All 8 questions from UPSC Civil Services Mains Mathematics
2023 Paper II (400 marks total). Every stem reproduced in full,
with directive-word analysis, marks, word limits, and answer-approach pointers.
8Questions
400Total marks
2023Year
Paper IIPaper
Topics covered
Group theory, ideals, series convergence, complex analysis, linear programming (1)Group theory, Lagrange multipliers, contour integration (1)Abstract algebra, complex analysis and linear programming (1)Real analysis, complex analysis and assignment problem (1)Partial differential equations, numerical methods, binary arithmetic, Hamiltonian mechanics, fluid dynamics (1)Partial differential equations, numerical linear algebra, Lagrangian mechanics (1)Boolean functions, mechanics, partial differential equations (1)PDE canonical form, numerical methods, Navier-Stokes equations (1)
A
Q1
50MCompulsoryexamineGroup theory, ideals, series convergence, complex analysis, linear programming
(a) Let G be a group of order 10 and G′ be a group of order 6. Examine whether there exists a homomorphism of G onto G′. (10 marks)
(b) Express the ideal 4Z + 6Z as a principal ideal in the integral domain Z. (10 marks)
(c) Test the convergence of the series
$$\sum\limits_{n=1}^{\infty} \frac{1.3.5...(2n-1)}{2.4.6...(2n)} \cdot \frac{x^{2n+1}}{(2n+1)}, \quad x > 0$$
(10 marks)
(d) State the sufficient conditions for a function $f(z) = f(x+iy) = u(x,y) + iv(x,y)$ to be analytic in its domain. Hence, show that $f(z) = \log z$ is analytic in its domain and find $\frac{df}{dz}$ (10 marks)
(e) A person requires 24, 24 and 20 units of chemicals A, B and C respectively for his garden. Product P contains 2, 4 and 1 units of chemicals A, B and C respectively per jar and product Q contains 2, 1 and 5 units of chemicals A, B and C respectively per jar. If a jar of P costs ₹ 30 and a jar of Q costs ₹ 50, then how many jars of each should be purchased in order to minimize the cost and meet the requirements? (10 marks)
हिंदी में पढ़ें
(a) मान लीजिए कोटि 10 का एक समूह G है तथा कोटि 6 का एक समूह G′ है। जाँच कीजिए कि क्या G से G′ पर एक आच्छादक समाकारिता का अस्तित्व है। (10 अंक)
(b) गुणजावली 4Z + 6Z को पूर्णांकीय प्रांत Z में एक मुख्य गुणजावली के रूप में व्यक्त कीजिए। (10 अंक)
(c) श्रेणी $\sum\limits_{n=1}^{\infty} \frac{1.3.5...(2n-1)}{2.4.6...(2n)} \cdot \frac{x^{2n+1}}{(2n+1)}$, $x > 0$ के अभिसरण का परीक्षण कीजिए। (10 अंक)
(d) एक फलन $f(z) = f(x+iy) = u(x,y) + iv(x,y)$ के इसके प्रांत में विलेखिक होने के लिए पर्याप्त प्रतिबंध लिखिए। तब दर्शाइए कि $f(z) = \log z$ अपने प्रांत में विलेखिक है तथा $\frac{df}{dz}$ ज्ञात कीजिए। (10 अंक)
(e) एक व्यक्ति को अपने उद्यान के लिए रसायन A, B तथा C की क्रमशः 24, 24 तथा 20 इकाई की आवश्यकता है। उत्पाद P के प्रत्येक मर्तबान में रसायन A, B तथा C की क्रमशः 2, 4 तथा 1 इकाई है तथा उत्पाद Q के प्रत्येक मर्तबान में रसायन A, B तथा C की क्रमशः 2, 1 तथा 5 इकाई है। यदि P के एक मर्तबान का मूल्य ₹ 30 है तथा Q के एक मर्तबान का मूल्य ₹ 50 है, तब न्यूनतम खर्च तथा आवश्यताओं की पूर्ति के लिए प्रत्येक उत्पाद के कितने मर्तबान खरीदे जाएँ? (10 अंक)
Answer approach & key points
Examine each sub-part systematically with equal time allocation (~20% per part) since all carry 10 marks. Begin with clear statements of theorems/conditions for (a), (b), (d); show complete computational steps for (c) and (e). Structure as: direct response to each sub-part labeled (a)-(e), with concise justification for theoretical parts and detailed working for computational parts. No separate introduction or conclusion needed.
