Q5 50M Compulsory solve PDE, linear algebra, Boolean algebra, mechanics, fluid dynamics
(a) Show that if f and g are arbitrary functions of their respective arguments, then u = f(x - kt + iαy) + g(x - kt - iαy), is a solution of ∂²u/∂x² + ∂²u/∂y² = (1/C²)∂²u/∂t², where α² = 1 - k²/C². (10 marks)
(b) Solve the following system of linear equations by Gauss-Jordan method: (10 marks)
2x + 3y - z = 5
4x + 4y - 3z = 3
2x - 3y + 2z = 2
(c) (i) Determine the decimal equivalent in sign magnitude form of (8D)₁₆ and (FF)₁₆.
(ii) Determine the decimal equivalent of (9B2.1A)₁₆. (10 marks)
(d) A rough uniform board of mass m and length 2a rests on a smooth horizontal plane and a man of mass M walks on it from one end to the other. Find the distance covered by the board during this time. (10 marks)
(e) The velocity potential φ of a flow is given by φ = (1/2)(x² + y² - 2z²). Determine the streamlines. (10 marks)
हिंदी में पढ़ें
(a) दर्शाइए कि यदि f और g उनके संबंधित स्वतंत्र चरों के स्वेच्छ फलन हैं, तब u = f(x - kt + iαy) + g(x - kt - iαy), ∂²u/∂x² + ∂²u/∂y² = (1/C²)∂²u/∂t² का एक हल है, जहाँ α² = 1 - k²/C² है। (10 अंक)
(b) गॉस-जॉर्डन विधि द्वारा निम्नलिखित रैखिक समीकरण निकाय को हल कीजिए: (10 अंक)
2x + 3y - z = 5
4x + 4y - 3z = 3
2x - 3y + 2z = 2
(c) (i) (8D)₁₆ और (FF)₁₆ के संचिह्न परिमाण रूप में दशमलव समतुल्य ज्ञात कीजिए।
(ii) (9B2.1A)₁₆ का दशमलव समतुल्य ज्ञात कीजिए। (10 अंक)
(d) द्रव्यमान m तथा लम्बाई 2a का एक खुरदुरा एकसमान बोर्ड एक चिकने क्षैतिज तल पर रखा है और M द्रव्यमान का एक व्यक्ति उस पर एक छोर से दूसरे छोर तक चलता है। इस दौरान बोर्ड द्वारा तय की गई दूरी ज्ञात कीजिए। (10 अंक)
(e) एक प्रवाह का वेग विभव φ, φ = (1/2)(x² + y² - 2z²) के द्वारा दिया गया है। धारा रेखाएँ ज्ञात कीजिए। (10 अंक)
Answer approach & key points
Solve each sub-part systematically with clear mathematical derivations. For (a), verify the PDE solution by computing partial derivatives and substituting; for (b), apply Gauss-Jordan elimination with complete row operations; for (c), convert hexadecimal to decimal using positional weights and sign-magnitude interpretation; for (d), apply conservation of momentum for the man-board system; for (e), derive velocity components from potential and integrate for streamlines. Allocate approximately 20% time to each part given equal marks distribution.
