Q6
(a) Solve the heat equation ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x), 0 ≤ x ≤ l. (20 marks) (b) Find a combinatorial circuit corresponding to the Boolean function f(x, y, z) = [x · (ȳ + z)] + y and write the input/output table for the circuit. (15 marks) (c) Find the moment of inertia of a right circular solid cone about one of its slant sides (generator) in terms of its mass M, height h and the radius of base as a. (15 marks)
हिंदी में प्रश्न पढ़ें
(a) उष्मा समीकरण ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 का शर्तों u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x), 0 ≤ x ≤ l से प्रतिबंधित हल ज्ञात कीजिये। (20 अंक) (b) बूलिय फलन f(x, y, z) = [x · (ȳ + z)] + y का संगत संयोजीव्यास परिपथ (कॉम्बिनेटोरियल सर्किट) ज्ञात कीजिये तथा परिपथ के लिये निवेश/निर्गत (इनपुट/आउटपुट) सारणी लिखिये। (15 अंक) (c) एक लम्ब वृत्तीय ठोस शंकु का उसकी संहति M, ऊँचाई h तथा आधार की त्रिज्या a के रूप में उसकी एक तिर्यक रेखा (जनक रेखा) के सापेक्ष जड़त्व-आघूर्ण ज्ञात कीजिये। (15 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve requires systematic derivation of exact solutions with complete mathematical rigor. Allocate approximately 40% of effort to part (a) as it carries 20 marks—apply separation of variables, determine Fourier coefficients for the parabolic initial condition, and write the complete series solution. Spend roughly 30% each on (b) and (c): for (b), construct the logic circuit using AND/OR/NOT gates and enumerate all 8 input combinations; for (c), set up proper coordinate system with cone vertex at origin, use parallel axis theorem or direct integration about slant generator, and express result purely in terms of M, h, a. Present each part distinctly with clear labeling.
Key points expected
- Part (a): Correct application of separation of variables u(x,t) = X(x)T(t), eigenvalues λ_n = n²π²/l², and Fourier sine series coefficients a_n = (4l²/n³π³)[1-(-1)ⁿ] for the initial condition x(l-x)
- Part (a): Final solution expressed as u(x,t) = Σ_{n=1}^∞ (8l²/n³π³)sin(nπx/l)exp(-n²π²t/l²) for odd n, with explicit recognition that even terms vanish
- Part (b): Correct Boolean simplification to f = xz + xȳ + y, or equivalent form, followed by circuit diagram with proper gate symbols (AND, OR, NOT) showing input/output flow
- Part (b): Complete truth table with all 8 rows (000 through 111) showing intermediate values ȳ, (ȳ+z), x·(ȳ+z), and final output f
- Part (c): Proper coordinate setup with cone geometry, density ρ = 3M/πa²h, and integration limits; correct application of perpendicular axis theorem or direct moment calculation about slant generator
- Part (c): Final expression I = (3M/20)(a² + 4h²) or equivalent simplified form involving slant height l = √(a²+h²), with dimensional verification [ML²]
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly identifies boundary conditions as Dirichlet homogeneous, chooses appropriate eigenfunction expansion; for (b): accurately parses Boolean expression with proper operator precedence; for (c): establishes cone geometry with vertex at origin, slant generator clearly identified, mass-density relation stated | Sets up most problems correctly but has minor errors in boundary condition interpretation for (a), or misplaces NOT gates in (b), or uses wrong axis for moment calculation in (c) | Fundamental setup errors: treats (a) as wave equation, ignores operator precedence in (b), calculates moment about wrong axis entirely in (c) |
| Method choice | 20% | 10 | For (a): separation of variables with Fourier series; for (b): De Morgan's laws or direct gate implementation with optimal circuit design; for (c): cylindrical coordinates or appropriate coordinate transformation with parallel/perpendicular axis theorem judiciously applied | Correct general methods but suboptimal choices—e.g., uses full Fourier series instead of sine series for (a), or brute-force truth table without simplification for (b), or direct integration without theorem assistance for (c) | Inappropriate methods: Laplace transform for (a) without justification, attempts K-map for 3-variable without completion, uses Cartesian coordinates without transformation for (c) leading to intractable integrals |
| Computation accuracy | 20% | 10 | For (a): exact Fourier coefficients with proper integration by parts; for (b): all 8 truth table rows correct, circuit logic verified; for (c): precise triple integration with correct limits, algebraic simplification error-free | Minor computational slips: sign errors in Fourier coefficients, 1-2 incorrect rows in truth table, or arithmetic errors in final moment expression that preserve dimensional correctness | Major computational failures: incorrect eigenvalue determination, multiple truth table errors, or integration yielding wrong powers of a and h with violated dimensional analysis |
| Step justification | 20% | 10 | Each mathematical step explicitly justified: orthogonality of eigenfunctions cited for (a), Boolean algebra laws named for (b), parallel axis theorem statement and application shown for (c); logical flow is examiner-friendly | Some steps justified but gaps remain—e.g., assumes orthogonality without mention, skips gate-level explanation, or omits theorem statement while applying it | Minimal or absent justification: series solution appears without derivation, circuit drawn without explanation, moment result stated without integration steps shown |
| Final answer & units | 20% | 10 | For (a): complete series solution with explicit coefficient formula and convergence note; for (b): simplified Boolean expression, clean circuit diagram, complete truth table; for (c): compact formula I = (3M/20)(a² + 4h²) or equivalent with [ML²] verification, physical interpretation noted | Correct final forms but incomplete: missing convergence discussion for (a), unlabeled circuit diagram for (b), or unsimplified algebraic expression for (c) without dimensional check | Missing or wrong final answers: no series summation for (a), incomplete/incorrect truth table for (b), or moment expressed with wrong powers or missing mass dependence for (c) |
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