Q8
(a)(i) Find the orthogonal trajectories of the family of confocal conics $$\frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1; \quad a > b > 0 \text{ are constants and } \lambda \text{ is a parameter.}$$ Show that the given family of curves is self orthogonal. (10 marks) (a)(ii) Find the general solution of the differential equation: $$x^2\frac{d^2y}{dx^2} - 2x(1+x)\frac{dy}{dx} + 2(1+x)y = 0.$$ Hence, solve the differential equation: $x^2\dfrac{d^2y}{dx^2} - 2x(1+x)\dfrac{dy}{dx} + 2(1+x)y = x^3$ by the method of variation of parameters. (10 marks) (b) Describe the motion and path of a particle of mass $m$ which is projected in a vertical plane through a point of projection with velocity $u$ in a direction making an angle $\theta$ with the horizontal direction. Further, if particles are projected from that point in the same vertical plane with velocity $4\sqrt{g}$, then determine the locus of vertices of their paths. (15 marks) (c) Using Stokes' theorem, evaluate $\displaystyle\iint_S (\nabla \times \vec{F})\cdot \hat{n}dS$, where $\vec{F} = (x^2+y-4)\hat{i} + 3xy\hat{j} + (2xy+z^2)\hat{k}$ and $S$ is the surface of the paraboloid $z = 4-(x^2+y^2)$ above the $xy$-plane. Here, $\hat{n}$ is the unit outward normal vector on $S$. (15 marks)
हिंदी में प्रश्न पढ़ें
(a)(i) संनाभि शंकु कुल $$\dfrac{x^2}{a^2+\lambda} + \dfrac{y^2}{b^2+\lambda} = 1; \quad a > b > 0 \text{ अचर हैं तथा } \lambda \text{ एक प्राचल है,}$$ के लम्बकोणीय संघेदी ज्ञात कीजिए। दर्शाइए कि दिया गया वक्र-कुल स्वलंबिक है। (10 अंक) (a)(ii) अवकल समीकरण: $x^2\dfrac{d^2y}{dx^2} - 2x(1+x)\dfrac{dy}{dx} + 2(1+x)y = 0$ का व्यापक हल ज्ञात कीजिए। अतः अवकल समीकरण: $x^2\dfrac{d^2y}{dx^2} - 2x(1+x)\dfrac{dy}{dx} + 2(1+x)y = x^3$ को प्राचल विचरण विधि द्वारा हल कीजिए। (10 अंक) (b) द्रव्यमान $m$ का एक कण, जो कि प्रक्षेपण बिन्दु से वेग $u$ के साथ क्षैतिज दिशा के साथ $\theta$ कोण बनाने वाली दिशा में प्रक्षेपण बिन्दु से गुजरने वाले उद्धवाधर समतल में प्रक्षेपित किया जाता है, उसकी गति तथा पथ का वर्णन कीजिए। यदि कणों को उसी बिन्दु से उसी उद्धवाधर समतल में वेग $4\sqrt{g}$ के साथ प्रक्षेपित किया जाता है, तो उनके पथों के शीर्षों के बिन्दुपथ को भी निर्धारित कीजिए। (15 अंक) (c) स्टोक्स प्रमेय का उपयोग करते हुए $\displaystyle\iint_S (\nabla \times \vec{F})\cdot \hat{n}dS$ का मान निकालिए, जहाँ पर $\vec{F} = (x^2+y-4)\hat{i} + 3xy\hat{j} + (2xy+z^2)\hat{k}$ तथा $S$, परवलयज $z = 4-(x^2+y^2)$ का $xy$-समतल से ऊपर का पृष्ठ है। यहाँ $\hat{n}$, $S$ पर एकक बहिर्मुखी अभिलम्ब सदिश है। (15 अंक)
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How this answer will be evaluated
Approach
Solve this multi-part numerical problem by allocating approximately 40% time to part (a) covering orthogonal trajectories and differential equations, 30% to part (b) on projectile dynamics and locus derivation, and 30% to part (c) applying Stokes' theorem. Begin with clear problem identification for each sub-part, show complete derivations with intermediate steps, and conclude with boxed final answers for each component.
