Q8
(a) Solve the differential equation $(x + 2)\frac{d^2y}{dx^2} - (2x + 5)\frac{dy}{dx} + 2y = (1 + x) e^x$ by the method of variation of parameters. (15 marks) (b) Verify Gauss's divergence theorem for $\vec{F} = [(x^2 - yz)\hat{i} + (y^2 - zx)\hat{j} + (z^2 - xy)\hat{k}]$, taken over the rectangular parallelopiped $0 \leq x \leq a$, $0 \leq y \leq b$, $0 \leq z \leq c$. (15 marks) (c) A particle is projected inside a fixed smooth cylinder with circular cross-section in a vertical plane from the lowest point with initial horizontal velocity u. Show that for (i) $(u^2 \leq 2ag)$; the particle oscillates about the mean position in the lower half, (ii) $(u^2 \geq 5ag)$; the particle executes complete circular motion, and (iii) $(2ag < u^2 < 5ag)$; the particle will leave the curve in a tangential direction, making an angle $\alpha$ with the horizontal such that $\cos \alpha = \frac{u^2 - 2ag}{3ag}$. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) अवकल समीकरण $(x + 2)\frac{d^2y}{dx^2} - (2x + 5)\frac{dy}{dx} + 2y = (1 + x) e^x$ को प्राचल विचरण विधि द्वारा हल कीजिए। (15 अंक) (b) समकोणिक समांतरपटलक $0 \leq x \leq a$, $0 \leq y \leq b$, $0 \leq z \leq c$ पर $\vec{F} = [(x^2 - yz)\hat{i} + (y^2 - zx)\hat{j} + (z^2 - xy)\hat{k}]$ के लिए गॉस अपसरण प्रमेय सत्यापित कीजिए। (15 अंक) (c) एक कण को उच्चाधर तल में वृत्ताकर अनुप्रस्थ-परिच्छेद वाले स्थिर चिकने बेलन के अंदर प्रारंभिक क्षैतिज वेग u के साथ सबसे निचले बिंदु से प्रक्षेपित किया जाता है। दर्शाइए कि (i) $(u^2 \leq 2ag)$ के लिए; कण निचले आधे भाग में माध्य स्थिति के आसपास (about) दोलन करता है, (ii) $(u^2 \geq 5ag)$ के लिए; कण पूर्णतः वृत्तीय गति करता है, और (iii) $(2ag < u^2 < 5ag)$ के लिए; कण, वक्र को एक स्पर्श की दिशा में, जो क्षैतिज के साथ कोण $\alpha$ बनाती है, छोड़ देगा, जबकि $\cos \alpha = \frac{u^2 - 2ag}{3ag}$ है। (20 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve all three parts systematically, allocating approximately 30% time to part (a) on variation of parameters, 30% to part (b) on divergence theorem verification, and 40% to part (c) on particle dynamics which carries the highest marks. Begin each part with clear identification of the method/theorem being applied, show complete working with intermediate steps, and conclude with explicit verification or derived conditions. For part (c), clearly distinguish the three energy regimes with proper energy conservation and normal reaction analysis.
Key points expected
- Part (a): Reduce to standard form, find complementary function by solving characteristic equation, apply variation of parameters correctly with Wronskian calculation, obtain particular integral and general solution
- Part (b): Compute divergence of F correctly as 2(x+y+z), evaluate volume integral over rectangular parallelopiped, calculate surface integral over all six faces with proper orientation, show equality of both integrals
- Part (c): Apply energy conservation between lowest point and arbitrary position, derive expression for normal reaction R in terms of angle and velocity, analyze R=0 condition for leaving the curve, establish critical velocity thresholds at u²=2ag and u²=5ag
- Part (c)(i): Show particle cannot reach horizontal diameter when u²≤2ag, prove oscillatory motion in lower half with amplitude determined by initial energy
- Part (c)(ii): Demonstrate complete circular motion requires R≥0 throughout, show u²≥5ag ensures positive normal reaction at topmost point
- Part (c)(iii): Derive leaving condition R=0, obtain cos α = (u²-2ag)/(3ag) by simultaneous solution of energy and force equations
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies standard form for (a) with proper reduction; sets correct limits and orientation for (b)'s rectangular parallelopiped; establishes valid coordinate system and energy reference for (c) with clear force diagram | Minor errors in standard form reduction or orientation signs; partially correct limits or missing one face in (b); energy equation present but reference point unclear in (c) | Wrong form for variation of parameters, incorrect divergence calculation, or fundamentally wrong energy setup; missing coordinate system or incorrect geometry |
| Method choice | 20% | 10 | Explicitly applies variation of parameters (not undetermined coefficients) for (a); uses direct volume integration and systematic face-by-face surface integration for (b); employs energy conservation and radial dynamics consistently for (c) | Correct method chosen but execution lacks clarity; some mixing of methods or incomplete application; uses alternative valid approaches but with extra computational burden | Uses wrong method (e.g., undetermined coefficients for (a)); attempts divergence theorem verification without computing both sides; applies kinematic equations instead of energy methods for (c) |
| Computation accuracy | 20% | 10 | Flawless integration in (a) with correct Wronskian and particular integral; exact equality of volume and surface integrals in (b) with all six faces computed correctly; precise algebraic derivation of cos α formula and correct critical values in (c) | Minor arithmetic errors that don't propagate significantly; one face integral incorrect in (b) or sign error in Wronskian; algebraic slips in (c) but correct final structure | Major integration errors; volume and surface integrals differ significantly; wrong critical values (2ag, 5ag) or incorrect final formula for cos α |
| Step justification | 20% | 10 | Clear justification for choosing basis solutions in (a); explicit statement of outward normals and orientation for each face in (b); physical reasoning for why R≥0 matters and rigorous derivation of inequalities in (c) | Steps shown but some logical gaps; missing explicit justification for sign conventions or inequality directions; physical interpretation present but not fully connected to mathematics | Missing crucial steps or 'magical' jumps in logic; no justification for why particle leaves when R=0; unexplained sign choices or missing case analysis |
| Final answer & units | 20% | 10 | Complete general solution with arbitrary constants for (a); explicit numerical verification that both sides equal abc(a+b+c) for (b); all three cases in (c) clearly stated with derived conditions and final formula boxed | Correct answers but not clearly highlighted; missing arbitrary constants in (a); verification stated but final equality not explicitly shown; one case in (c) incompletely treated | Incomplete or wrong final answers; missing verification statement; no clear conclusion for any of the three cases in (c); answers without proper mathematical closure |
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