Q6
(a) Show that the solution of the two-dimensional Laplace's equation ∂²φ(x,y)/∂x² + ∂²φ(x,y)/∂y² = 0, x ∈ (-∞, ∞), y ≥ 0 subject to the boundary condition φ(x,0) = f(x), x ∈ (-∞, ∞), along with φ(x,y) → 0 for |x| → ∞ and y → ∞ can be written in the form φ(x,y) = (y/π)∫_{-∞}^{∞} [f(ξ)dξ]/[y² + (x-ξ)²]. (20 marks) (b) Draw the logical circuit for the Boolean expression Y = ABC̄ + BC̄ + ĀB. Also, obtain the output Y (truth table) for the three input bit sequences: A = 10001111, B = 00111100, C = 11000100 (15 marks) (c) Find the moment of inertia of a quadrant of an elliptic disk x²/a² + y²/b² = 1, of mass M about the line passing through its centre and perpendicular to its plane. Given that the density at any point is proportional to xy. (15 marks)
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(a) दर्शाइए कि द्विविम लाप्लास समीकरण ∂²φ(x,y)/∂x² + ∂²φ(x,y)/∂y² = 0, x ∈ (-∞, ∞), y ≥ 0 का हल, परिसीमा प्रतिबंध φ(x,0) = f(x), x ∈ (-∞, ∞) के अधीन तथा φ(x,y) → 0 जब |x| → ∞ और y → ∞, φ(x,y) = (y/π)∫_{-∞}^{∞} [f(ξ)dξ]/[y² + (x-ξ)²] के रूप में लिखा जा सकता है। (20 अंक) (b) बूलिय व्यंजक Y = ABC̄ + BC̄ + ĀB के लिए तर्कसंगत परिपथ (लॉजिकल सर्किट) खींचिए। तीन निवेश द्वयक अनुक्रमों A = 10001111, B = 00111100, C = 11000100 के लिए निर्गत Y (सत्यमान सारणी) भी प्राप्त कीजिए। (15 अंक) (c) दीर्घवृत्तीय डिस्क x²/a² + y²/b² = 1, के एक चतुर्थांश, जिसका द्रव्यमान M है, का उसके तल के लंबवत तथा उसके केंद्र से गुजरने वाली रेखा के सापेक्ष, जड़त्व आघूर्ण ज्ञात कीजिए। दिया गया है कि किसी भी बिंदु पर घनत्व xy के समानुपाती है। (15 अंक)
Directive word: Derive
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How this answer will be evaluated
Approach
Derive the Poisson integral formula for Laplace's equation in part (a) using Fourier transform or Green's function method, spending approximately 40% of time due to its 20 marks weightage. For part (b), simplify the Boolean expression using Boolean algebra laws, design the logic circuit with AND/OR/NOT gates, and construct the truth table for the given bit sequences (~30% time). For part (c), set up the double integral for moment of inertia with variable density ρ = kxy, transform to elliptical coordinates, and evaluate carefully (~30% time). Present solutions with clear section headings for each part.
Key points expected
- Part (a): Apply Fourier transform in x to convert PDE to ODE, solve the resulting equation φ̂(k,y) = A(k)e^{-|k|y}, apply boundary condition to find A(k) = f̂(k), then invert using convolution theorem with Poisson kernel
- Part (a): Alternatively use Green's function method with image source for half-plane to derive the integral representation directly
- Part (b): Simplify Y = ABC̄ + BC̄ + ĀB to Y = BC̄ + ĀB using absorption law, or further to minimal form, then draw circuit with 2 NOT gates, 2 AND gates, 1 OR gate
- Part (b): Construct truth table showing A,B,C inputs and Y output, then evaluate Y for given 8-bit sequences by bitwise operation: Y = 00110000
- Part (c): Set up I = ∫∫ ρ(x,y)(x²+y²)dA with ρ = kxy over first quadrant, use transformation x = arcosθ, y = brsinθ with Jacobian abr
- Part (c): Determine k from total mass condition M = k∫∫xy dA = kab²/8, then evaluate I = (4M/π)(a²+b²) or equivalent simplified form
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly states Fourier transform pair and decay condition; for (b): proper Boolean variable identification and gate symbols; for (c): correct density function ρ=kxy, proper quadrant limits (x≥0,y≥0), and correct moment of inertia integrand ρ(x²+y²) | Minor errors in transform definition or gate labeling; density or limits partially correct but Jacobian missed in (c) | Wrong transform direction, incorrect gate types, or fundamental setup errors like using uniform density or wrong quadrant |
| Method choice | 20% | 10 | For (a): Fourier transform or Green's function method executed with recognition of Poisson kernel; for (b): optimal Boolean simplification (K-map or algebraic) before circuit design; for (c): elliptical coordinates or smart substitution reducing to standard integrals | Correct but inefficient methods, e.g., direct integration without coordinate transformation in (c), or unsimplified Boolean expression | Inappropriate methods like separation of variables without transform in (a), or random gate connections without simplification |
| Computation accuracy | 20% | 10 | For (a): correct inversion integral and residue/contour evaluation; for (b): accurate bitwise operations yielding Y=00110000; for (c): exact evaluation of ∫∫r⁵cosθsinθ drdθ with correct limits and final coefficient | Minor arithmetic slips in coefficients or exponents, or one bit error in truth table evaluation | Major computational errors like wrong Fourier inversion, incorrect bit-wise AND/OR, or missing Jacobian leading to wrong dimension |
| Step justification | 20% | 10 | Each mathematical step annotated: why e^{-|k|y} not e^{|k|y} (boundedness), which Boolean law applied (absorption: A+ĀB=A+B), why polar-like coordinates suit ellipse, with convergence arguments for improper integrals | Some steps justified but gaps in reasoning, e.g., assuming solution form without decay argument, or stating Boolean results without law names | Unjustified leaps, missing decay conditions, or purely mechanical computation without mathematical reasoning |
| Final answer & units | 20% | 10 | Part (a): integral formula exactly as stated; part (b): clear circuit diagram with labeled inputs/outputs and verified truth table; part (c): I in terms of M,a,b with correct dimensional analysis [ML²] | Correct form but minor notational inconsistencies or missing diagram labels | Incomplete answers, wrong formula structure, missing circuit diagram, or dimensional inconsistency in moment of inertia |
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