For (a): Apply First Isomorphism Theorem to show no onto homomorphism exists since gcd(10,6)=2 but 6∤10, or use Lagrange's theorem on kernel/image orders
For (b): Use property that 4Z + 6Z = gcd(4,6)Z = 2Z, proving the sum is principal ideal generated by 2
For (c): Apply Raabe's test or Gauss test to determine convergence at x=1 (diverges) and x<1 (converges), with careful handling of the limit
For (d): State Cauchy-Riemann equations as sufficient conditions; verify u=log|z|, v=arg(z) satisfy them; derive df/dz = 1/z
For (e): Formulate LPP with objective minimize Z=30x+50y subject to 2x+2y≥24, 4x+y≥24, x+5y≥20, x,y≥0; solve by graphical method or simplex to get optimal solution at (8,4)
(a) Prove that a non-commutative group of order 2p, where p is an odd prime, must have a subgroup of order p. (15 marks)
(b) Using the method of Lagrange's multipliers, find the minimum and maximum distances of the point P(2, 6, 3) from the sphere x² + y² + z² = 4. (15 marks)
(c) Evaluate $\int_{0}^{2\pi} \frac{\cos 2\theta}{5+4\cos\theta} d\theta$ using contour integration. (20 marks)
हिंदी में पढ़ें
(a) सिद्ध कीजिए कि 2p कोटि के एक अक्रमविनिमेय समूह, जहाँ p एक विषम अभाज्य संख्या है, में p कोटि का एक उपसमूह होना आवश्यक है। (15 अंक)
(b) लग्रांज गुणक विधि के उपयोग से बिंदु P(2, 6, 3) की गोले x² + y² + z² = 4 से न्यूनतम तथा अधिकतम दूरियाँ ज्ञात कीजिए। (15 अंक)
(c) कंटूर समाकलन का उपयोग कर $\int_{0}^{2\pi} \frac{\cos 2\theta}{5+4\cos\theta} d\theta$ का मान ज्ञात कीजिए। (20 अंक)
Answer approach & key points
Begin with a clear statement of what is to be proved for part (a), then solve for (b) and (c). Allocate approximately 30% time to part (a) (group theory proof), 30% to part (b) (Lagrange multipliers), and 40% to part (c) (contour integration, highest marks). Structure as: (a) statement → application of Sylow/Cauchy theorems → conclusion; (b) setup of distance function and constraint → Lagrange equations → solving critical points; (c) contour setup → residue calculation → final evaluation. Conclude with boxed final answers for each part.
Part (a): Application of Cauchy's theorem or Sylow theorems to establish existence of element of order p, hence cyclic subgroup of order p
Part (a): Explicit use of non-commutativity to rule out cyclic group of order 2p, ensuring the subgroup of order p is proper
Part (b): Correct formulation of distance squared function D = (x-2)² + (y-6)² + (z-3)² with constraint g = x² + y² + z² - 4 = 0
Part (b): Setting up and solving the system ∇D = λ∇g with geometric interpretation that extrema occur along line through origin and P
Part (c): Substitution z = e^(iθ) to convert real integral to complex contour integral over unit circle |z|=1
Part (c): Correct identification of poles inside unit circle (at z = -1/2) and calculation of residue
Part (c): Evaluation of contour integral using residue theorem and extraction of real part to obtain final answer
50MsolveAbstract algebra, complex analysis and linear programming
(a) Prove that x² + 1 is an irreducible polynomial in Z₃[x]. Further show that the quotient ring $\frac{Z_3[x]}{\langle x^2+1 \rangle}$ is a field of 9 elements. 15 marks
(b) Prove that u(x, y) = eˣ(x cos y - y sin y) is harmonic. Find its conjugate harmonic function v(x, y) and express the corresponding analytic function f(z) in terms of z. 15 marks
(c) Solve the following linear programming problem by Big M method :
Minimize Z = 2x₁ + 3x₂
subject to
x₁ + x₂ ≥ 9
x₁ + 2x₂ ≥ 15
2x₁ - 3x₂ ≤ 9
x₁, x₂ ≥ 0
Is the optimal solution unique? Justify your answer. 20 marks
हिंदी में पढ़ें
(a) सिद्ध कीजिए कि x² + 1, Z₃[x] में एक अविभाज्य बहुपद है। यह भी दर्शाइए कि विभाग वलय $\frac{Z_3[x]}{\langle x^2+1 \rangle}$, 9 अवयवों का एक क्षेत्र है। 15
