Q5
(a) Find the orthogonal trajectories of the family of curves r = c(sec θ + tan θ), where c is a parameter. (10 marks) (b) Solve the integral equation y(t) = cos t + ∫₀ᵗ y(x) cos(t-x)dx using Laplace transform. (10 marks) (c) At any time t (in seconds), the coterminous edges of a variable parallelepiped are represented by the vectors ᾱ = tî + (t+1)ĵ + (2t+1)k̂ β̄ = 2tî + (3t-1)ĵ + tk̂ γ̄ = î + 3tĵ + k̂ What is the rate of change of the vectorial area of the parallelogram, whose coterminous edges are ᾱ and γ̄? Also find the rate of change of the volume of the parallelepiped at t = 1 second. (10 marks) (d) A solid hemisphere rests in equilibrium on a solid sphere of equal radius. Determine the stability of the equilibrium in the two situations—(i) when the curved surface and (ii) when the flat surface of the hemisphere rests on the sphere. (10 marks) (e) (i) Let C be a plane curve r̄(t) = f(t)î + g(t)ĵ, where f and g have second-order derivatives. Show that the curvature at a point is given by κ = |f'(t)g''(t) - g'(t)f''(t)| / ([f'(t)]² + [g'(t)]²)^(3/2) What is the value of torsion τ at any point of this curve? (5 marks) (ii) Show that the principal normals at two consecutive points of a curve do not intersect unless torsion τ is zero. (5 marks)
हिंदी में प्रश्न पढ़ें
(a) वक्र-कुल r = c(sec θ + tan θ) के लंबकोणीय संवेधी ज्ञात कीजिए, जहाँ c एक प्राचल है। (10 अंक) (b) लाप्लास रूपांतर का प्रयोग करते हुए समाकल समीकरण y(t) = cos t + ∫₀ᵗ y(x) cos(t-x)dx को हल कीजिए। (10 अंक) (c) किसी समय t (सेकंड में) पर एक चर समांतर-षट्फलक की सहावसानी किनारे सदिशों ᾱ = tî + (t+1)ĵ + (2t+1)k̂ β̄ = 2tî + (3t-1)ĵ + tk̂ γ̄ = î + 3tĵ + k̂ द्वारा निरूपित हैं। समांतर-चतुर्भुज, जिसकी सहावसानी किनारे ᾱ और γ̄ हैं, के सदिशीय क्षेत्रफल की परिवर्तन दर क्या है? t = 1 सेकंड पर समांतर-षट्फलक के आयतन की परिवर्तन दर भी ज्ञात कीजिए। (10 अंक) (d) एक ठोस गोले के ऊपर समान त्रिज्या का एक ठोस गोलार्ध साम्यावस्था में रखा है। दो स्थितियों में—(i) जब गोलार्ध का वक्रीय पृष्ठ तथा (ii) जब गोलार्ध का समतलीय पृष्ठ गोले पर स्थित है, साम्यावस्था का स्थायित्व ज्ञात कीजिए। (10 अंक) (e) (i) माना C एक समतल वक्र r̄(t) = f(t)î + g(t)ĵ है, जहाँ f और g के द्वितीय कोटि के अवकलज हैं। दर्शाइए कि वक्र के किसी बिंदु पर वक्रता κ = |f'(t)g''(t) - g'(t)f''(t)| / ([f'(t)]² + [g'(t)]²)^(3/2) है। इस वक्र के किसी बिंदु पर ऐंठन (टॉर्शन) τ का मान क्या है? (5 अंक) (ii) दर्शाइए कि किसी वक्र के दो क्रमागत बिंदुओं पर मुख्य अभिलंब प्रतिच्छेद नहीं करते जब तक कि ऐंठन (टॉर्शन) τ शून्य न हो। (5 अंक)
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How this answer will be evaluated
Approach
Solve each sub-part systematically, allocating approximately 15-18 minutes per 10-mark section. For (a), convert to Cartesian or use polar differential equation for orthogonal trajectories; for (b), apply Laplace transform with convolution theorem; for (c), use vector cross and triple products with time differentiation; for (d), apply potential energy method for stability analysis; for (e)(i)-(ii), derive curvature formula and analyze principal normal geometry. Present solutions with clear headings for each sub-part.
