Q5 50M Compulsory solve Differential equations, linear algebra, numerical methods, classical mechanics, fluid dynamics
(a) It is given that the equation of any cone with vertex at (a, b, c) is f((x-a)/(z-c), (y-b)/(z-c)) = 0. Find the differential equation of the cone. (10 marks)
(b) Solve, by Gauss elimination method, the system of equations
2x + 2y + 4z = 18
x + 3y + 2z = 13
3x + y + 3z = 14 (10 marks)
(c) (i) Convert the number (1093.21875)₁₀ into octal and the number (1693.0628)₁₀ into hexadecimal systems.
(ii) Express the Boolean function F(x, y, z) = xy + x'z in a product of maxterms form. (10 marks)
(d) A particle at a distance r from the centre of force moves under the influence of the central force F = -k/r², where k is a constant. Obtain the Lagrangian and derive the equations of motion. (10 marks)
(e) The velocity components of an incompressible fluid in spherical polar coordinates (r, θ, ψ) are (2Mr⁻³cosθ, Mr⁻²sinθ, 0), where M is a constant. Show that the velocity is of the potential kind. Find the velocity potential and the equations of the streamlines. (10 marks)
हिंदी में पढ़ें
(a) दिया गया है कि शीर्ष (a, b, c) वाले किसी शंकु का समीकरण f((x-a)/(z-c), (y-b)/(z-c)) = 0 है। शंकु का अवकल समीकरण ज्ञात कीजिए। (10 अंक)
(b) गॉस विलोपन विधि द्वारा समीकरण निकाय
2x + 2y + 4z = 18
x + 3y + 2z = 13
3x + y + 3z = 14
को हल कीजिए। (10 अंक)
(c) (i) संख्या (1093.21875)₁₀ को अष्टाधारी तथा संख्या (1693.0628)₁₀ को षोडश-आधारी पद्धति में बदलिए।
(ii) बूलीय फलन F(x, y, z) = xy + x'z को योगद (मैक्सटर्म) के गुणन के रूप में अभिव्यक्त कीजिए। (10 अंक)
(d) एक कण, जो बल-केंद्र से r दूरी पर है, केंद्रीय बल F = -k/r², जहाँ k एक स्थिरांक है, के प्रभाव में गतिमान है। लैग्रांजियन निकालिये तथा गति के समीकरणों को व्युत्पन्न कीजिये। (10 अंक)
(e) किसी असंपीड्य तरल के गोलीय ध्रुवी निर्देशांकों (r, θ, ψ) में वेग-घटक (2Mr⁻³cosθ, Mr⁻²sinθ, 0) है, जहाँ M एक स्थिरांक है। दर्शाइये कि वेग, विभव प्रकार का है। वेग विभव तथा धारारेखाओं के समीकरण ज्ञात कीजिये। (10 अंक)
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) calculate: given > formula > substitution > result with units > interpretation | (c) calculate: given > formula > substitution > result with units > interpretation | (d) derive: given > assumptions > stepwise derivation > result > check | (e) derive: given > assumptions > stepwise derivation > result > check Full marks: Complete derivations with all steps, correct results, and verification.
