Q6
(a) Solve the wave equation a²∂²u/∂x² = ∂²u/∂t², 0<x<L, t>0 subject to the conditions u(0,t)=0, u(L,t)=0 u(x,0)=(1/4)x(L-x), ∂u/∂t|ₜ₌₀=0 (20 marks) (b) Obtain the Boolean function F(x, y, z) based on the table given below. Then simplify F(x, y, z) and draw the corresponding GATE network: | x | y | z | F(x, y, z) | |---|---|---|------------| | 1 | 1 | 1 | 1 | | 1 | 1 | 0 | 1 | | 1 | 0 | 1 | 1 | | 1 | 0 | 0 | 0 | | 0 | 1 | 1 | 1 | | 0 | 1 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 0 | 0 | 0 | (15 marks) (c) Obtain the Lagrangian equation for the motion of a system of two particles of unequal masses connected by an inextensible string passing over a small smooth pulley. (15 marks)
हिंदी में प्रश्न पढ़ें
(a) तरंग समीकरण a²∂²u/∂x² = ∂²u/∂t², 0<x<L, t>0 का शर्तों u(0,t)=0, u(L,t)=0 u(x,0)=(1/4)x(L-x), ∂u/∂t|ₜ₌₀=0 से प्रतिबंधित हल प्राप्त कीजिए। (20 अंक) (b) नीचे दी गई सारणी पर आधारित बूलियन फलन F(x,y,z) को निकालिए और तब F(x,y,z) को सरल कीजिए तथा उसके अनुरूप GATE परिपथ खींचिए : | x | y | z | F(x,y,z) | |:---:|:---:|:---:|:----------:| | 1 | 1 | 1 | 1 | | 1 | 1 | 0 | 1 | | 1 | 0 | 1 | 1 | | 1 | 0 | 0 | 0 | | 0 | 1 | 1 | 1 | | 0 | 1 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 0 | 0 | 0 | (15 अंक) (c) एक छोटी चिकनी घिरनी के ऊपर से गुजरने वाली एक अवितान्य डोरी के सिरों से बंधे असमान संहति वाले दो कणों के निकाय की गति के लिए लग्रांजी समीकरण प्राप्त कीजिए। (15 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve requires systematic derivation and computation across all three sub-parts. Allocate approximately 40% of effort to part (a) given its 20 marks weightage—apply separation of variables for the wave equation with proper Fourier series expansion of the initial displacement. Spend roughly 30% each on parts (b) and (c): for (b), construct the canonical SOP form, apply Karnaugh map or Boolean algebra simplification, then design the gate network; for (c), set up generalized coordinates, write kinetic and potential energies, and derive Lagrange's equations of motion for the Atwood machine variant.
Key points expected
- Part (a): Correct separation of variables u(x,t)=X(x)T(t), application of boundary conditions to obtain eigenvalues λn=(nπ/L)², and Fourier sine series expansion of initial displacement φ(x)=(1/4)x(L-x)
- Part (a): Accurate computation of Fourier coefficients bn=(2/L)∫₀ᴸ φ(x)sin(nπx/L)dx using integration by parts, yielding the complete solution u(x,t)=Σ bn sin(nπx/L)cos(anπt/L)
- Part (b): Correct Boolean function F=Σm(1,3,5,7) from truth table, simplification to F=xz+yz or F=z(x+y) using K-map or algebraic manipulation
- Part (b): Proper gate network diagram showing OR gate for (x+y) feeding into AND gate with z, or equivalent NAND-NAND realization
- Part (c): Selection of generalized coordinate (vertical displacement of one mass), expression of kinetic energy T=(1/2)(m₁+m₂)ẋ² and potential energy V=m₁gx+m₂g(l-x)
- Part (c): Derivation of Lagrange's equation leading to (m₁+m₂)ẍ=(m₁-m₂)g, showing correct equation of motion for the Atwood machine
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly identifies separation ansatz and boundary conditions leading to standing wave modes; for (b): accurately reads minterms from truth table; for (c): properly defines generalized coordinate and constraint of inextensible string | Sets up most problems correctly but has minor errors in boundary condition application or misidentifies one minterm; generalized coordinate chosen but energy expressions partially flawed | Fundamental misunderstanding of separation of variables, incorrect minterm extraction, or fails to identify appropriate generalized coordinates for the constrained system |
| Method choice | 20% | 10 | For (a): Fourier series method with eigenfunction expansion; for (b): K-map or systematic Boolean algebra leading to minimal SOP/POS; for (c): Lagrangian formalism with proper treatment of constraints | Correct general approach but suboptimal method choice (e.g., direct algebraic simplification without K-map, or Newtonian approach instead of Lagrangian) | Inappropriate methods such as Laplace transform for (a), truth table without simplification for (b), or force balance without energy formulation for (c) |
| Computation accuracy | 20% | 10 | For (a): exact Fourier coefficients bn=(2L²/(n³π³))[1-(-1)ⁿ]; for (b): correct simplification to F=z(x+y); for (c): accurate derivation of acceleration ẍ=(m₁-m₂)g/(m₁+m₂) | Correct computational steps with minor arithmetic errors in integration, simplification, or final algebra; partial credit for correct setup with wrong execution | Major computational errors in integration by parts, Boolean simplification, or differentiation of Lagrangian; incorrect final expressions |
| Step justification | 20% | 10 | Clear justification of orthogonality for Fourier coefficients, explains why certain minterms group together in K-map, and physically interprets Lagrange equations; cites d'Alembert's solution connection for wave equation | Shows most steps with occasional gaps in reasoning; computational steps present but lacks physical interpretation or theorem citations | Unjustified leaps between steps, missing derivation of key results, or presents final answers without showing intermediate working |
| Final answer & units | 20% | 10 | Complete series solution for (a) with proper notation; minimal Boolean expression and correctly labeled gate diagram for (b); explicit Lagrange equations and physical interpretation of normal modes for (c) | Correct final forms but missing summation limits, incomplete gate labels, or lacks discussion of physical significance of results | Missing final answers, incorrect units in physical quantities, incomplete gate networks, or fails to state the equation of motion explicitly |
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