- Part (a): Correct computation of ∂u/∂x, ∂²u/∂x², ∂u/∂y, ∂²u/∂y², ∂u/∂t, ∂²u/∂t² using chain rule and verification that α² = 1 - k²/C² satisfies the wave equation
- Part (b): Construction of augmented matrix, systematic row reduction to reduced row echelon form, and correct final values x = 1, y = 2, z = 3
- Part (c)(i): Correct sign-magnitude interpretation: (8D)₁₆ = +141, (FF)₁₆ = -127 (8-bit) or -255 (extended); handling of MSB as sign bit
- Part (c)(ii): Accurate hexadecimal conversion: (9B2.1A)₁₆ = 9×256 + 11×16 + 2 + 1/16 + 10/256 = 2482.1015625
- Part (d): Application of center of mass conservation: board displacement = 2aM/(m+M) in opposite direction to man's motion
- Part (e): Derivation of velocity components u = x, v = y, w = -2z and integration to obtain parametric streamlines x = x₀eᵗ, y = y₀eᵗ, z = z₀e⁻²ᵗ
Q6 50M derive Laplace equation, Boolean algebra, moment of inertia
(a) Show that the solution of the two-dimensional Laplace's equation ∂²φ(x,y)/∂x² + ∂²φ(x,y)/∂y² = 0, x ∈ (-∞, ∞), y ≥ 0 subject to the boundary condition φ(x,0) = f(x), x ∈ (-∞, ∞), along with φ(x,y) → 0 for |x| → ∞ and y → ∞ can be written in the form φ(x,y) = (y/π)∫_{-∞}^{∞} [f(ξ)dξ]/[y² + (x-ξ)²]. (20 marks)
(b) Draw the logical circuit for the Boolean expression Y = ABC̄ + BC̄ + ĀB. Also, obtain the output Y (truth table) for the three input bit sequences: A = 10001111, B = 00111100, C = 11000100 (15 marks)
(c) Find the moment of inertia of a quadrant of an elliptic disk x²/a² + y²/b² = 1, of mass M about the line passing through its centre and perpendicular to its plane. Given that the density at any point is proportional to xy. (15 marks)
हिंदी में पढ़ें
(a) दर्शाइए कि द्विविम लाप्लास समीकरण ∂²φ(x,y)/∂x² + ∂²φ(x,y)/∂y² = 0, x ∈ (-∞, ∞), y ≥ 0 का हल, परिसीमा प्रतिबंध φ(x,0) = f(x), x ∈ (-∞, ∞) के अधीन तथा φ(x,y) → 0 जब |x| → ∞ और y → ∞, φ(x,y) = (y/π)∫_{-∞}^{∞} [f(ξ)dξ]/[y² + (x-ξ)²] के रूप में लिखा जा सकता है। (20 अंक)
(b) बूलिय व्यंजक Y = ABC̄ + BC̄ + ĀB के लिए तर्कसंगत परिपथ (लॉजिकल सर्किट) खींचिए। तीन निवेश द्वयक अनुक्रमों A = 10001111, B = 00111100, C = 11000100 के लिए निर्गत Y (सत्यमान सारणी) भी प्राप्त कीजिए। (15 अंक)
(c) दीर्घवृत्तीय डिस्क x²/a² + y²/b² = 1, के एक चतुर्थांश, जिसका द्रव्यमान M है, का उसके तल के लंबवत तथा उसके केंद्र से गुजरने वाली रेखा के सापेक्ष, जड़त्व आघूर्ण ज्ञात कीजिए। दिया गया है कि किसी भी बिंदु पर घनत्व xy के समानुपाती है। (15 अंक)
Answer approach & key points
Derive the Poisson integral formula for Laplace's equation in part (a) using Fourier transform or Green's function method, spending approximately 40% of time due to its 20 marks weightage. For part (b), simplify the Boolean expression using Boolean algebra laws, design the logic circuit with AND/OR/NOT gates, and construct the truth table for the given bit sequences (~30% time). For part (c), set up the double integral for moment of inertia with variable density ρ = kxy, transform to elliptical coordinates, and evaluate carefully (~30% time). Present solutions with clear section headings for each part.
