Q7
(a)(i) Find the solution of the differential equation : $\dfrac{dy}{dx}=-\dfrac{2xy^3+2}{3x^2y^2+8e^{4y}}$ 10 (a)(ii) Reduce the equation $x^2p^2+y(2x+y)p+y^2=0$ to Clairaut's form by the substitution $y=u$ and $xy=v$. Hence solve the equation and show that $y+4x=0$ is a singular solution of the differential equation. 10 (b) A solid hemisphere is supported by a string fixed to a point on its rim and to a point on a smooth vertical wall with which the curved surface is in contact. If $\theta$ is the angle of inclination of the string with vertical and $\phi$ is the angle of inclination of the plane base of the hemisphere to the vertical, then find the value of $(\tan\phi-\tan\theta)$. 15 (c) If the tangent to a curve makes a constant angle θ with a fixed line, then prove that the ratio of radius of torsion to radius of curvature is proportional to tanθ. Further prove that if this ratio is constant, then the tangent makes a constant angle with a fixed direction. 15
हिंदी में प्रश्न पढ़ें
(a)(i) अवकल समीकरण : $\dfrac{dy}{dx}=-\dfrac{2xy^3+2}{3x^2y^2+8e^{4y}}$ का हल ज्ञात कीजिए। 10 (a)(ii) समीकरण $x^2p^2+y(2x+y)p+y^2=0$ का प्रतिस्थापन $y=u$ और $xy=v$ द्वारा क्लेरो रूप में समान्यन कीजिए। अतः समीकरण का हल निकालिए और दर्शाइए कि $y+4x=0$ अवकल समीकरण का एक विचित्र हल है। 10 (b) एक ठोस अर्ध-गोलक एक डोरी द्वारा, जिसका एक सिरा एक चिकनी उच्चाधर दीवार पर एक बिंदु से और दूसरा सिरा अर्धगोलक के किनारे (रिम) पर स्थित एक बिंदु से बंधा है, उच्चाधर दीवार के सहारे टिका है। ठोस अर्धगोलक का वक्रित पृष्ठ दीवार को स्पर्श करता है। अगर उच्चाधर के साथ डोरी का आनति कोण $\theta$ है और अर्धगोलक के समतल आधार (बेस) का आनति कोण $\phi$ है तो $(\tan\phi-\tan\theta)$ का मान ज्ञात कीजिए। 15 (c) अगर एक वक्र की स्पर्श रेखा एक नियत रेखा के साथ एक स्थिर कोण θ बनाती है तो सिद्ध कीजिए कि वक्रता की त्रिज्या के साथ व्यवर्तन त्रिज्या का अनुपात tanθ के समानुपाती है। और आगे सिद्ध कीजिए कि अगर यह अनुपात एक स्थिरांक है, तो स्पर्श रेखा एक नियत दिशा के साथ एक स्थिर कोण बनाती है। 15
Directive word: Solve
This question asks you to solve. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.
See our UPSC directive words guide for a full breakdown of how to respond to each command word.
How this answer will be evaluated
Approach
Solve this multi-part problem by allocating approximately 40% time to part (a) covering both differential equations (20 marks), 30% to the mechanics problem (b) involving equilibrium of a hemisphere (15 marks), and 30% to the differential geometry proof in part (c) on curvature and torsion (15 marks). Begin with clear identification of the solution method for each sub-part, execute calculations systematically with proper justification, and conclude with verification of results including the singular solution in (a)(ii) and the constant angle property in (c).
