Q5
(a) Find the solution of the equation $(D^2 + DD' - 2D'^2)z = y\sin x$, where $D \equiv \frac{\partial}{\partial x}$ and $D' \equiv \frac{\partial}{\partial y}$. (10 marks) (b) Solve the following system of linear equations by Gauss-Seidel method : $$\begin{align} 10x + 2y + z &= 9\\ 2x + 20y - 2z &= -44\\ -2x + 3y + 10z &= 22 \end{align}$$ (10 marks) (c) (i) Convert the number $(3479)_{10}$ into binary system and the number $(7AE \cdot 9F)_{16}$ into decimal system. (ii) Determine the truth table for the Boolean function $$F(x, y, z) = (x + y + z')(x' + y')$$ Also derive the full disjunctive normal form of $F(x, y, z)$ from the truth table. (10 marks) (d) A bead of mass $m$ slides on a frictionless wire in the shape of a cycloid given by $x = a(\theta - \sin\theta)$, $y = a(1 + \cos\theta)$, $(0 \leq \theta \leq 2\pi)$. Find the Lagrangian function. Hence show that the equation of motion can be written as $$\frac{d^2u}{dt^2} + \frac{g}{4a}u = 0$$ where $u = \cos\left(\frac{\theta}{2}\right)$. (4+6=10 marks) (e) A source and a sink of equal strength are placed at points $\left(\pm\frac{a}{2}, 0\right)$ within a fixed circular boundary $x^2 + y^2 = a^2$. Show that the streamlines are given by $$\left(r^2 - \frac{a^2}{4}\right)(r^2 - 4a^2) - 4a^2y^2 = ky(r^2 - a^2)$$ where $k$ is a constant and $r^2 = x^2 + y^2$. (10 marks)
हिंदी में प्रश्न पढ़ें
(a) समीकरण $(D^2 + DD' - 2D'^2)z = y\sin x$, जहाँ $D \equiv \frac{\partial}{\partial x}$ और $D' \equiv \frac{\partial}{\partial y}$ है, का हल ज्ञात कीजिए। (10 अंक) (b) निम्न रैखिक समीकरण निकाय को गाउस-सीडल विधि से हल कीजिए : $$\begin{align} 10x + 2y + z &= 9\\ 2x + 20y - 2z &= -44\\ -2x + 3y + 10z &= 22 \end{align}$$ (10 अंक) (c) (i) संख्या $(3479)_{10}$ को द्वि-आधारी पद्धति और संख्या $(7AE \cdot 9F)_{16}$ को दशमलव पद्धति में बदलिए। (ii) बूलियन फलन $F(x, y, z) = (x + y + z')(x' + y')$ के लिए सत्यमान सारणी ज्ञात कीजिए। सत्यमान सारणी से $F(x, y, z)$ का पूर्ण वियोजनीय प्रसामान्य रूप भी प्राप्त कीजिए। (10 अंक) (d) $x = a(\theta - \sin\theta)$, $y = a(1 + \cos\theta)$, $(0 \leq \theta \leq 2\pi)$ द्वारा दिए गए एक चक्रज के रूप में एक घर्षणहीन तार पर $m$ द्रव्यमान का एक मनका फिसलता है। लैग्रांजी फलन ज्ञात कीजिए। अतः दर्शाइए कि गति का समीकरण $$\frac{d^2u}{dt^2} + \frac{g}{4a}u = 0$$ के रूप में लिखा जा सकता है, जहाँ $u = \cos\left(\frac{\theta}{2}\right)$ है। (4+6=10 अंक) (e) बराबर सामर्थ्य के एक स्रोत और एक अभिगम एक निश्चित वृत्तीय सीमा $x^2 + y^2 = a^2$ के अंतर्गत बिंदुओं $\left(\pm\frac{a}{2}, 0\right)$ पर रखे हैं। दर्शाइए कि धारारेखाएँ $$\left(r^2 - \frac{a^2}{4}\right)(r^2 - 4a^2) - 4a^2y^2 = ky(r^2 - a^2)$$ द्वारा दी जाती हैं, जहाँ $k$ एक अचर है और $r^2 = x^2 + y^2$ है। (10 अंक)
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How this answer will be evaluated
Approach
Solve each of the five independent parts systematically, presenting clear working for PDE solution, iterative numerical method, number system conversions with Boolean algebra, Lagrangian derivation with substitution, and complex potential streamlines. Structure as five distinct sections with proper labeling (a) through (e), showing complete derivations before stating final results.