Key points expected
- (a) Convert r = c(sec θ + tan θ) to r(1 - sin θ) = c or equivalent, find dr/dθ, replace dθ/dr by -r²(dr/dθ) for orthogonal trajectories, integrate to get final family
- (b) Recognize convolution structure, apply Laplace transform to get Y(s) = s/(s²+1) + Y(s)·s/(s²+1), solve for Y(s), invert to get y(t) = 1 + t²/2 or equivalent closed form
- (c) Compute ᾱ × γ̄, differentiate with respect to t for rate of change of vectorial area; compute [ᾱ β̄ γ̄] as scalar triple product, differentiate and evaluate at t=1 for volume rate
- (d) For curved surface down: hemisphere center below sphere center, small displacement raises CG, stable equilibrium; for flat surface down: hemisphere rocks with CG rising then falling, analyze restoring torque, unstable equilibrium
- (e)(i) Derive curvature using κ = |r̄' × r̄''|/|r̄'|³ for plane curve, show numerator reduces to |f'g'' - g'f''|, note τ = 0 for plane curves
- (e)(ii) Parametrize consecutive points as r̄(s) and r̄(s+δs), write principal normal lines, show intersection condition requires coplanarity implying τ = 0
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 12 | Correctly identifies differential equation for orthogonal trajectories in (a), recognizes convolution integral in (b), sets up proper vector products in (c), draws accurate diagrams with CG positions in (d), and correctly parametrizes plane curves in (e) | Basic setup correct but misses subtle points like proper differential form in (a) or convolution identification in (b); vector products attempted but notation inconsistent | Wrong differential equation in (a), treats (b) as standard ODE without Laplace, confuses cross and dot products in (c), no diagram in (d), incorrect curve parametrization in (e) |
| Method choice | 20% | 12 | Uses polar differential equation method for (a), applies convolution theorem efficiently in (b), employs vector calculus identities for (c), uses potential energy/perturbation method for (d), applies Serret-Frenet framework for (e) | Correct general methods but inefficient choices (e.g., Cartesian conversion in (a) leading to messy algebra); Laplace used but with unnecessary partial fractions | Inappropriate methods like variation of parameters in (a), numerical approach in (b), component-wise brute force in (c), force balance without energy consideration in (d) |
| Computation accuracy | 20% | 12 | Flawless algebra: correct integration in (a) yielding r = c₁e^(-θ), accurate partial fraction and inversion in (b), correct derivatives of vector products in (c), precise CG height calculations in (d), exact curvature formula derivation in (e) | Minor arithmetic slips like sign errors in integration constants, coefficient mistakes in partial fractions, or evaluation errors at t=1 in (c) | Major computational errors: wrong integration leading to incorrect trajectory family, failure to solve algebraic equation for Y(s), incorrect determinant evaluation, wrong stability conclusion due to calculation error |
| Step justification | 20% | 12 | Justifies orthogonal trajectory condition dr/dθ → -r²(dθ/dr), explains convolution theorem application, shows vector differentiation steps, proves stability via second derivative of potential energy, rigorously derives torsion condition | Steps shown but key justifications missing (e.g., why negative reciprocal slope, why τ=0 for plane curves stated without proof) | Gaps in reasoning: jumps from integral equation to solution without Laplace steps, asserts stability without mathematical test, states curvature formula without derivation |
| Final answer & units | 20% | 12 | Complete answers: (a) r = c₁e^(-θ) or equivalent explicit form, (b) y(t) = 1 + t²/2, (c) vectorial area rate as vector, volume rate with units m³/s at t=1, (d) clear stable/unstable verdicts with reasons, (e) τ=0 stated, non-intersection condition proved | Correct forms but missing units in (c), or incomplete specification (e.g., 'stable' without specifying which case) | Answers missing, wrong final forms, or no conclusion on stability; curvature formula incomplete or torsion value omitted |
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