(b) सिद्ध कीजिए कि u(x, y) = eˣ(x cos y - y sin y) प्रसंवादी है। इसका संयुग्मी प्रसंवादी फलन v(x, y) ज्ञात कीजिए तथा संगत विश्लेषिक फलन f(z) को z के पदों में व्यक्त कीजिए। 15
(c) बड़ा M (बिग M) विधि से निम्नलिखित रैखिक प्रोग्रामन समस्या को हल कीजिए :
न्यूनतमीकरण कीजिए Z = 2x₁ + 3x₂
बशर्ते कि
x₁ + x₂ ≥ 9
x₁ + 2x₂ ≥ 15
2x₁ - 3x₂ ≤ 9
x₁, x₂ ≥ 0
क्या इष्टतम हल अद्वितीय है? अपने उत्तर का तर्क प्रस्तुत कीजिए। 20
Answer approach & key points
Solve this multi-part problem by proving results in (a) and (b) before applying computational methods in (c). Allocate approximately 30% time to part (a) on irreducibility and field construction, 30% to part (b) on harmonic functions and analytic construction, and 40% to part (c) on the Big M method LP solution including uniqueness analysis. Structure as: direct proofs for (a)-(b), systematic tableau operations for (c), with clear final verification of all claims.
For (a): Prove x²+1 has no roots in Z₃ (hence irreducible), then apply the theorem that F[x]/⟨p(x)⟩ is a field when p(x) is irreducible, counting 9 elements explicitly
For (b): Verify uₓₓ + uᵧᵧ = 0 using product rule differentiation, then find v(x,y) via Cauchy-Riemann equations or Milne-Thomson method, finally express f(z) = z·e^z
For (c): Convert minimization to standard form using surplus and artificial variables with Big M penalty, construct initial simplex tableau
For (c): Execute simplex iterations correctly, identifying entering and leaving variables at each pivot
For (c): State optimal solution with Z_min = 15 at (x₁, x₂) = (6, 3), and justify uniqueness by checking no alternative optimal solutions exist in final tableau
50MproveReal analysis, complex analysis and assignment problem
(a) Prove that the oscillation of a real-valued bounded function f defined on [a, b] is the supremum of the set {|f(x₁)-f(x₂)| : x₁, x₂ ∈ [a, b]}. 15 marks
(b) Classify the singular point z = 0 of the function f(z) = e^z/(z-sin z) and obtain the principal part of its Laurent series expansion. 15 marks
(c) A department head has 5 subordinates and 5 jobs to be performed. The time (in hours) that each subordinate will take to perform each job is given in the matrix below :
How should the jobs be assigned, one to each subordinate, so as to minimize the total time? Also, obtain the total minimum time to perform all the jobs if the subordinate IV cannot be assigned job C. 20 marks
हिंदी में पढ़ें
(a) सिद्ध कीजिए कि [a, b] पर परिभाषित एक वास्तविक मान परिबद्ध फलन f का दोलन, समुच्चय {|f(x₁)-f(x₂)| : x₁, x₂ ∈ [a, b]} का उच्चक है। 15
(b) फलन f(z) = e^z/(z-sin z) के विचित्र बिंदु z = 0 का वर्गीकरण कीजिए तथा इसके लॉरेंट श्रेणी प्रसार का मुख्य भाग ज्ञात कीजिए। 15
(c) एक विभाग के अध्यक्ष के अधीन 5 कर्मचारी हैं तथा उसके पास 5 कार्य हैं। प्रत्येक कर्मचारी के लिए प्रत्येक कार्य को करने का समय (घंटों में) नीचे आव्यूह में दिया गया है :
कुल समय के न्यूनतमीकरण के लिए, प्रत्येक कर्मचारी को एक कार्य किस प्रकार दिया जाए? यदि कर्मचारी IV को कार्य C नहीं दिया जा सकता है, तो सभी कार्यों को करने में लगने वाला कुल न्यूनतम समय भी ज्ञात कीजिए। 20
Answer approach & key points
Begin with a clear statement of definitions for oscillation, supremum, and bounded variation. For part (a), establish the equivalence through careful set-theoretic arguments showing ω(f) = sup|f(x₁)-f(x₂)|. For part (b), use Taylor series expansion to identify the order of the pole at z=0 and extract coefficients for the principal part. For part (c), apply the Hungarian algorithm to solve the 5×5 assignment problem, then re-optimize with the additional constraint. Allocate approximately 30% time to (a), 30% to (b), and 40% to (c) given its higher weightage and computational demand.
