Q6
(a) Solve $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ for a rectangular plate subject to the boundary conditions $$u(0,y) = 0, \quad u(a,y) = 0$$ $$u(x,0) = 0, \quad u(x,b) = f(x)$$ (20 marks) (b) Simplify the Boolean function $$F(x,y,z) = xyz + x'yz + xy'z + xyz'$$ and draw the corresponding GATE network. (15 marks) (c) Calculate the moment of inertia of a uniform solid cylinder of mass $M$, radius $R$ and length $L$ with respect to a set of axes passing through the centre of the cylinder, where $z$-axis is the axis of the cylinder and $\rho$ is the constant density at any point of the cylinder. Also find $\frac{L}{R}$ for which the moment of inertia about $x$- or $y$-axis will be minimum for a given mass of the cylinder. (15 marks)
हिंदी में प्रश्न पढ़ें
(a) एक आयताकार प्लेट के लिए $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ को परिसीमा प्रतिबंधों $$u(0,y) = 0, \quad u(a,y) = 0$$ $$u(x,0) = 0, \quad u(x,b) = f(x)$$ के अधीन हल कीजिए। (20 अंक) (b) बूलिय फलन $$F(x,y,z) = xyz + x'yz + xy'z + xyz'$$ का सरलीकरण कीजिए और संगत GATE परिपथ को रेखांकित कीजिए। (15 अंक) (c) द्रव्यमान $M$, त्रिज्या $R$ और लंबाई $L$ के एक एकसमान ठोस बेलन का, बेलन के केंद्र से होकर जाने वाले अक्षों के एक समुच्चय के सापेक्ष जड़त्व आघूर्ण की गणना कीजिए, जहाँ $z$-अक्ष, बेलन का अक्ष है और $\rho$, बेलन के किसी भी बिंदु पर अचर घनत्व है। बेलन के एक दिए गए द्रव्यमान के लिए $\frac{L}{R}$, जिसके लिए $x$- या $y$-अक्ष के सापेक्ष जड़त्व आघूर्ण न्यूनतम होगा, भी ज्ञात कीजिए। (15 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve this three-part numerical problem by allocating approximately 40% time to part (a) given its 20 marks weightage, and roughly 30% each to parts (b) and (c). Begin with clear identification of the mathematical technique for each sub-part: separation of variables for Laplace equation, Boolean algebraic simplification with K-map or algebraic manipulation, and integration for moment of inertia. Present solutions sequentially with proper mathematical derivations, diagrams where requested, and concluding with boxed final answers for each part.
Key points expected
- Part (a): Apply separation of variables u(x,y) = X(x)Y(y), derive eigenvalues λ_n = nπ/a, obtain Fourier sine series coefficients for the non-homogeneous boundary condition u(x,b) = f(x)
- Part (b): Simplify F = yz + xz + xy using Boolean algebra or K-map, identify minimal sum-of-products form, draw AND-OR or NAND-NAND gate network with proper labeling
- Part (c): Set up triple integral in cylindrical coordinates for I_z = ½MR² and I_x = I_y = M(3R² + L²)/12 using ρ = M/(πR²L), minimize I_x by differentiating with respect to L/R ratio to get L/R = √3
- Correct handling of boundary conditions in (a): three homogeneous and one non-homogeneous condition leading to Sturm-Liouville problem
- Proper gate-level diagram in (b) showing input variables, logic gates, and output with standard IEEE/ANSI symbols
- Dimensional consistency check in (c): all moments of inertia having units [ML²] and the optimal ratio being dimensionless
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies u = X(x)Y(y) for (a), recognizes common factor patterns or K-map setup for (b), and establishes proper coordinate system and density relation ρ = M/V for (c); all boundary conditions and geometric constraints accurately transcribed | Partially correct setups with minor errors such as wrong sign convention in separation constant, incomplete K-map labeling, or incorrect limits of integration | Fundamental setup errors like assuming wrong solution form for Laplace equation, algebraic factorization mistakes in Boolean simplification, or using wrong coordinate system for cylinder |
| Method choice | 20% | 10 | Selects separation of variables with proper eigenfunction expansion for (a), uses efficient K-map or systematic Quine-McCluskey approach for (b), employs cylindrical coordinates with correct volume element r dr dθ dz for (c); demonstrates awareness of alternative methods | Correct but inefficient methods, such as direct algebraic simplification without K-map for (b) or Cartesian coordinates for (c) leading to complicated integrals | Inappropriate methods like direct integration for Laplace equation without separation, truth table without simplification for Boolean function, or incorrect application of parallel axis theorem |
| Computation accuracy | 20% | 10 | Flawless execution: correct eigenvalues λ_n = nπ/a, accurate Fourier coefficient formula B_n = (2/a)∫f(x)sin(nπx/a)dx; correct Boolean simplification to F = xy + xz + yz; exact integrals yielding I_z = ½MR², I_x = M(3R²+L²)/12, and optimal L/R = √3 | Minor computational slips such as arithmetic errors in integration constants, one incorrect minterm in K-map, or algebraic mistakes in moment of inertia calculation that don't propagate catastrophically | Major computational errors: wrong eigenvalue determination, incorrect Boolean reduction leading to wrong minimal form, or integration errors producing dimensionally inconsistent results |
| Step justification | 20% | 10 | Clear justification for choosing negative separation constant -λ², explicit reasoning for discarding exponential growth terms using boundary conditions, logical steps in Boolean algebra with cited theorems (absorption, consensus), explicit differentiation and second derivative test for minimization in (c) | Some steps shown but with gaps in reasoning, such as stating results without explaining why certain terms vanish or how minimization condition is derived | Missing crucial justifications: no explanation for sign choice in separation constant, unjustified term elimination, or assertion of minimum without verification |
| Final answer & units | 20% | 10 | Complete series solution u = ΣB_n sin(nπx/a) sinh(nπy/a)/sinh(nπb/a) with coefficient formula; minimal Boolean expression F = xy + xz + yz with correctly drawn AND-OR gate network; I_z = ½MR², I_x = M(3R²+L²)/12, and exact optimal ratio L/R = √3 ≈ 1.732 with proper units [kg·m²] | Correct final forms but missing explicit coefficient formulas, incomplete gate diagrams, or correct expressions without numerical evaluation of optimal ratio | Missing final answers, wrong boxed results, no units, or completely incorrect expressions due to accumulated errors from previous parts |
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