Q6
(a) Solve the differential equation: d³y/dx³ - 3d²y/dx² + 4dy/dx - 2y = eˣ + cos x. (15 marks) (b) When a particle is projected from a point O₁ on the sea level with a velocity v and angle of projection θ with the horizon in a vertical plane, its horizontal range is R₁. If it is further projected from a point O₂, which is vertically above O₁ at a height h in the same vertical plane, with the same velocity v and same angle θ with the horizon, its horizontal range is R₂. Prove that R₂ > R₁ and (R₂-R₁):R₁ is equal to (1/2){√(1 + 2gh/v²sin²θ) - 1}:1. (15 marks) (c) Evaluate the integral ∬ₛ (3y²z²î + 4z²x²ĵ + z²y²k̂)·n̂ dS, where S is the upper part of the surface 4x² + 4y² + 4z² = 1 above the plane z = 0 and bounded by the xy-plane. Hence, verify Gauss-Divergence theorem. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) अवकल समीकरण : d³y/dx³ - 3d²y/dx² + 4dy/dx - 2y = eˣ + cos x का हल कीजिए। (15 अंक) (b) एक कण को समुद्र तल पर बिन्दु O₁ से वेग v तथा क्षैतिज से प्रक्षेप कोण θ पर उद्वाधर तल में प्रक्षेपित किया जाता है तो क्षैतिज परास R₁ है। यदि इसको पुनः बिन्दु O₂, जो उसी उद्वाधर तल में O₁ के उद्वाधरतः h ऊँचाई पर है, से उसी वेग v तथा क्षैतिज से समान कोण θ पर प्रक्षेपित किया जाता है तो क्षैतिज परास R₂ है। सिद्ध कीजिए R₂ > R₁ तथा (R₂ - R₁) : R₁ = (1/2){√(1 + 2gh/v²sin²θ) - 1} : 1. (15 अंक) (c) समाकल ∬ₛ (3y²z²î + 4z²x²ĵ + z²y²k̂)·n̂ dS का मान ज्ञात कीजिए; जहाँ S समतल z = 0 के ऊपर पृष्ठ 4x² + 4y² + 4z² = 1 का ऊपरी भाग है और xy-समतल द्वारा परिबद्ध है। अतः गॉस-अपसरण प्रमेय को सत्यापित कीजिए। (20 अंक)
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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) on third-order linear ODE, 30% to part (b) on projectile motion derivation, and 40% to part (c) on surface integral evaluation and Gauss-Divergence theorem verification. Begin each part with clear problem identification, show complete working with intermediate steps, and conclude with boxed final answers. For part (c), explicitly compute both the surface integral directly and the volume integral via divergence theorem to demonstrate verification.
Key points expected
- Part (a): Correctly find complementary function by solving characteristic equation m³ - 3m² + 4m - 2 = 0 with roots m = 1, 1±i, and construct particular integrals for both eˣ and cos x terms using appropriate methods
- Part (a): Apply correct operator methods or undetermined coefficients to obtain PI for eˣ (resonance case) and PI for cos x, then combine for complete general solution
- Part (b): Set up projectile equations from sea level O₁: derive range R₁ = v²sin(2θ)/g using standard trajectory equations y = xtanθ - gx²/(2v²cos²θ)
- Part (b): Set up modified equations from height h at O₂, find new range by solving when projectile hits sea level (y = -h), derive R₂ = (v²cosθ/g)[sinθ + √(sin²θ + 2gh/v²)], prove R₂ > R₁ and establish required ratio
- Part (c): Identify S as upper hemisphere 4x² + 4y² + 4z² = 1, z ≥ 0 with radius 1/2, correctly compute divergence ∇·F = 2yz² + 2z²x + 2zy² for verification
- Part (c): Evaluate surface integral directly using spherical coordinates or projection method, then compute volume integral ∭(∇·F)dV over hemisphere, show equality to verify Gauss-Divergence theorem
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly writes characteristic equation and identifies all three roots including complex conjugate pair; for (b): properly sets coordinate systems at O₁ and O₂ with correct initial conditions; for (c): correctly identifies hemisphere geometry with radius 1/2 and bounds, writes proper divergence expression | Identifies most roots in (a) with minor errors; sets up projectile equations in (b) but with inconsistent coordinate choices; identifies surface in (c) but with incorrect radius or bounds | Major errors in characteristic equation roots; confuses projectile setup between level and elevated launch; fails to identify hemisphere geometry or computes divergence incorrectly |
| Method choice | 20% | 10 | For (a): uses operator D-method or variation of parameters correctly with proper handling of resonance for eˣ term; for (b): chooses energy method or direct integration appropriately; for (c): selects optimal spherical coordinates for surface integral and cylindrical/spherical for volume integral | Uses standard methods but with suboptimal choices (e.g., undetermined coefficients with avoidable complexity); completes projectile derivation but with longer algebraic route; uses Cartesian coordinates in (c) making computation tedious | Inappropriate method selection (e.g., Laplace transforms for (a)); uses incorrect energy conservation in (b); attempts direct surface parametrization without coordinate transformation in (c) |
| Computation accuracy | 20% | 10 | Accurate arithmetic throughout: correct factorization (m-1)(m²-2m+2)=0 in (a); precise algebraic manipulation leading to clean ratio in (b); exact evaluation of both surface and volume integrals in (c) with matching numerical results | Minor computational slips (e.g., sign errors in PI coefficients, arithmetic errors in range formula simplification, one incorrect term in divergence or integration bounds) that don't fundamentally derail the solution | Serious computational errors: wrong partial fraction decomposition, incorrect quadratic formula application, major integration errors, or failure to obtain matching values in theorem verification |
| Step justification | 20% | 10 | Every non-trivial step explained: why particular integral form changes for resonance in (a); physical reasoning for trajectory intersection in (b); explicit statement of Gauss theorem conditions and verification of orientations in (c); clear logical flow between steps | Most key steps justified but some gaps (e.g., assumes PI form without explanation, skips algebraic simplification justification, states theorem without checking applicability conditions) | Minimal reasoning provided; jumps between steps without explanation; no physical interpretation in (b); applies Gauss theorem without verifying closed surface or simply asserts equality without demonstration |
| Final answer & units | 20% | 10 | Complete general solution with arbitrary constants for (a); fully derived ratio with clear proof of R₂ > R₁ for (b); explicit numerical values for both integrals in (c) with clear verification statement; all answers properly boxed and dimensionally consistent | Correct final forms but missing arbitrary constants in (a), or incomplete simplification of ratio in (b), or correct methodology in (c) but with unresolved final numerical comparison | Missing final answers, incorrect boxed results, or failure to complete any part; no verification statement in (c); dimensional inconsistencies in projectile results |
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