For (a): Correct definition of oscillation ω(f,[a,b]) = M-m where M=sup f, m=inf f; proof that this equals sup{|f(x₁)-f(x₂)|} through showing both inequalities
For (a): Demonstration that for any ε>0, there exist x₁,x₂ with f(x₁)>M-ε/2 and f(x₂)<m+ε/2, establishing the supremum attainment
For (b): Taylor expansion of sin z = z - z³/6 + z⁵/120 - ... and e^z = 1 + z + z²/2 + ... to determine z-sin z = z³/6 - z⁵/120 + ...
For (b): Identification of z=0 as a pole of order 2 (simple pole cancellation leaves z³ in denominator, e^z≈1), computation of Laurent series principal part coefficients a₋₁, a₋₂
For (c): Correct setup of cost matrix, application of Hungarian algorithm (row/column reduction, minimum lines covering zeros, matrix adjustments)
For (c): Optimal assignment without constraints and verification of optimality; modified solution with constraint IV≠C using branch-and-bound or re-optimization
For (c): Clear statement of minimum total time in both cases with proper units (hours)
(a) By eliminating the arbitrary functions f and g from z = f(x² - y) + g(x² + y), form partial differential equation. (10 marks)
(b) Given dy/dx = (y² - x)/(y² + x) with initial condition y = 1 at x = 0. Find the value of y for x = 0.4 by Euler's method, correct to 4 decimal places, taking step length h = 0.1. (10 marks)
(c) Evaluate, using the binary arithmetic, the following numbers in their given system:
(i) (634.235)₈ - (132.223)₈
(ii) (7AB.432)₁₆ - (5CA.D61)₁₆ (10 marks)
(d) A planet of mass m is revolving around the sun of mass M. The kinetic energy T and the potential energy V of the planet are given by T = ½m(ṙ² + r²θ̇²) and V = GMm(1/2a - 1/r), where (r, θ) are the polar coordinates of the planet at time t, G is the gravitational constant and 2a is the major axis of the ellipse (the path of the planet). Find the Hamiltonian and the Hamilton equations of the planet's motion. (10 marks)
(e) In a fluid motion, there is a source of strength 2m placed at z = 2 and two sinks of strength m are placed at z = 2 + i and z = 2 - i. Find the streamlines. (10 marks)
हिंदी में पढ़ें
(a) z = f(x² - y) + g(x² + y) से स्वेच्छिक फलनों f तथा g का विलोपन कर अंशिक अवकल समीकरण बनाइए। (10 अंक)
(b) दिया है dy/dx = (y² - x)/(y² + x) तथा प्रारंभिक प्रतिबंध x = 0 पर y = 1 है। ऑयलर की विधि से पद लंबाई (स्टेप लैंथ) h = 0.1 लेते हुए x = 0.4 के लिए y का मान, दशमलव के 4 स्थानों तक सही, ज्ञात कीजिए। (10 अंक)
(c) द्वि-आधारी अंकगणित का उपयोग कर निम्नलिखित संख्याओं का मूल्यांकन उनकी दी गई पद्धति में कीजिए:
(i) (634.235)₈ - (132.223)₈
(ii) (7AB.432)₁₆ - (5CA.D61)₁₆ (10 अंक)
(d) m द्रव्यमान का एक ग्रह M द्रव्यमान के सूर्य की परिक्रमा कर रहा है। ग्रह की गतिज ऊर्जा T तथा स्थितिज ऊर्जा V, T = ½m(ṙ² + r²θ̇²) तथा V = GMm(1/2a - 1/r) द्वारा दी गई हैं, जहाँ t समय पर ग्रह के ध्रुवीय निर्देशांक (r, θ) हैं, गुरुत्वीय स्थिरांक G है तथा दीर्घवृत्त (ग्रह का पथ) का दीर्घ अक्ष 2a है। ग्रह की गति के लिए हैमिल्टोनी तथा हैमिल्टन समीकरणों को ज्ञात कीजिए। (10 अंक)
(e) एक तरल प्रवाह में, 2m सामर्थ्य का एक स्रोत z = 2 पर स्थित है तथा m सामर्थ्य के दो अभिगम (सिंक) z = 2 + i और z = 2 - i पर स्थित हैं। प्रवाह-रेखाएँ ज्ञात कीजिए। (10 अंक)
Answer approach & key points
Solve each sub-part systematically with clear working: for (a) apply partial differentiation to eliminate arbitrary functions; for (b) execute Euler's method with 4 iterations; for (c)(i)-(ii) perform octal and hexadecimal subtraction with borrowing; for (d) construct Hamiltonian from T-V and derive canonical equations; for (e) use complex potential for source-sink superposition. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e), ensuring all numerical answers are boxed with correct precision.