- Part (a): Apply Fourier transform in x to convert PDE to ODE, solve the resulting equation φ̂(k,y) = A(k)e^{-|k|y}, apply boundary condition to find A(k) = f̂(k), then invert using convolution theorem with Poisson kernel
- Part (a): Alternatively use Green's function method with image source for half-plane to derive the integral representation directly
- Part (b): Simplify Y = ABC̄ + BC̄ + ĀB to Y = BC̄ + ĀB using absorption law, or further to minimal form, then draw circuit with 2 NOT gates, 2 AND gates, 1 OR gate
- Part (b): Construct truth table showing A,B,C inputs and Y output, then evaluate Y for given 8-bit sequences by bitwise operation: Y = 00110000
- Part (c): Set up I = ∫∫ ρ(x,y)(x²+y²)dA with ρ = kxy over first quadrant, use transformation x = arcosθ, y = brsinθ with Jacobian abr
- Part (c): Determine k from total mass condition M = k∫∫xy dA = kab²/8, then evaluate I = (4M/π)(a²+b²) or equivalent simplified form
Q7 50M solve PDE, numerical integration, fluid dynamics
(a) Find the integral surface of the following quasi-linear equation $$(y - \phi) \frac{\partial \phi}{\partial x} + (\phi - x) \frac{\partial \phi}{\partial y} = x - y,$$ which passes through the curve $\phi = 0$, $xy = 1$ and through the circle $x + y + \phi = 0$, $x^2 + y^2 + \phi^2 = a^2$. (15 marks)
(b) Integrate $f(x) = 5x^3 - 3x^2 + 2x + 1$ from $x = -2$ to $x = 4$ using (i) Simpson's $\frac{3}{8}$ rule with width h = 1, and (ii) Trapezoidal rule with width h = 1. (15 marks)
(c) Let the velocity field $$u(x, y) = \frac{B(x^2 - y^2)}{(x^2 + y^2)^2}, \quad v(x, y) = \frac{2Bxy}{(x^2 + y^2)^2}, \quad w(x, y) = 0$$ satisfy the equations of motion for inviscid incompressible flow, where B is a constant. Determine the pressure associated with this velocity field. (20 marks)
हिंदी में पढ़ें
(a) निम्नलिखित रैखिक-कल्प समीकरण $$(y - \phi) \frac{\partial \phi}{\partial x} + (\phi - x) \frac{\partial \phi}{\partial y} = x - y,$$ का वह समाकल पृष्ठ ज्ञात कीजिए, जो कि वक्र $\phi = 0$, $xy = 1$ और वृत्त $x + y + \phi = 0$, $x^2 + y^2 + \phi^2 = a^2$ से होकर गुजरता है। (15 अंक)
(b) (i) चौड़ाई h = 1 के साथ सिम्पसन के $\frac{3}{8}$ नियम, और (ii) चौड़ाई h = 1 के साथ समलंबी (ट्रेपिजॉइडल) नियम का उपयोग करके $f(x) = 5x^3 - 3x^2 + 2x + 1$ का $x = -2$ से $x = 4$ तक समाकलन कीजिए। (15 अंक)
(c) मान लीजिए कि वेग क्षेत्र $$u(x, y) = \frac{B(x^2 - y^2)}{(x^2 + y^2)^2}, \quad v(x, y) = \frac{2Bxy}{(x^2 + y^2)^2}, \quad w(x, y) = 0,$$ जहाँ B एक अचर है, अश्यान असंपीड्य प्रवाह के लिए गति समीकरणों को संतुष्ट करता है। इस वेग क्षेत्र से सहचारी (एसोसिएटेड) दाब का निर्धारण कीजिए। (20 अंक)
Answer approach & key points
Solve all three parts systematically, allocating approximately 30% time to part (a) on Lagrange's auxiliary equations and integral surfaces, 30% to part (b) on numerical integration with proper tabulation for both Simpson's 3/8 and Trapezoidal rules, and 40% to part (c) on applying Euler's equations for inviscid flow to determine pressure distribution. Present each part with clear headings, show all working steps, and verify boundary conditions are satisfied.
