Q6
(a) If F(s) and G(s) are Laplace transforms of f(t) and g(t) respectively, then prove that $$\mathcal{L}\left\{\int_{0}^{t} f(x) g(t-x) dx\right\} = F(s) G(s).$$ Using this result, solve the equation $$y(t) = t + \int_{0}^{t} y(x) \sin(t-x) dx.$$ 15 marks (b) One end of an elastic string, having natural length a, is fixed at some point O and a heavy particle is attached to the other end of the string. The string is drawn vertically downward till it is four times its natural length at the point C and then released. If the modulus of elasticity of the string is equal to the weight of the particle, then show that the particle will return to the same point C in the time $$\sqrt{\frac{a}{g}}\left(2\sqrt{3} + \frac{4\pi}{3}\right).$$ 15 marks (c) (i) Find the absolute value of the directional derivative of $\phi(x, y, z) = x^2y^2z^2$ at the point $(1, 1, -1)$ in the direction of the tangent to the curve $x = e^t$, $y = 2\sin t + 1$, $z = t - \cos t$, at $t = 0$. 10 marks (ii) If $\nabla \cdot \overrightarrow{\mathrm{E}}=0$, $\nabla \cdot \overrightarrow{\mathrm{H}}=0$, $\nabla \times \overrightarrow{\mathrm{E}}=-\frac{\partial \overrightarrow{\mathrm{H}}}{\partial t}$ and $\nabla \times \overrightarrow{\mathrm{H}}=\frac{\partial \overrightarrow{\mathrm{E}}}{\partial t}$, then show that $\nabla^{2} \overrightarrow{\mathrm{H}}=\frac{\partial^{2} \overrightarrow{\mathrm{H}}}{\partial t^{2}}$ and $\nabla^{2} \overrightarrow{\mathrm{E}}=\frac{\partial^{2} \overrightarrow{\mathrm{E}}}{\partial t^{2}}$. 10 marks
हिंदी में प्रश्न पढ़ें
(a) यदि f(t) और g(t) के लाप्लास रूपान्तर क्रमशः F(s) और G(s) हैं, तो सिद्ध कीजिए कि $$\mathcal{L}\left\{\int_{0}^{t} f(x) g(t-x) dx\right\} = F(s) G(s)$$ है। इस परिणाम का प्रयोग करते हुए, समीकरण $$y(t) = t + \int_{0}^{t} y(x) \sin(t-x) dx$$ को हल कीजिए। 15 अंक (b) एक प्रत्यास्थ डोरी, जिसकी प्राकृतिक लंबाई a है, का एक छोर किसि बिंदु O पर स्थिर है और डोरी के दूसरे छोर पर एक भारी कण जुड़ा हुआ है। डोरी को उर्ध्वाधर नीचे की ओर बिंदु C तक तब तक खींचा जाता है जब तक वह अपनी प्राकृतिक लंबाई से चार गुना न हो जाए तथा फिर छोड़ दिया जाता है। यदि डोरी का प्रत्यास्थता गुणांक कण के भार के बराबर है, तो दर्शाइए कि कण $$\sqrt{\frac{a}{g}}\left(2\sqrt{3} + \frac{4\pi}{3}\right)$$ समय में उसी बिंदु C पर वापस आ जाएगा। 15 अंक (c) (i) $\phi(x, y, z) = x^2y^2z^2$ का बिंदु $(1, 1, -1)$ पर, वक्र $x = e^t$, $y = 2\sin t + 1$, $z = t - \cos t$, के बिंदु $t = 0$ पर स्पर्श-रेखा की दिशा में दिक्-अवकलज का निरपेक्ष मान ज्ञात कीजिए। 10 अंक (ii) यदि $\nabla \cdot \overrightarrow{\mathrm{E}}=0$, $\nabla \cdot \overrightarrow{\mathrm{H}}=0$, $\nabla \times \overrightarrow{\mathrm{E}}=-\frac{\partial \overrightarrow{\mathrm{H}}}{\partial t}$ और $\nabla \times \overrightarrow{\mathrm{H}}=\frac{\partial \overrightarrow{\mathrm{E}}}{\partial t}$ है, तो दर्शाइए कि $\nabla^{2} \overrightarrow{\mathrm{H}}=\frac{\partial^{2} \overrightarrow{\mathrm{H}}}{\partial t^{2}}$ और $\nabla^{2} \overrightarrow{\mathrm{E}}=\frac{\partial^{2} \overrightarrow{\mathrm{E}}}{\partial t^{2}}$ है। 10 अंक
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How this answer will be evaluated
Approach
Begin with the proof of the convolution theorem in part (a) using Fubini's theorem and the definition of Laplace transform, then apply it to solve the integral equation via algebraic manipulation in s-domain and partial fractions. For part (b), establish the equation of motion in two phases (stretched string SHM and free fall), carefully handling the transition conditions at natural length. Part (c)(i) requires computing the gradient and unit tangent vector before taking the dot product, while (c)(ii) demands vector calculus identities to derive the wave equations. Allocate approximately 30% time to (a), 35% to (b), 20% to (c)(i), and 15% to (c)(ii) based on mark distribution and computational complexity.
