Q5
(a) Obtain the partial differential equation by eliminating arbitrary function f from the equation f(x+y+z, x²+y²+z²) = 0. (10 marks) (b) Find a positive root of the equation 3x = 1+cosx by a numerical technique using initial values 0, π/2; and further improve the result using Newton-Raphson method correct to 8 significant figures. (10 marks) (c) (i) Convert (3798·3875)₁₀ into octal and hexadecimal equivalents. (ii) Obtain the principal conjunctive normal form of (⌐P → R) ∧ (Q ⇔ P). (10 marks) (d) A particle is constrained to move along a circle lying in the vertical xy-plane. With the help of the D'Alembert's principle, show that its equation of motion is ẍy - ÿx - gx = 0, where g is the acceleration due to gravity. (10 marks) (e) What arrangements of sources and sinks can have the velocity potential w=logₑ(z-a²/z)? Draw the corresponding sketch of the streamlines and prove that two of them subdivide into the circle r=a and the axis of y. (10 marks)
हिंदी में प्रश्न पढ़ें
(a) समीकरण f(x+y+z, x²+y²+z²) = 0 से स्वेच्छिक फलन f का विलोपन कर आंशिक अवकल समीकरण को प्राप्त कीजिए। (10 अंक) (b) प्रारंभिक मानों 0, π/2 का उपयोग करके एक संख्यात्मक तकनीक के द्वारा समीकरण 3x = 1+cosx का एक धनात्मक मूल ज्ञात कीजिए, तथा न्यूटन-राप्सन विधि के द्वारा परिणाम को 8 सार्थक अंकों तक और शुद्ध मान के निकट लाइए। (10 अंक) (c) (i) (3798·3875)₁₀ को अष्टाधारी तथा षोडशाधारी तुल्यमानों में बदलिए। (ii) (⌐P → R) ∧ (Q ⇔ P) का मुख्य संयोजक सामान्य रूप (प्रिंसिपल कंजक्टिव नॉर्मल फॉर्म) प्राप्त कीजिए। (10 अंक) (d) उर्ध्वाधर xy-तल में स्थित एक वृत्त के अनुदिश एक कण गति के लिए बंधक है। डी'एलंबर्ट के नियम की सहायता से दर्शाइए कि इसकी गति का समीकरण ẍy - ÿx - gx = 0 है, जहाँ g गुरुत्वीय त्वरण है। (10 अंक) (e) उदगमों (स्रोतों) व अभिगमों (सिंकों) के किस विन्यास से वेग विभव w=logₑ(z-a²/z) हो सकता है? संगत धारा-रेखाओं का खाका खींचिए और सिद्ध कीजिए कि उनमें से दो, वृत्त r=a तथा y-अक्ष में प्रतिभाजित होती हैं। (10 अंक)
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How this answer will be evaluated
Approach
Solve each sub-part systematically with clear mathematical derivations. Allocate approximately 20% time to each part: (a) PDE formation by differentiating and eliminating f; (b) Bisection/Regula-Falsi followed by Newton-Raphson iteration; (c)(i) base conversion with fractional parts, (c)(ii) Boolean algebra simplification to PCNF; (d) D'Alembert's principle with constraint equations; (e) complex potential analysis for source-sink systems. Present solutions in sequence with proper notation and diagrams where required.
Key points expected
- Part (a): Correct identification of arguments u=x+y+z, v=x²+y²+z²; proper differentiation to obtain p=∂z/∂x, q=∂z/∂y; elimination of f to get (y-z)p + (z-x)q = x-y
- Part (b): Application of Regula-Falsi or Bisection method between 0 and π/2 to get initial approximation ~0.607; Newton-Raphson iteration with f(x)=3x-1-cosx, f'(x)=3+sinx; convergence to 8 significant figures: 0.60710163
- Part (c)(i): Correct octal conversion: (7306.3107)₈ and hexadecimal: (E86.6300)₁₆ with proper handling of fractional part by repeated multiplication
- Part (c)(ii): Conversion of (¬P→R)∧(Q↔P) to (P∨R)∧((Q∧P)∨(¬Q∧¬P)); expansion to maxterms; final PCNF as Π(0,2,3,4,5) or equivalent canonical form
- Part (d): Application of D'Alembert's principle with constraint x²+y²=a²; virtual work formulation; proper differentiation of constraints to derive ẍy-ÿx-gx=0
- Part (e): Identification of source at origin and sink at infinity with dipole-like term; stream function ψ; proof that ψ=0 on r=a and y-axis; sketch showing circular streamline and dividing streamlines
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies all mathematical setups: proper substitution for PDE elimination, appropriate bracketing for root-finding with valid initial conditions, correct base conversion algorithms, accurate Boolean expression parsing, proper constraint identification for D'Alembert's principle, and correct complex potential decomposition for fluid flow | Most setups are correct but with minor errors in one or two parts: wrong initial interval choice in (b), incorrect handling of fractional parts in (c)(i), or missing constraint equations in (d) | Fundamental setup errors across multiple parts: incorrect identification of arguments in (a), wrong numerical method chosen, invalid base conversion procedure, major errors in Boolean expression interpretation, or failure to apply D'Alembert's principle correctly |
| Method choice | 20% | 10 | Selects optimal methods: Lagrange's method for PDE elimination, Regula-Falsi followed by Newton-Raphson for rapid convergence, standard base conversion algorithms, truth table or algebraic manipulation for PCNF, generalized coordinates for constrained motion, and complex potential theory for source-sink analysis | Acceptable methods chosen but suboptimal: using only Bisection instead of combined method in (b), or direct expansion without simplification in (c)(ii); methods work but lack efficiency | Inappropriate methods selected: attempting direct elimination without differentiation in (a), using Newton-Raphson without bracketing in (b), or completely wrong approach to any major part |
| Computation accuracy | 20% | 10 | Precise calculations throughout: correct partial derivatives and elimination in (a), 8 significant figure accuracy (0.60710163) in (b), exact octal and hexadecimal conversions with proper rounding, correct maxterm expansion in (c)(ii), accurate algebraic manipulation in (d), and correct streamline equations in (e) | Generally correct computations with minor arithmetic errors: 6-7 significant figures in (b), small conversion errors in (c)(i), or one incorrect maxterm in PCNF; errors don't propagate catastrophically | Significant computational errors: wrong derivatives in (a), failure to converge or wrong root in (b), completely wrong base conversions, major errors in Boolean expansion, or algebraic mistakes in (d) and (e) |
| Step justification | 20% | 10 | Clear logical progression with explicit justification: shows differentiation steps for PDE formation, displays iteration tables for numerical method, explains each base conversion step, provides truth table or algebraic laws for Boolean simplification, derives constraint equations for D'Alembert's principle, and proves streamline properties rigorously | Most steps shown but with gaps: missing intermediate algebraic steps, iteration table incomplete, or skipping some Boolean simplification rules; logic is followable but not fully transparent | Major logical gaps or missing steps: jumps from problem statement to final answer without working, omits crucial differentiation or elimination steps, or presents unsubstantiated claims about streamlines |
| Final answer & units | 20% | 10 | All final answers clearly stated with proper format: PDE in standard form, root to 8 significant figures with verification, bases indicated for conversions, PCNF in proper Π notation, equation of motion as specified, and streamline sketch with key features labeled; includes verification where possible | Final answers present but with formatting issues: missing subscripts for bases, incomplete PCNF notation, sketch without proper labeling, or root with insufficient significant figures | Missing or incorrect final answers: wrong PDE form, incorrect numerical value, wrong base representations, invalid PCNF, incorrect final equation, or no diagram for part (e) |
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