- Part (a): Formulate Lagrange's auxiliary equations dx/(y-φ) = dy/(φ-x) = dφ/(x-y) and find multipliers to obtain first integrals; apply both boundary conditions (hyperbola φ=0, xy=1 and circle x+y+φ=0, x²+y²+φ²=a²) to determine the integral surface
- Part (b)(i): Apply Simpson's 3/8 rule with h=1 requiring 3n intervals; construct table for f(x)=5x³-3x²+2x+1 from x=-2 to x=4 (7 points), apply weights 1,3,3,2,3,3,1 and compute integral value
- Part (b)(ii): Apply Trapezoidal rule with h=1; use same tabulated values with weights 1,2,2,2,2,2,1 and compare accuracy with exact analytical integration
- Part (c): Verify incompressibility (∂u/∂x + ∂v/∂y = 0), identify this as a 2D dipole flow in potential flow theory; apply Euler equations ∂u/∂t + u∂u/∂x + v∂u/∂y = -1/ρ ∂p/∂x (similarly for y) to obtain pressure field p(x,y)
- Part (c) continued: Integrate to find p = p∞ - ρ/2 (u²+v²) + C/r⁴ terms, or express pressure coefficient; handle singularity at origin appropriately and express final pressure in terms of B, ρ, and r²=x²+y²
Q8 50M solve PDE canonical form, interpolation, vortex dynamics
(a) Solve the partial differential equation $$\frac{\partial}{\partial y}\left(\frac{\partial \phi}{\partial x} + \phi\right) + 2x^2y\left(\frac{\partial \phi}{\partial x} + \phi\right) = 0$$ by transforming it to the canonical form. (15 marks)
(b) Using Newton's forward difference formula for interpolation, estimate the value of f(2·5) from the following data: x : 1 2 3 4 5 6, f(x) : 0 1 8 27 64 125. (15 marks)
(c) Suppose an infinite liquid contains two parallel, equal and opposite rectilinear vortices at a distance 2a. Show that the streamlines relative to the vortex are given by the equation $$\log \frac{x^2 + (y-a)^2}{x^2 + (y+a)^2} + \frac{y}{a} = C,$$ where C is a constant, the origin is the middle point of the join, and the line joining the vortices is the axis of y. (20 marks)
हिंदी में पढ़ें
(a) आंशिक अवकल समीकरण $$\frac{\partial}{\partial y}\left(\frac{\partial \phi}{\partial x} + \phi\right) + 2x^2y\left(\frac{\partial \phi}{\partial x} + \phi\right) = 0$$ को विहित रूप में रूपांतरित करके हल कीजिए। (15 अंक)
(b) अंतर्वेशन के लिए न्यूटन के अग्रांतर सूत्र का उपयोग करके निम्नलिखित आँकड़ों से f(2·5) के मान का आकलन कीजिए: x : 1 2 3 4 5 6, f(x) : 0 1 8 27 64 125. (15 अंक)
(c) मान लीजिए कि एक अनंत द्रव में दो समानांतर, समान तथा विपरीत सरलरेखीय भ्रमिल 2a की दूरी पर हैं। दर्शाइए कि भ्रमिल के सापेक्ष धारा रेखाएँ समीकरण $$\log \frac{x^2 + (y-a)^2}{x^2 + (y+a)^2} + \frac{y}{a} = C$$ द्वारा दी गई हैं, जहाँ C एक अचर है, मूल-बिंदु जुड़ाव का मध्य बिंदु है, और भ्रमिलों को जोड़ने वाली रेखा y का अक्ष है। (20 अंक)
Answer approach & key points
Solve all three sub-parts systematically, allocating time proportionally to marks: approximately 18 minutes for part (a) on PDE canonical transformation, 18 minutes for part (b) on Newton's forward interpolation, and 24 minutes for part (c) on vortex dynamics derivation. Begin each part with clear identification of the method, show complete working with proper mathematical notation, and conclude with boxed final answers.
- Part (a): Substitute u = ∂φ/∂x + φ to reduce the PDE to first-order form, then use integrating factor or characteristic method to obtain canonical form
- Part (a): Correctly identify the canonical transformation and solve for φ in terms of arbitrary functions
- Part (b): Construct forward difference table with proper spacing (h=1), identify correct node (x₀=1) and calculate u = (2.5-1)/1 = 1.5
- Part (b): Apply Newton's forward formula with correct binomial coefficients and compute f(2.5) = 0 + 1.5(1) + 0.375(7) + ... leading to value 15.625 or exact verification
- Part (c): Set up complex potential for two opposite vortices at (0,a) and (0,-a) with circulations ±κ
- Part (c): Transform to moving frame with vortex velocity V = κ/(4πa), derive stream function ψ, and obtain streamline equation through algebraic manipulation
- Part (c): Verify the final logarithmic form matches the required expression with constant C