Q5
(a) Solve $\left(1-y^{2}+\frac{y^{4}}{x^{2}}\right)\left(\frac{dy}{dx}\right)^{2}-2\frac{y}{x}\frac{dy}{dx}+\frac{y^{2}}{x^{2}}=0$. 10 marks (b) Form the differential equation of all ellipses whose axes coincide with coordinate axes. 10 marks (c) Prove that the time taken by the Earth to travel over half of its orbit, which is separated by the minor axis and is remote from the Sun, when the Sun is at the focus of the elliptic orbit, is two days more than half of the year. The eccentricity of the orbit is taken as $\frac{1}{60}$. 10 marks (d) Given that A and B are two points in the same horizontal line distant 2a apart. AO and BO are two equal heavy strings tied together at O and carrying their weight at O. If $l$ is length of each string and $d$ is depth of O below AB, then show that the parameter $c$ of this catenary, in which the strings hang, is given by $$l^{2}-d^{2}=2c^{2}\left[\cosh\left(\frac{a}{c}\right)-1\right].$$ 10 marks (e) If $u=x+y+z$, $v=x^{2}+y^{2}+z^{2}$ and $w=xy+yz+zx$, then show that grad u, grad v and grad w are coplanar. 10 marks
हिंदी में प्रश्न पढ़ें
(a) $\left(1-y^{2}+\frac{y^{4}}{x^{2}}\right)\left(\frac{dy}{dx}\right)^{2}-2\frac{y}{x}\frac{dy}{dx}+\frac{y^{2}}{x^{2}}=0$ को हल कीजिए । 10 अंक (b) सभी दीर्घवृत्तों, जिनके अक्ष निर्देशांक अक्षों के संपाती हैं, का अवकल समीकरण बनाइए । 10 अंक (c) सिद्ध कीजिए कि पृथ्वी को अपनी कक्षा के आधे भाग, जो कि लघु अक्ष द्वारा अलग किया गया है और सूर्य से सुदूर है, जब सूर्य दीर्घवृत्तीय कक्षा की नाभि (फोकस) पर है, की यात्रा करने में लगने वाला समय आधे वर्ष से दो दिन अधिक है। कक्षा की उत्केन्द्रता $\frac{1}{60}$ ली गई है। 10 अंक (d) दिया गया है कि A और B एक ही क्षैतिज रेखा पर स्थित दो बिंदु हैं, जिनके बीच की दूरी 2a है। AO और BO दो समान भारी डोरी हैं जो O पर एक साथ बंधी हैं और जिनका भार O पर है। यदि प्रत्येक डोरी की लंबाई $l$ है तथा $d$, AB से नीचे O की गहराई है, तो दर्शाइए कि इस कैटनरी, जिसमें डोरी लटकी है, का प्राचल $c$ $$l^{2}-d^{2}=2c^{2}\left[\cosh\left(\frac{a}{c}\right)-1\right]$$ द्वारा दिया गया है। 10 अंक (e) यदि $u=x+y+z$, $v=x^{2}+y^{2}+z^{2}$ और $w=xy+yz+zx$ है, तो दर्शाइए कि grad u, grad v और grad w समतलीय हैं। 10 अंक
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How this answer will be evaluated
Approach
Solve each sub-part systematically, allocating approximately equal time (~20%) to each 10-mark section. For (a), identify the substitution v = y/x to reduce to Clairaut's form; for (b), eliminate parameters from the standard ellipse equation; for (c), apply Kepler's second law with area integration; for (d), use catenary boundary conditions at symmetric points; for (e), compute gradients and verify scalar triple product vanishes. Present solutions with clear headings for each part.
Key points expected
- Part (a): Recognize homogeneous structure, substitute v = y/x, transform to Clairaut's equation v = px + f(p), obtain complete primitive and singular solution
- Part (b): Start with ellipse equation x²/a² + y²/b² = 1, eliminate two arbitrary constants a and b to get second-order differential equation xy(d²y/dx²) + x(dy/dx)² - y(dy/dx) = 0
- Part (c): Apply Kepler's second law (equal areas in equal times), compute sector area from minor axis to aphelion using polar ellipse equation r = a(1-e²)/(1+e cos θ), integrate and compare with half-year
- Part (d): Set up symmetric catenary y = c cosh(x/c) with origin at lowest point O, apply boundary conditions at x = ±a with y = c + d, eliminate parameter to derive required relation
- Part (e): Compute ∇u, ∇v, ∇w explicitly, form scalar triple product [∇u ∇v ∇w] or show linear dependence via ∇v = x∇u + z∇w-type relation, verify coplanarity condition
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies substitution v=y/x for (a), recognizes two-parameter family needing second-order DE for (b), sets up Keplerian area integral with proper limits for (c), establishes symmetric catenary with correct coordinate system for (d), and chooses appropriate coplanarity test for (e) | Identifies correct general approach for most parts but has minor errors in substitution choice, parameter counting, or coordinate setup in one or two sections | Wrong substitutions (e.g., treating (a) as linear), incorrect order of DE in (b), confuses sector areas in (c), wrong catenary orientation in (d), or attempts determinant of gradients without proper justification in (e) |
| Method choice | 20% | 10 | Selects optimal methods: Clairaut's equation treatment for (a), systematic parameter elimination for (b), elliptic integral/series expansion for (c), hyperbolic identity manipulation for (d), and elegant linear dependence argument for (e) | Uses workable but suboptimal methods, such as direct integration instead of standard form recognition, or brute-force determinant calculation instead of structural insight | Inappropriate methods like treating (a) as exact equation, attempting first-order DE for (b), using circular approximation for (c), or failing to use catenary properties in (d) |
| Computation accuracy | 20% | 10 | Flawless algebraic manipulation, correct differentiation of implicit relations, accurate elliptic integral evaluation with e=1/60 approximation, precise hyperbolic identity application, and exact gradient calculations | Minor computational slips such as sign errors in chain rule, arithmetic mistakes in series expansion, or algebraic slips in eliminating parameters that don't affect final structure | Major errors like incorrect expansion of (dy/dx)² terms, wrong elimination leading to incorrect DE, order-of-magnitude errors in time calculation, or incorrect partial derivatives |
| Step justification | 20% | 10 | Explicitly states why v=y/x is chosen, justifies parameter elimination steps, explains Kepler's law application with area element derivation, proves catenary parameter relations, and shows why scalar triple product zero implies coplanarity | Shows key steps with minimal justification, assumes standard results without citation, or skips intermediate algebraic verification | Unjustified leaps between steps, asserts results without derivation, or fails to connect physical reasoning to mathematical formalism especially in (c) and (d) |
| Final answer & units | 20% | 10 | Presents complete primitive and singular solution for (a), clean second-order DE for (b), verified 'two days more' result with clear approximation for (c), fully derived catenary parameter relation for (d), and concise coplanarity proof for (e) | Correct final forms but incomplete (missing singular solution in (a), unsimplified DE in (b), or unverified numerical claim in (c)) | Missing answers, incorrect final forms, or failure to demonstrate the required proofs; particularly leaving results in implicit form when explicit required |
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