Q6
(a) A cable of weight $w$ per unit length and length $2l$ hangs from two points P and Q in the same horizontal line. Show that the span of the cable is $2l\left(1 - \dfrac{2h^2}{3l^2}\right)$, where $h$ is the sag in the middle of the tightly stretched position. 20 (b) Solve the following differential equation by using the method of variation of parameters : $(x^2 - 1)\dfrac{d^2y}{dx^2} - 2x\dfrac{dy}{dx} + 2y = (x^2 - 1)^2$, given that $y = x$ is one solution of the reduced equation. 15 (c) Verify Green's theorem in the plane for $\displaystyle\oint_C (3x^2 - 8y^2)\,dx + (4y - 6xy)\,dy$, where C is the boundary curve of the region defined by $x = 0$, $y = 0$, $x + y = 1$. 15
हिंदी में प्रश्न पढ़ें
(a) $2l$ लम्बाई का एक तार (केबल) जिसका भार $w$ प्रति इकाई (यूनिट) लम्बाई है, एक क्षैतिज रेखा के दो बिन्दुओं P तथा Q से लटकी हुई है। दर्शाइए कि तार की विस्तृति (स्पैन) $2l\left(1 - \dfrac{2h^2}{3l^2}\right)$ है, जहाँ $h$ तार के कसकर खींची हुई स्थिति में मध्य का झोल है। 20 (b) प्राचल-विचरण विधि का उपयोग करके, निम्नलिखित अवकल समीकरण : $$(x^2 - 1)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + 2y = (x^2 - 1)^2$$ को हल कीजिए, जहाँ समानीत समीकरण का एक हल $y = x$ दिया गया है। 15 (c) समतल में ग्रीन के प्रमेय को $\displaystyle\oint_C (3x^2 - 8y^2)\,dx + (4y - 6xy)\,dy$ के लिए सत्यापित कीजिए, जहाँ C, $x = 0$, $y = 0$, $x + y = 1$ द्वारा परिभाषित क्षेत्र का सीमा वक्र है। 15
Directive word: Prove
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How this answer will be evaluated
Approach
This question demands rigorous mathematical proof and solution across three distinct areas: cable mechanics, differential equations, and vector calculus. Allocate approximately 40% of time to part (a) as it carries 20 marks, requiring careful derivation of the catenary approximation; spend 30% each on parts (b) and (c). Begin each part with clear statement of given data, proceed through systematic derivation/solution, and conclude with explicit verification of results.
Key points expected
- Part (a): Correct setup of catenary equation y = c cosh(x/c), Taylor expansion for small sag-to-span ratio, and derivation of span formula 2l(1 - 2h²/3l²) using arc length constraint
- Part (b): Identification of second solution y₂ using reduction of order or Wronskian, correct construction of particular integral via variation of parameters formula, and complete integration with arbitrary constants
- Part (c): Proper parameterization of triangular boundary C with three segments, correct evaluation of line integral ∮(Mdx + Ndy), computation of double integral ∬(∂N/∂x - ∂M/∂y)dA over region, and explicit equality verification
- Clear labeling of all three parts with logical flow between steps, proper use of mathematical notation and conventions
- Explicit statement of assumptions: tightly stretched cable (small sag approximation), continuous differentiability for Green's theorem, and non-homogeneous term handling in (b)
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly identifies catenary differential equation and boundary conditions y(±a)=0, sag condition y(0)=-h; for (b): properly reduces order to find second solution y₂ = 1 + x²/2 · ln|(x-1)/(x+1)| or equivalent; for (c): accurately sketches triangular region with vertices (0,0), (1,0), (0,1) and orients boundary counter-clockwise | Sets up most equations correctly but misses one boundary condition in (a), or finds second solution with minor errors in (b), or misidentifies one segment of triangle in (c) | Incorrect fundamental setup: wrong curve equation in (a), fails to find second solution in (b), or completely wrong region description in (c) |
| Method choice | 20% | 10 | For (a): employs Taylor expansion cosh(t) ≈ 1 + t²/2 for small sag; for (b): systematic variation of parameters with Wronskian W(y₁,y₂); for (c): judicious choice of integration order (dx dy or dy dx) and efficient parameterization of line segments | Uses correct general methods but with suboptimal choices, e.g., direct integration without exploiting symmetry in (a), or correct but cumbersome parameterization in (c) | Wrong method entirely: attempts parabolic approximation without justification in (a), uses undetermined coefficients instead of variation of parameters in (b), or confuses Green's theorem with Stokes/Gauss in (c) |
| Computation accuracy | 20% | 10 | Flawless algebraic manipulation: correct expansion of arc length integral s = ∫√(1+y'²)dx ≈ ∫(1+y'²/2)dx in (a), accurate integration of rational functions in (b), precise evaluation yielding -1/3 for both sides in (c) | Minor computational slips: sign errors in Taylor coefficients, arithmetic mistakes in Wronskian evaluation, or one segment integral calculated incorrectly but others correct | Major computational failures: incorrect series expansion leading to wrong coefficient, fundamental integration errors, or both sides of Green's theorem differing significantly without reconciliation |
| Step justification | 20% | 10 | Explicitly justifies: small sag approximation h << l in (a), linear independence of solutions and non-vanishing Wronskian in (b), satisfaction of Green's theorem hypotheses (continuous partial derivatives, simple closed curve) in (c); cites relevant theorems | Shows most key steps with partial justification, mentions approximation without quantifying error, or states theorem conditions without verifying them explicitly | Bare assertion of results without justification, missing crucial steps like reduction of order derivation, or no mention of why Green's theorem applies to the given region |
| Final answer & units | 20% | 10 | Precise final forms: exact span formula with proper bracketing in (a), complete general solution y = c₁x + c₂y₂ + yₚ with explicit particular integral in (b), clear statement that both sides equal -1/3 confirming Green's theorem in (c); proper dimensional consistency throughout | Correct final answers but with minor presentation issues: unsimplified expressions, missing arbitrary constants, or correct numerical verification without explicit conclusion statement | Missing or wrong final answers, incorrect boxed results, failure to verify Green's theorem equality, or dimensional inconsistencies in physical quantities |
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