- Put u = (x-a)/(z-c), v = (y-b)/(z-c), p = ∂z/∂x, q = ∂z/∂y
- Differentiate f(u, v) = 0 partially with respect to x and y and eliminate f_u, f_v
- Result: (x-a)p + (y-b)q = z-c
- Note the PDE is the condition that the surface is a cone with vertex (a, b, c)
- Augmented matrix for 2x+2y+4z=18, x+3y+2z=13, 3x+y+3z=14
- Row operations to upper-triangular form (pivot on row 1, eliminate x from rows 2 and 3, then y from row 3)
- Back-substitution
- Final values x = 1, y = 2, z = 3, verified in all three equations
- Octal conversion of (1093.21875)₁₀
- Hexadecimal conversion of (1693.0628)₁₀
- Truth table or K-map for F(x,y,z)
- Maxterm form: ΠM(0, 2, 4, 5)
- Kinetic energy in polar coordinates
- Potential energy V = -k/r
- Lagrangian L = T - V
- Euler–Lagrange equations m(r̈ − rθ̇²) = −k/r² and r²θ̇ = h
- Show curl of velocity is zero
- Integrate to find velocity potential φ
- Derive equations of streamlines
- Note the printed v_θ = Mr⁻² sinθ fails both tests and use the intended Mr⁻³ sinθ, giving φ = −M cosθ/r² and r = C sin²θ
Q6 50M solve PDE heat equation, Boolean algebra, moment of inertia
(a) Solve the heat equation ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x), 0 ≤ x ≤ l. (20 marks)
(b) Find a combinatorial circuit corresponding to the Boolean function f(x, y, z) = [x · (ȳ + z)] + y and write the input/output table for the circuit. (15 marks)
(c) Find the moment of inertia of a right circular solid cone about one of its slant sides (generator) in terms of its mass M, height h and the radius of base as a. (15 marks)
हिंदी में पढ़ें
(a) ऊष्मा समीकरण ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 का शर्तों u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x), 0 ≤ x ≤ l से प्रतिबंधित हल ज्ञात कीजिये। (20 अंक)
(b) बूलीय फलन f(x, y, z) = [x · (ȳ + z)] + y का संगत संचयविन्यास परिपथ (कॉम्बिनेटोरियल सर्किट) ज्ञात कीजिये तथा परिपथ के लिये निवेश/निर्गत (इनपुट/आउटपुट) सारणी लिखिये। (15 अंक)
(c) एक लम्ब वृत्तीय ठोस शंकु का उसकी संहति M, ऊँचाई h तथा आधार की त्रिज्या a के रूप में उसकी एक तिर्यक रेखा (जनक रेखा) के सापेक्ष जड़त्व-आघूर्ण ज्ञात कीजिये। (15 अंक)
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) describe: define > structure or process in order > labelled diagram > significance | (c) derive: given > assumptions > stepwise derivation > result > check Full marks: Complete derivations with all steps, correct results, and verification
- Apply separation of variables to find general solution
- Satisfy boundary conditions u(0,t)=u(l,t)=0
- Expand initial condition x(l-x) in Fourier sine series
- Bₙ = 8l²/(n³π³) for odd n and 0 for even n, giving the final series
- Draw circuit with correct logic gates
- Implement function f(x,y,z) = [x·(ȳ+z)]+y
- Provide complete truth table for all 8 inputs
- Label inputs and outputs clearly
- Set up coordinate system with slant side as axis
- Use integration to find moment of inertia
- Final I = 3Ma²(a² + 6h²)/[20(a² + h²)]
- Show all integration steps clearly
Q7 50M solve Partial differential equations and fluid dynamics
(a) Find the general solution of the partial differential equation (D² + DD′ - 6D′²)z = x² sin(x+y) where D ≡ ∂/∂x and D′ ≡ ∂/∂y. (15 marks)
(b) The velocity of a train which starts from rest is given by the following table, the time being reckoned in minutes from the start and the velocity in km/hour:
t (minutes): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
v (km/hour): 16, 28.8, 40, 46.4, 51.2, 32, 17.6, 8, 3.2, 0
Using Simpson's 1/3rd rule, estimate approximately in km the total distance run in 20 minutes. (15 marks)
(c) Two point vortices each of strength k are situated at (± a, 0) and a point vortex of strength -k/2 is situated at the origin. Show that the fluid motion is stationary and also find the equations of streamlines. If the streamlines, which pass through the stagnation points, meet the x-axis at (± b, 0), then show that 3√3(b² − a²)² = 16a³b. (20 marks)
हिंदी में पढ़ें
(a) आंशिक अवकल समीकरण (D² + DD′ - 6D′²)z = x² sin(x+y) जहाँ D ≡ ∂/∂x तथा D′ ≡ ∂/∂y, का व्यापक हल ज्ञात कीजिये। (15 अंक)
(b) एक रेलगाड़ी, जो कि विश्राम से चलना प्रारंभ करती है, का वेग निम्नलिखित सारणी द्वारा दिया गया है। प्रस्थान से समय की गणना मिनट में तथा वेग की कि० मी०/घंटा में की गयी है:
t (मिनट): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
v (कि० मी०/घंटा): 16, 28.8, 40, 46.4, 51.2, 32, 17.6, 8, 3.2, 0
सिम्पसन के 1/3 नियम का उपयोग करके 20 मिनट में तय की गयी कुल दूरी (लगभग) का आकलन कि० मी० में कीजिये। (15 अंक)
(c) दो बिंदु भ्रमिल, जहाँ प्रत्येक का सामर्थ्य k है, (± a, 0) पर स्थित हैं तथा -k/2 सामर्थ्य का एक बिंदु भ्रमिल, मूलबिंदु पर स्थित है। दर्शाइये कि तरल गति अचल है तथा धारारेखाओं के समीकरण भी ज्ञात कीजिये। यदि धारारेखाएँ, जो कि प्रगतिरोध बिंदुओं (स्टैगनेशन पॉइंट) से गुजरती हैं, x-अक्ष पर (± b, 0) पर मिलती हैं, तब दर्शाइये कि 3√3(b² − a²)² = 16a³b। (20 अंक)
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) calculate: given > formula > substitution > result with units > interpretation | (c) derive: given > assumptions > stepwise derivation > result > check Full marks: All parts fully derived with correct methods, units, and verification; no algebraic errors.