For (a): Correctly identify arguments u = x² - y and v = x² + y, compute partial derivatives p = ∂z/∂x and q = ∂z/∂y, then eliminate f' and g' to obtain PDE: x(∂z/∂x) = 2(∂z/∂y) or equivalent second-order form
For (b): Apply Euler's formula y_{n+1} = y_n + hf(x_n, y_n) for 4 steps (x=0 to 0.4), showing each iteration with h=0.1, final answer y(0.4) ≈ 1.4615 (to 4 decimal places)
For (c)(i): Perform octal subtraction (634.235)₈ - (132.223)₈ = (502.012)₈ with proper borrowing across radix point
For (c)(ii): Perform hexadecimal subtraction (7AB.432)₁₆ - (5CA.D61)₁₆ = (1E0.871)₁₆ handling borrow from 16's complement
For (d): Form Hamiltonian H = T + V = ½m(p_r²/m² + p_θ²/(m²r²)) + GMm(1/2a - 1/r), derive ṙ = ∂H/∂p_r, θ̇ = ∂H/∂p_θ, ṗ_r = -∂H/∂r, ṗ_θ = -∂H/∂θ = 0
For (e): Construct complex potential W = 2m ln(z-2) - m ln(z-2-i) - m ln(z-2+i), extract stream function ψ = Im(W), obtain streamline equation or implicit form
50MsolvePartial differential equations, numerical linear algebra, Lagrangian mechanics
(a) Find the surface passing through the two lines z = x = 0 and z-1 = x-y = 0, and satisfying the partial differential equation ∂²z/∂x² - 4∂²z/∂x∂y + 4∂²z/∂y² = 0. (15 marks)
(b) Solve the system of linear equations 7x₁ - x₂ + 2x₃ = 11, 2x₁ + 8x₂ - x₃ = 9, x₁ - 2x₂ + 9x₃ = 7 correct up to 4 significant figures by the Gauss-Seidel iterative method. Take initially guessed solution as x₁ = x₂ = x₃ = 0. (15 marks)
(c) A mechanical system with 2 degrees of freedom has the Lagrangian L = ½m(ẋ² + ẏ²) - ½m(w₁²x² + w₂²y²) + kxy where m, w₁, w₂, k are constants. Find the parameter θ so that under the transformation x = q₁ cos θ - q₂ sin θ, y = q₁ sin θ + q₂ cos θ the Lagrangian in terms of q₁, q₂ will not contain the product term q₁q₂. Find the Lagrange's equations w.r.t. q₁ and q₂ independent of parameter θ. (20 marks)
हिंदी में पढ़ें
(a) दो रेखाओं z = x = 0 तथा z-1 = x-y = 0 से होकर जाने वाला और आंशिक अवकल समीकरण ∂²z/∂x² - 4∂²z/∂x∂y + 4∂²z/∂y² = 0 को संतुष्ट करने वाला पृष्ठ ज्ञात कीजिए। (15 अंक)
(b) गॉस-सीडेल पुनरावृत्ति विधि से रैखिक समीकरण निकाय 7x₁ - x₂ + 2x₃ = 11, 2x₁ + 8x₂ - x₃ = 9, x₁ - 2x₂ + 9x₃ = 7 का 4 सार्थक अंकों तक सही हल ज्ञात कीजिए। आरंभिक अनुमानित हल x₁ = x₂ = x₃ = 0 लीजिए। (15 अंक)
(c) स्वतंत्रता की कोटि 2 के एक यांत्रिक तंत्र का लैग्रांजियन L = ½m(ẋ² + ẏ²) - ½m(w₁²x² + w₂²y²) + kxy है, जहाँ m, w₁, w₂, k अचर हैं। वह प्राचल θ ज्ञात कीजिए, जिसके लिए रूपांतरण x = q₁ cos θ - q₂ sin θ, y = q₁ sin θ + q₂ cos θ के अंतर्गत q₁, q₂ के पदों में लैग्रांजियन में गुणन पद q₁q₂ नहीं होगा। प्राचल θ से स्वतंत्र, q₁ तथा q₂ के सापेक्ष लग्रांज समीकरणों को ज्ञात कीजिए। (20 अंक)
Answer approach & key points
Solve all three sub-parts systematically: for (a) use Monge's auxiliary equations to find the general solution and apply boundary conditions; for (b) apply Gauss-Seidel iteration with proper convergence check to 4 significant figures; for (c) transform coordinates, eliminate cross terms by choosing θ = ½ tan⁻¹(2k/(m(w₁²-w₂²))), then derive uncoupled Lagrange equations. Allocate time proportionally: ~30% to (a), ~30% to (b), and ~40% to (c) given mark distribution.