Key points expected
- Part (a): Correct proof of convolution theorem using change of order of integration and identification of Laplace kernel; proper setup of subsidiary equation ȳ(s) = 1/s² + ȳ(s)/(s²+1) and its solution yielding y(t) = t + t³/6
- Part (b): Correct derivation of SHM equation ẍ = -g(x-a)/a for x > a with solution x = a + 3a cos(√(g/a)t); determination of time to reach natural length t₁ = √(a/g)·π/3 and velocity v₁ = 3√(ag)/2
- Part (b) continued: Analysis of free motion phase with initial conditions, time to reach maximum height and return, and total time calculation showing the required expression √(a/g)(2√3 + 4π/3)
- Part (c)(i): Computation of ∇φ = (2xy²z², 2x²yz², 2x²y²z) = (2, 2, -2) at (1,1,-1); tangent vector (1, 2, 1) at t=0; unit vector and directional derivative magnitude |8/√6| = 4√(2/3) or equivalent simplified form
- Part (c)(ii): Application of curl to Faraday's and Ampère's equations, use of vector identity ∇×(∇×H) = ∇(∇·H) - ∇²H with divergence-free condition to obtain wave equations for both E and H fields
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly identifies the convolution integral structure and sets up the double integral with proper limits; for (b): establishes correct differential equations for both phases with accurate initial/boundary conditions; for (c)(i): correctly evaluates gradient and tangent vector at the specified point; for (c)(ii): properly applies vector identities with divergence-free constraints | Sets up most equations correctly but has minor errors in initial conditions for (b) or gradient evaluation for (c)(i); may miss divergence-free simplification in (c)(ii) | Fundamental errors in setting up the convolution proof, wrong differential equation for string motion, incorrect gradient or tangent vector calculation, or failure to use vector calculus identities properly |
| Method choice | 20% | 10 | For (a): uses Fubini's theorem with justified change of integration order; for (b): correctly identifies need for piecewise analysis (SHM vs free motion) with proper matching conditions; for (c)(i): selects directional derivative formula with unit vector normalization; for (c)(ii): chooses curl operation and vector triple product identity efficiently | Uses appropriate methods but may miss optimal approaches like partial fractions in (a) or may not clearly separate phases in (b); some inefficiency in vector calculations | Inappropriate method selection such as attempting time-domain direct solution for (a), treating entire motion as single phase in (b), or using gradient alone without direction for (c)(i) |
| Computation accuracy | 20% | 10 | Flawless algebraic manipulation in Laplace inversion including partial fraction decomposition; precise trigonometric calculations for phase times in (b); exact arithmetic for directional derivative magnitude; correct application of curl-curl identity with all terms accounted for | Minor computational slips such as sign errors in partial fractions, slight inaccuracies in trigonometric values, or arithmetic errors in final simplification that don't fundamentally alter the structure | Major computational errors including wrong partial fraction decomposition, incorrect integration constants, failure to normalize direction vector, or algebraic mistakes in vector identity expansion |
| Step justification | 20% | 10 | Explicitly justifies change of integration order in (a); clearly states matching conditions at natural length for (b); shows unit tangent vector derivation in (c)(i); provides complete reasoning for dropping divergence terms in (c)(ii) using given conditions | Shows most key steps but may omit justification for some transitions or assume results without explicit verification of continuity conditions at phase boundaries | Missing crucial justifications such as why convolution theorem applies, how phase transitions are handled, or why specific vector identities are valid; presents results without intermediate reasoning |
| Final answer & units | 20% | 10 | All four sub-parts yield complete, simplified answers: y(t) = t + t³/6 for (a); exact time expression √(a/g)(2√3 + 4π/3) for (b); simplified exact value for directional derivative magnitude in (c)(i); both wave equations explicitly derived with proper vector notation in (c)(ii) | Correct final forms but unsimplified or with minor notational issues; may leave answers in intermediate form like unsimplified radicals or missing final vector wave equation statement | Missing final answers, wrong functional forms, incorrect dimensional analysis (especially for time expression in b), or failure to present both required results in (c)(ii) |
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