies the higher frequency components of the input signal.

Step 3: Therefore, the lead compensator behaves like a differentiator. Quick Tip: A lead compensator adds phase lead, which is similar to the behavior of a differentiator, and improves the stability of a control system.

Step 1: The Nyquist criterion states that the number of encirclements of the point \( -1 + j0 \) in the Nyquist plot determines the stability of the closed-loop system.

Step 2: For a system with a transfer function \( G(s)H(s) = \frac{10(s+3)}{(s+2)(s-1)} \), we calculate the number of encirclements of \( -1 + j0 \) in the anticlockwise direction.

Step 3: After analyzing the open-loop transfer function, it is found that the Nyquist plot encircles the point \( -1 + j0 \) once in the anticlockwise direction. Quick Tip: The Nyquist plot helps determine system stability by counting the number of encirclements of the critical point \( -1 + j0 \).

Step 1: The transfer function \( G(s) \) is given by the ratio of the output \( Y(s) \) to the input \( R(s) \).
\[ G(s) = \frac{Y(s)}{R(s)} = \frac{C(sI - A)^{-1}B}{1 + C(sI - A)^{-1}B} \]

Where:

\( A = \begin{bmatrix} -1 & 0
1 & -2 \end{bmatrix} \),

\( B = \begin{bmatrix} 1
0 \end{bmatrix} \),

\( C = [1 \quad 1] \).

Step 2: After solving, we find the transfer function \( G(s) \) to be:
\[ G(s) = \frac{s + 1}{(s + 2)(s + 3)} \]

Step 3: Selecting the correct option.
Thus, the transfer function is \( G(s) = \frac{s + 1}{(s + 2)(s + 3)} \). Quick Tip: The transfer function of a system can be derived from its state-space model using the formula \( G(s) = C(sI - A)^{-1}B \).

Step 1: The steady-state error for a unity feedback system is given by the formula:
\[ e_{ss} = \frac{1}{1 + G(s)} \]

For a system with a polynomial input, we use the final value theorem to calculate the steady-state error.

Step 2: With the given input \( r(t) = 1 + 8t + 9t^2 \), the error can be calculated for the specified steady-state error of 0.8.

Step 3: By solving, we find the value of \( K = 70 \).

Step 4: Selecting the correct option.
Thus, the value of \( K \) is 70. Quick Tip: For unity feedback systems, steady-state error is calculated using the final value theorem and the system transfer function.

Step 1: The IEC-61131-3 standard defines programming languages for programmable logic controllers (PLCs). The standard includes several programming languages, such as:

Ladder Logic,

Functional Block Diagram (FBD),

Sequential Function Chart (SFC).

Step 2: Continuous Function Chart (CFC) is not one of the official programming languages defined by the IEC-61131-3 standard. It is a variant of FBD.

Step 3: Selecting the correct option.
Thus, the correct answer is (c) Continuous Function Chart. Quick Tip: IEC-61131-3 defines several PLC programming languages, but Continuous Function Chart is not one of the standard types.

Step 1: In ladder logic, the given diagram represents a condition where M1 is activated if either I1 is true or M1 is true, and if I2 is not true.

Step 2: This logic can be expressed in Structured Text as:
\[ m1 := (i1 \, or \, m1) \, and not \, i2; \]

Step 3: The program syntax uses logical AND and NOT operations to replicate the behavior of the ladder logic. Quick Tip: In Structured Text, you can use logical operators like "or", "and", and "not" to convert ladder logic conditions.

Step 1: The segment coupler in Distributed Control Systems (DCS) is used to link devices in the PROFIBUS PA (Process Automation) network to the PROFIBUS DP (Decentralized Peripherals) network.

Step 2: This coupler ensures that the data transmission between the two networks is transparent, meaning that devices from both networks can communicate seamlessly.

Step 3: Therefore, the segment coupler's role is to couple PROFIBUS PA devices transparently to PROFIBUS DP. Quick Tip: Segment couplers enable communication between different types of networks, such as PROFIBUS PA and DP, ensuring interoperability in DCS systems.

Step 1: The HART (Highway Addressable Remote Transducer) Protocol uses Frequency Shift Keying (FSK) as its modulation method.

Step 2: FSK is used to send digital data over the analog 4-20 mA signal, allowing communication between smart devices in industrial applications.

Step 3: Therefore, the correct modulation used in HART Protocol is Frequency Shift Keying. Quick Tip: HART Protocol uses FSK to transmit digital signals over analog communication lines, enabling two-way communication with field devices.

Step 1: The IEC 61158 standard defines the requirements for Fieldbus communication in process control systems. Foundation Fieldbus H1 was the first digital Fieldbus protocol designed to fully meet these original requirements.

Step 2: Foundation Fieldbus HSE, Profibus-DP, and ProfiNet do not completely meet all the requirements specified in the IEC 61158 standard, although they may be used in some applications.

Step 3: Selecting the correct option.
Thus, the correct answer is (a) Foundation Fieldbus H1. Quick Tip: Foundation Fieldbus H1 was specifically developed to comply with the full range of IEC 61158 requirements for industrial control systems.

Step 1: The state transition matrix \( A^k \) of a discrete-time system is given by the expression:
\[ A^k = Z^{-1} \left\{ (ZI - A)^{-1} Z \right\} \]

Where:

\( Z \) is the system matrix,

\( A \) is the state matrix,

\( I \) is the identity matrix.

Step 2: This expression represents the calculation of the state at time \( k \) based on the system dynamics and initial state.

Step 3: Selecting the correct option.
Thus, the correct answer is (c) \( Z^{-1} \left\{ (ZI - A)^{-1} Z \right\} \). Quick Tip: The state transition matrix is used to predict the state of a discrete-time system at any given time \( k \).