For (a): Recognize PDE as parabolic type (B²-AC=0), use characteristic coordinates ξ=y+2x, η=y to reduce to canonical form and obtain general solution z = f(y+2x) + x·g(y+2x)
For (a): Apply boundary conditions on lines z=x=0 and z-1=x-y=0 to determine arbitrary functions f and g, yielding specific surface equation
For (b): Verify diagonal dominance of coefficient matrix, rewrite equations in iterative form, execute Gauss-Seidel iterations until convergence to 4 significant figures
For (c): Transform kinetic and potential energy terms using rotation matrix, collect coefficients of q₁q₂ and set to zero to find tan(2θ) = 2k/(m(w₁²-w₂²))
For (c): Obtain diagonalized Lagrangian L = ½m(Ω₁²q̇₁² + Ω₂²q̇₂²) - ½m(Ω₁²q₁² + Ω₂²q₂²) and derive two independent harmonic oscillator equations
For (c): Express final frequencies Ω₁² = ω₁²cos²θ + ω₂²sin²θ - (k/m)sin2θ and Ω₂² = ω₁²sin²θ + ω₂²cos²θ + (k/m)sin2θ in terms of original parameters
(a) (i) Find the conjunctive normal form (CNF) of the following Boolean function: f(x, y, z, t) = x · y · z + x̄ · y · (t + z̄) (15 marks)
(ii) Express the Boolean function f(x, y, z) = x + (x̄ · ȳ + x̄ · z) + z in disjunctive normal form (DNF) and construct the truth table for the function. (15 marks)
(b) A perfectly rough ball is at rest within a hollow cylindrical roller. The roller is drawn along a level path with uniform velocity V. Let a and b be the radii of the ball and the roller respectively. If V² > 27/7 g(b-a), then show that the ball will roll completely round the inside of the roller. (15 marks)
(c) Solve the partial differential equation a² ∂²u/∂x² = ∂²u/∂t², 0 < x < L, t > 0 subject to the conditions u(0,t) = 0, u(L,t) = 0, t > 0; u(x,0) = x, (∂u/∂t)ₜ₌₀ = 1, 0 < x < L. (20 marks)
हिंदी में पढ़ें
(क) (i) निम्न बूलिय फलन का योगात्मक प्रसामान्य स्वरूप (CNF) ज्ञात कीजिए : f(x, y, z, t) = x · y · z + x̄ · y · (t + z̄) (15 अंक)
(ii) बूलिय फलन f(x, y, z) = x + (x̄ · ȳ + x̄ · z) + z को विभोजनीय (डिस्जंक्टिव) प्रसामान्य स्वरूप (DNF) में व्यक्त कीजिए तथा इस फलन के लिए सत्यमान सारणी बनाइए। (15 अंक)
(ख) एक आदर्श रक्ष गेंद एक खोखले बेलनाकार रोलर में विराम की स्थिति में है। रोलर को एक समतल पथ के अनुदिश एकसमान वेग V से खींचा जाता है। मान लीजिए कि a तथा b क्रमशः गेंद तथा रोलर की त्रिज्याएँ हैं। यदि V² > 27/7 g(b-a) है, तब दर्शाइए कि गेंद रोलर के अन्दर पूर्ण रूप से घूम जाएगी। (15 अंक)
(ग) अंशिक अवकल समीकरण a² ∂²u/∂x² = ∂²u/∂t², 0 < x < L, t > 0 का शर्तों u(0,t) = 0, u(L,t) = 0, t > 0; u(x,0) = x, (∂u/∂t)ₜ₌₀ = 1, 0 < x < L से प्रतिबंधित हल ज्ञात कीजिए। (20 अंक)
Answer approach & key points
Solve this multi-part numerical problem by allocating approximately 30% time to part (a)(i) CNF conversion, 30% to part (a)(ii) DNF and truth table, 20% to part (b) mechanics proof, and 20% to part (c) wave equation solution. Begin with Boolean algebra manipulations using De Morgan's laws and canonical forms, then proceed to the constrained motion analysis with energy and angular momentum conservation, and finally apply separation of variables with Fourier series for the PDE. Present each sub-part clearly with proper mathematical notation and logical flow.
