Q7
(a) (i) Find the conjunctive normal form (CNF) of the following Boolean function: f(x, y, z, t) = x · y · z + x̄ · y · (t + z̄) (15 marks) (ii) Express the Boolean function f(x, y, z) = x + (x̄ · ȳ + x̄ · z) + z in disjunctive normal form (DNF) and construct the truth table for the function. (15 marks) (b) A perfectly rough ball is at rest within a hollow cylindrical roller. The roller is drawn along a level path with uniform velocity V. Let a and b be the radii of the ball and the roller respectively. If V² > 27/7 g(b-a), then show that the ball will roll completely round the inside of the roller. (15 marks) (c) Solve the partial differential equation a² ∂²u/∂x² = ∂²u/∂t², 0 < x < L, t > 0 subject to the conditions u(0,t) = 0, u(L,t) = 0, t > 0; u(x,0) = x, (∂u/∂t)ₜ₌₀ = 1, 0 < x < L. (20 marks)
हिंदी में प्रश्न पढ़ें
(क) (i) निम्न बूलिय फलन का योगात्मक प्रसामान्य स्वरूप (CNF) ज्ञात कीजिए : f(x, y, z, t) = x · y · z + x̄ · y · (t + z̄) (15 अंक) (ii) बूलिय फलन f(x, y, z) = x + (x̄ · ȳ + x̄ · z) + z को विभोजनीय (डिस्जंक्टिव) प्रसामान्य स्वरूप (DNF) में व्यक्त कीजिए तथा इस फलन के लिए सत्यमान सारणी बनाइए। (15 अंक) (ख) एक आदर्श रक्ष गेंद एक खोखले बेलनाकार रोलर में विराम की स्थिति में है। रोलर को एक समतल पथ के अनुदिश एकसमान वेग V से खींचा जाता है। मान लीजिए कि a तथा b क्रमशः गेंद तथा रोलर की त्रिज्याएँ हैं। यदि V² > 27/7 g(b-a) है, तब दर्शाइए कि गेंद रोलर के अन्दर पूर्ण रूप से घूम जाएगी। (15 अंक) (ग) अंशिक अवकल समीकरण a² ∂²u/∂x² = ∂²u/∂t², 0 < x < L, t > 0 का शर्तों u(0,t) = 0, u(L,t) = 0, t > 0; u(x,0) = x, (∂u/∂t)ₜ₌₀ = 1, 0 < x < L से प्रतिबंधित हल ज्ञात कीजिए। (20 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve this multi-part numerical problem by allocating approximately 30% time to part (a)(i) CNF conversion, 30% to part (a)(ii) DNF and truth table, 20% to part (b) mechanics proof, and 20% to part (c) wave equation solution. Begin with Boolean algebra manipulations using De Morgan's laws and canonical forms, then proceed to the constrained motion analysis with energy and angular momentum conservation, and finally apply separation of variables with Fourier series for the PDE. Present each sub-part clearly with proper mathematical notation and logical flow.
Key points expected
- For (a)(i): Apply De Morgan's laws and distributive laws to convert f(x,y,z,t) = xyz + x̄y(t+z̄) into CNF; identify maxterms where f=0 and express as product of sums
- For (a)(ii): Simplify f(x,y,z) = x + (x̄ȳ + x̄z) + z using Boolean identities, convert to DNF as sum of minterms, and construct complete 8-row truth table
- For (b): Set up Lagrangian for ball in rolling cylinder with constraint of rolling without slipping; derive energy equation and condition for complete loop using effective potential
- For (c): Apply separation of variables u(x,t) = X(x)T(t) to wave equation; determine eigenvalues λn = nπ/L and eigenfunctions sin(nπx/L); apply initial conditions u(x,0)=x and ut(x,0)=1 to find Fourier coefficients
- For (c): Compute Fourier sine series coefficients bn = (2/L)∫₀ᴸ x sin(nπx/L)dx and cn = (2/nπa)∫₀ᴸ sin(nπx/L)dx for the complete solution
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies Boolean variables and operations for (a); properly sets up rotating reference frame and constraint equations for (b); correctly identifies wave equation type and boundary conditions as Dirichlet homogeneous for (c) | Minor errors in Boolean expression parsing or missing one constraint in mechanics setup; partially correct PDE classification but boundary conditions stated incompletely | Fundamental misunderstanding of CNF/DNF requirements; incorrect force diagram or missing rolling constraint; wrong PDE type identification or boundary condition errors |
| Method choice | 20% | 10 | Uses maxterm identification for CNF, systematic truth table construction for DNF, energy conservation with rotating frame for mechanics, and standard separation of variables with Fourier expansion for wave equation | Correct general approach but inefficient methods chosen, such as truth table expansion without simplification for (a), or direct integration without eigenfunction expansion for (c) | Inappropriate methods like K-map for 4-variable CNF without justification, or attempting Newton's laws without constraints for (b), or Laplace transform for wave equation with given boundary conditions |
| Computation accuracy | 20% | 10 | Flawless algebraic manipulation in Boolean expansions; correct derivation of V² > 27g(b-a)/7 condition with proper handling of centripetal and rotational energy terms; accurate Fourier coefficient integration yielding bn = 2L(-1)^(n+1)/nπ | Minor arithmetic slips in Boolean term expansion or sign errors in Fourier coefficients; correct final inequality but with missing factor in energy derivation | Major computational errors like incorrect De Morgan application, wrong moment of inertia usage leading to incorrect numerical factor, or integration errors preventing series solution completion |
| Step justification | 20% | 10 | Explicitly cites De Morgan's laws, distributive laws, and absorption laws for Boolean steps; clearly states conservation principles and constraint equations for mechanics; justifies eigenfunction orthogonality and term-by-term differentiation for PDE solution | States key theorems without full justification, or skips obvious algebraic steps while showing critical derivations; mentions separation of variables without proving validity | Missing justification for critical steps like canonical form conversion, or asserts results without derivation; no explanation for why Fourier series applies or unjustified term operations |
| Final answer & units | 20% | 10 | Presents complete CNF as product of maxterms, complete DNF with verified truth table, proven inequality with physical interpretation, and closed-form series solution u(x,t) = Σ[bn cos(nπat/L) + cn sin(nπat/L)]sin(nπx/L) with explicit coefficients | Correct final forms but incomplete truth table, or correct inequality without boxed final statement, or series solution without simplified coefficient expressions | Incomplete answers missing required components like truth table for (a)(ii), or missing final inequality statement for (b), or general solution without applying initial conditions for (c) |
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