Key points expected
- For (a)(i): Recognize the equation as exact after rewriting in differential form Mdx + Ndy = 0, verify ∂M/∂y = ∂N/∂x, and find the potential function ψ(x,y) = x²y³ + 2x + 2e^(4y) = c
- For (a)(ii): Apply substitution y = u, xy = v to transform to Clairaut's form v = pu + f(p) where p = dv/du, obtain general solution v = cu + c²/(c+1), and verify y + 4x = 0 as singular solution via c-discriminant or envelope method
- For (b): Draw correct free-body diagram with three forces (weight W at center of mass, tension T along string, normal reaction R perpendicular to wall), apply equilibrium conditions ΣFx = 0, ΣFy = 0, ΣM = 0 about appropriate point, use geometry relating θ and φ through hemisphere radius a, and derive tanφ - tanθ = 1/2
- For (c): Use Frenet-Serret formulas with fixed direction making angle θ with tangent, express d𝐭/ds = κ𝐧 and d𝐛/ds = -τ𝐧, show that 𝐭·𝐚 = cosθ implies 𝐧·𝐚 = 0 and 𝐛·𝐚 = sinθ, derive τ/κ = tanθ, and prove converse by showing constant τ/κ implies 𝐭·𝐚 is constant
- Cross-cutting: Maintain dimensional consistency, state all assumptions explicitly (smooth wall, uniform hemisphere, inextensible string), and verify boundary conditions or special cases where applicable
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies exactness condition for (a)(i), proper substitution variables for (a)(ii), accurate force diagram with correct moment arm for (b), and appropriate fixed direction with Frenet frame for (c); all initial conditions and geometric constraints properly stated | Identifies basic approach for most parts but misses subtle conditions like verifying exactness or misplaces center of mass; force diagram essentially correct but moment arm or geometry has minor errors | Fails to recognize equation type in (a), uses wrong substitution, draws incorrect free-body diagram with wrong force directions, or sets up wrong coordinate system for (c) |
| Method choice | 20% | 10 | Selects optimal methods: exact differentials for (a)(i), Clairaut substitution with c-discriminant for (a)(ii), moment equilibrium about strategic point for (b), and elegant vector proof using Frenet-Serret for (c); avoids unnecessary computation | Uses workable but suboptimal methods, such as integrating factor instead of recognizing exactness, or solving (b) by resolving all forces without moment equation; completes proofs but with extraneous steps | Applies completely wrong methods like treating (a)(i) as linear or Bernoulli, fails to use Clairaut form in (a)(ii), attempts force resolution without moments for (b), or uses coordinate-based rather than intrinsic geometry for (c) |
| Computation accuracy | 20% | 10 | Flawless algebraic manipulation: correct partial derivatives in exactness verification, accurate chain rule application in Clairaut reduction, precise trigonometric identities yielding tanφ - tanθ = 1/2, and error-free vector calculus in Frenet derivations | Minor computational slips like sign errors in partial derivatives, arithmetic mistakes in solving for constants, or algebraic errors in trigonometric simplification that don't fundamentally derail the solution | Major computational errors: incorrect integration of exact differentials, wrong discriminant calculation missing singular solution, fundamental errors in equilibrium equations, or incorrect differentiation of Frenet formulas |
| Step justification | 20% | 10 | Every non-trivial step justified: explicit verification of exactness condition, clear demonstration of why substitution yields Clairaut form, physical reasoning for equilibrium conditions with reference to rigid body statics, and rigorous proof of both directions in (c) with clear logical flow | Most key steps explained but some gaps in reasoning, such as asserting without proof that envelope gives singular solution, or stating equilibrium conditions without explaining why particular moment center was chosen | Bare assertions without justification, missing verification steps, unexplained jumps in algebraic manipulation, or proofs that assert conclusions without establishing necessary implications |
| Final answer & units | 20% | 10 | Complete final answers: implicit solution x²y³ + 2x + 2e^(4y) = c for (a)(i), explicit general and singular solutions for (a)(ii), exact value tanφ - tanθ = 1/2 for (b), and fully stated proportional relationship τ/κ = tanθ with rigorous converse proof for (c); dimensional consistency checked | Correct final forms but incomplete presentation, such as missing arbitrary constant, not explicitly verifying y + 4x = 0 is singular, or stating answer without showing derivation; minor dimensionless quantity oversights | Missing or wrong final answers, incorrect form of solution (explicit when should be implicit), failure to identify singular solution, wrong numerical value for (b), or incomplete proof of converse statement in (c) |
Practice this exact question
Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.
Evaluate my answer →More from Mathematics 2023 Paper I
- Q1 (a) Let V₁ = (2, -1, 3, 2), V₂ = (-1, 1, 1, -3) and V₃ = (1, 1, 9, -5) be three vectors of the space ℝ⁴. Does (3, -1, 0, -1) ∈ span {V₁, V₂…
- Q2 (a) If the matrix of a linear transformation T : IR³→IR³ relative to the basis {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is $$\begin{bmatrix} 1 & 1…
- Q3 Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ (i) Verify the Cayley-Hamilton theorem for the matrix $A$. (ii)…
- Q4 Find the rank of the matrix $A = \begin{bmatrix} 1 & 2 & -1 & 0 \\ -1 & 3 & 0 & -4 \\ 2 & 1 & 3 & -2 \\ 1 & 1 & 1 & -1 \end{bmatrix}$ by re…
- Q5 (a) Obtain the solution of the initial-value problem dy/dx - 2xy = 2, y(0) = 1 in the form y = eˣ²[1 + √π erf(x)]. (10 marks) (b) Given tha…
- Q6 (a) Solve the differential equation: d³y/dx³ - 3d²y/dx² + 4dy/dx - 2y = eˣ + cos x. (15 marks) (b) When a particle is projected from a poin…
- Q7 (a)(i) Find the solution of the differential equation : $\dfrac{dy}{dx}=-\dfrac{2xy^3+2}{3x^2y^2+8e^{4y}}$ 10 (a)(ii) Reduce the equation $…
- Q8 (a) Solve the following initial value problem by using Laplace transform technique : $$\frac{d^2y}{dt^2} - 4\frac{dy}{dt} + 3y(t) = f(t),$$…