Key points expected
- For (a)(i): Derive differential equation of confocal conics family, find orthogonal trajectories by replacing dy/dx with -dx/dy, and verify self-orthogonality by showing the family coincides with its orthogonal trajectories
- For (a)(ii): Transform the homogeneous equation using y = vx substitution or identify one solution by inspection, find second linearly independent solution, then apply variation of parameters for the particular integral
- For (b): Derive parametric equations of projectile motion x = ut cos θ, y = ut sin θ - ½gt², eliminate t to get trajectory equation y = x tan θ - (gx²)/(2u²cos²θ), find vertex coordinates, and eliminate θ to obtain locus
- For (c): Verify Stokes' theorem applicability, compute curl of F, identify boundary curve C as circle x²+y²=4 at z=0, parametrize C as r(t) = (2cos t, 2sin t, 0), evaluate line integral ∮F·dr directly
- For (c) alternative: Compute surface integral of curl F over paraboloid with proper normal orientation, verify consistency with line integral result
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies the differential equation for confocal conics in (a)(i), recognizes the structure of Cauchy-Euler equation in (a)(ii), sets up proper coordinate system and initial conditions for projectile in (b), and correctly identifies the boundary curve and orientation for Stokes' theorem in (c) | Sets up most problems correctly but makes minor errors in differential equation formation, misses the reduction of order technique in (a)(ii), or incorrectly identifies the surface boundary in (c) | Fundamental errors in setting up differential equations, wrong approach to projectile motion setup, or complete misunderstanding of Stokes' theorem application with wrong boundary curve |
| Method choice | 20% | 10 | Uses standard orthogonal trajectory method with proper substitution, applies reduction of order or inspection method appropriately for (a)(ii), uses vertex formula and parameter elimination for locus in (b), and chooses efficient line integral evaluation over surface integral in (c) | Uses correct general methods but inefficiently, such as attempting general solution formulas without exploiting equation structure, or computes both line and surface integrals in (c) without recognizing equivalence | Chooses inappropriate methods like undetermined coefficients for Cauchy-Euler equation, uses Cartesian instead of parametric approach for projectile, or attempts direct surface integration without using Stokes' theorem |
| Computation accuracy | 20% | 10 | Flawless algebraic manipulation in eliminating λ for orthogonal trajectories, correct Wronskian calculation and integration in variation of parameters, accurate trigonometric identities in vertex locus derivation, and precise vector calculus operations in curl and line integral | Minor computational slips like sign errors in integration constants, arithmetic errors in Wronskian evaluation, or incorrect limits in parameter integration that don't affect final structure | Major computational errors leading to wrong family of curves, incorrect particular integral, wrong locus equation, or magnitude errors in final vector calculation |
| Step justification | 20% | 10 | Explicitly justifies the orthogonal trajectory substitution rule, explains why y = vx or inspection works for (a)(ii), physically interprets vertex conditions in (b), and states why line integral suffices for (c) with reference to curl being well-defined | Shows key steps with minimal justification, assumes standard results without citation, or provides physical interpretation without mathematical rigor | Missing crucial steps, unjustified leaps in logic, or no explanation of why methods are valid for the given problems |
| Final answer & units | 20% | 10 | Presents clean final answers: explicit orthogonal trajectory equation showing self-orthogonality, complete general solution with particular integral for (a)(ii), clear locus equation (typically parabola or ellipse) for vertices in (b), and numerical value with proper vector notation for (c) | Correct final forms but poorly presented, missing particular solution in (a)(ii), or un simplified locus equation in (b) | Incomplete answers, missing components of general solution, wrong final locus, or no numerical evaluation for (c) |
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