Key points expected
- Part (a): Factorize operator as (D-D')(D+2D'), find CF using f1(y+x)+f2(y-2x), and PI using 1/(D²+DD'-2D'²) y sin x with proper shifting
- Part (b): Check diagonal dominance, rearrange if needed, apply Gauss-Seidel iteration formula with at least 3-4 iterations showing convergence to x=1, y=-2, z=3
- Part (c)(i): Binary conversion of 3479 using division method giving 110110010111, and hexadecimal to decimal with positional weights for 7AE.9F
- Part (c)(ii): Complete 8-row truth table for F(x,y,z), identify minterms where F=1, and express as Σm(1,2,4,6) or expanded sum of products
- Part (d): Express arc length element ds² = 4a²sin²(θ/2)dθ², form L = 2ma²sin²(θ/2)(dθ/dt)² - mga(1+cosθ), substitute u = cos(θ/2) to obtain SHM equation
- Part (e): Construct complex potential W = m ln[(z-a/2)/(z+a/2)] + m ln[(z-a²/2z̄)/(z+a²/2z̄)] for circle theorem, extract imaginary part for stream function ψ, and manipulate to given form
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies PDE type and factorizes operator; verifies diagonal dominance for Gauss-Seidel; sets up correct conversion bases (2, 10, 16); establishes proper cycloid parameterization with arc length; applies circle theorem correctly for source-sink in circular boundary | Minor errors in operator factorization or missing diagonal dominance check; correct bases but arithmetic errors in setup; partially correct Lagrangian setup with wrong kinetic energy form; attempts circle theorem but misses image system | Wrong operator factorization leading to incorrect CF; fails to check convergence condition; confuses binary/hexadecimal conversion methods; uses Cartesian instead of generalized coordinates; no recognition of image systems needed |
| Method choice | 20% | 10 | Selects appropriate particular integral method for PDE (shifting or undetermined coefficients); uses efficient Gauss-Seidel iteration scheme; applies standard conversion algorithms; chooses energy method with proper generalized coordinate; employs complex potential and circle theorem elegantly | Correct methods but inefficient execution (e.g., Jacobi instead of Gauss-Seidel); workable but lengthy conversion methods; correct Lagrangian approach but clumsy substitution; correct method for streamlines but messy algebra | Wrong method for PI (treating as ODE); uses Gauss elimination instead of iterative method; random trial for conversions; Newtonian approach instead of Lagrangian; attempts direct integration of velocity field without complex potential |
| Computation accuracy | 20% | 10 | Accurate PI calculation yielding z = f1(y+x) + f2(y-2x) - y sin x - cos x; converged values x=1.000, y=-2.000, z=3.000; exact binary 110110010111 and decimal 1966.62109375; correct algebraic manipulation to SHM form; verified streamline equation matching given form | Correct CF but sign errors in PI; iterations show correct trend but stopped early or arithmetic slips; one conversion correct, other with minor error; correct Lagrangian but algebraic slip in substitution; correct approach to streamlines but coefficient errors | Major computational errors in PI evaluation; divergent iterations due to wrong formula; both conversions wrong; incorrect energy expression; algebraic errors preventing derivation of required form |
| Step justification | 20% | 10 | Shows complete working: operator factorization steps, shifting rule application, iteration formulas with substitution steps, division/positional algorithms with remainders/powers, arc length derivation with chain rule, circle theorem statement and application; each algebraic manipulation justified | Shows key steps but skips some algebraic details; iteration steps shown but without explicit substitution; conversion steps partially shown; some calculus steps implied; major steps in streamline derivation present but condensed | Jumps from problem statement to answer with minimal working; iterations listed as final values only; conversions stated without method; Lagrangian written without derivation; streamlines asserted without complex potential setup |
| Final answer & units | 20% | 10 | All five parts with complete final answers: explicit z(x,y) with arbitrary functions, converged numerical values with iteration count, exact binary and decimal representations, derived SHM equation with substitution clearly shown, streamlined equation in required form with constant k identified | Most answers correct but incomplete (missing arbitrary functions, unconverged iterations, unverified streamline form); or correct answers with wrong format | Missing multiple final answers; answers without proper functional form; numerical values clearly wrong; no attempt at parts (d) or (e) |
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