For (a)(i): Apply De Morgan's laws and distributive laws to convert f(x,y,z,t) = xyz + x̄y(t+z̄) into CNF; identify maxterms where f=0 and express as product of sums
For (a)(ii): Simplify f(x,y,z) = x + (x̄ȳ + x̄z) + z using Boolean identities, convert to DNF as sum of minterms, and construct complete 8-row truth table
For (b): Set up Lagrangian for ball in rolling cylinder with constraint of rolling without slipping; derive energy equation and condition for complete loop using effective potential
For (c): Apply separation of variables u(x,t) = X(x)T(t) to wave equation; determine eigenvalues λn = nπ/L and eigenfunctions sin(nπx/L); apply initial conditions u(x,0)=x and ut(x,0)=1 to find Fourier coefficients
For (c): Compute Fourier sine series coefficients bn = (2/L)∫₀ᴸ x sin(nπx/L)dx and cn = (2/nπa)∫₀ᴸ sin(nπx/L)dx for the complete solution
(a) Reduce the partial differential equation ∂²z/∂y² - ∂²z/∂x∂y + ∂z/∂x - ∂z/∂y(1+1/x) + z/x = 0 to canonical form. (15 marks)
(b) Compute a root of the equation log₁₀(2x+1) - x² + 3 = 0, in the interval [0, 3], by Regula-Falsi method, correct to 6 decimal places. (15 marks)
(c) Determine under what conditions the velocity field u = c(x² - y²), v = -2cxy, w = 0 is a solution to the Navier-Stokes momentum equations. Assuming that the conditions are met, determine the resulting pressure distribution, when z is up and the external body forces are Bₓ = 0 = Bᵧ, Bᵤ = -g. (20 marks)
हिंदी में पढ़ें
(क) आंशिक अवकल समीकरण ∂²z/∂y² - ∂²z/∂x∂y + ∂z/∂x - ∂z/∂y(1+1/x) + z/x = 0 को विहित रूप में समानीत कीजिए। (15 अंक)
(ख) मिथ्या-स्थिति (रेगुला-फाल्सि) विधि से अंतराल [0, 3] में, समीकरण log₁₀(2x+1) - x² + 3 = 0 के एक मूल का, दशमलव के 6 स्थानों तक सही, अभिकलन कीजिए। (15 अंक)
(ग) ज्ञात कीजिए कि किन शर्तों के अंतर्गत वेग क्षेत्र (velocity field) u = c(x² - y²), v = -2cxy, w = 0 नेवियर-स्टोक्स संवेग समीकरणों का एक हल है। यह मानते हुए कि शर्तें मान्य हैं, परिणामी दाब बंटन ज्ञात कीजिए, जब z ऊपर है तथा बाह्य पिंड बल Bₓ = 0 = Bᵧ, Bᵤ = -g हैं। (20 अंक)
Answer approach & key points
Solve this three-part numerical problem by allocating approximately 30% time to part (a) on PDE canonical reduction, 30% to part (b) on Regula-Falsi root-finding, and 40% to part (c) on Navier-Stokes verification and pressure determination. Begin with clear identification of equation types, proceed through systematic derivations and iterative calculations, and conclude with boxed final answers for each sub-part.
Part (a): Correct classification of the second-order PDE and identification of characteristic curves to transform to canonical form
Part (a): Proper substitution of new variables and reduction to standard canonical form (parabolic/hyperbolic/elliptic)
Part (b): Verification that f(0)·f(3) < 0 for root existence and correct Regula-Falsi iteration formula setup
Part (b): Iterative computation showing convergence to 6 decimal places with clear tabulation of iterations
Part (c): Verification of continuity equation (∇·u = 0) as necessary condition for Navier-Stokes solution
Part (c): Substitution into momentum equations to determine constraints on c and fluid properties
Part (c): Integration of pressure gradients to obtain p(x,y,z) with proper incorporation of body force B_z = -g