- Complementary function from auxiliary equation m² + m - 6 = 0
- Particular integral via operator method for x² sin(x+y)
- z = f(y − 3x) + g(y + 2x) + (x²/4 − 13/32)sin(x+y) − (3x/8)cos(x+y)
- Verification of PI by substitution
- Simpson's 1/3rd rule formula stated with h = 2
- Correct identification of y0, y1, ..., y10 from table
- Substitution into (h/3)[y0 + y10 + 4(odd) + 2(even)]
- Divide by 60 (minutes to hours): distance ≈ 8.25 km
- Velocity potential or stream function for three vortices
- Proof of stationarity (time-independence of flow field)
- Equation of streamlines derived from ψ = const
- Stagnation points found and used to derive relation
Q8 50M solve PDE canonical forms and numerical methods
(a) Reduce the following partial differential equation to a canonical form and hence solve it: yu_xx + (x+y)u_xy + xu_yy = 0 (15 marks)
(b) Using Runge-Kutta method of fourth order, solve the differential equation dy/dx = x + y² with y(0) = 1, at x = 0.2. Use four decimal places for calculation and step length 0.1. (15 marks)
(c) Verify that w = ik log{(z − ia)/(z + ia)} is the complex potential of a steady flow of fluid about a circular cylinder, where the plane y = 0 is a rigid boundary. Find also the force exerted by the fluid on unit length of the cylinder. (20 marks)
हिंदी में पढ़ें
(a) निम्नलिखित आंशिक अवकल समीकरण yu_xx + (x+y)u_xy + xu_yy = 0 को विहित रूप में समानीत कीजिये और अतएव इसको हल कीजिये। (15 अंक)
(b) चतुर्थ कोटि की रुन्गे-कुट्टा विधि का उपयोग करके अवकल समीकरण dy/dx = x + y², जबकि y(0) = 1 है, को x = 0.2 पर हल कीजिये। परिकलन में दशमलव के चार स्थानों तक तथा पग लम्बाई (स्टेप लैंथ) 0.1 का उपयोग कीजिये। (15 अंक)
(c) सत्यापित कीजिये कि एक वृत्ताकार बेलन के इर्द-गिर्द एक तरल के अपरिवर्ती प्रवाह का सम्मिश्र विभव w = ik log{(z − ia)/(z + ia)} है, जहाँ समतल y = 0 एक दृढ़ सीमा है। बेलन की एकक लम्बाई (यूनिट लैंथ) पर तरल द्वारा लगाये गये बल को भी ज्ञात कीजिये। (20 अंक)
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) calculate: given > formula > substitution > result with units > interpretation | (c) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete derivations with all steps shown, correct final answers, and clear justification of methods.
- Identify A, B, C coefficients from the PDE
- Solve the characteristic equation for m
- Define new variables ξ and η
- Canonical form u_ξη + (1/ξ)u_η = 0 and solution u = f(y² − x²)/(y − x) + g(y − x)
- State the RK4 formulas for k1, k2, k3, k4
- Perform calculation for the first step (x=0 to 0.1)
- Perform calculation for the second step (x=0.1 to 0.2)
- Four-decimal working giving y(0.1) = 1.1165 and y(0.2) = 1.2736
- Show that y=0 is a streamline (rigid boundary)
- Identify the circular cylinder geometry from the potential
- Apply Blasius theorem or equivalent for force calculation
- Force πρk²/a per unit length towards the wall (X = 0, Y = −πρk²/a)