All 8 questions from UPSC Civil Services Mains Mathematics 2024 Paper II (400 marks total). Every stem reproduced in full, with directive-word analysis, marks, word limits, and answer-approach pointers.
8Questions
400Total marks
2024Year
Paper IIPaper
Topics covered
Group theory, complex analysis, convergence, linear programming (1)Convergence, group theory, complex analysis (1)Complex analysis, series differentiation, linear programming (1)Ring theory, Riemann integration, assignment problem (1)PDE, linear algebra, Boolean algebra, mechanics, fluid dynamics (1)Laplace equation, Boolean algebra, moment of inertia (1)PDE, numerical integration, fluid dynamics (1)PDE canonical form, interpolation, vortex dynamics (1)
A
Q1
50MCompulsoryproveGroup theory, complex analysis, convergence, linear programming
(a) Let G be a finite group of order mn, where m and n are prime numbers with m > n. Show that G has at most one subgroup of order m. 10 marks
(b) If w = f(z) is an analytic function of z, then show that
((∂²)/(∂ x²) + (∂²)/(∂ y²)) log |f'(z)| = 0. 10 marks
(c) Test the convergence of ∫limits₀² (log x)/(√((2-x))) dx . 10 marks
(d) If φ and ψ are functions of x and y satisfying Laplace equation, then show that f(z) = p + iq, i = √−1 is an analytic function, where p = (∂ φ)/(∂ y) - (∂ ψ)/(∂ x) and q = (∂ φ)/(∂ x) + (∂ ψ)/(∂ y) . 10 marks
(e) Use two phase method to solve the following linear programming problem :
Maximize z = x₁ + 2x₂
subject to x₁ - x₂ ≥ 3
2x₁ + x₂ ≤ 10
x₁, x₂ ≥ 0
10 marks
हिंदी में पढ़ें
(a) मान लीजिए कि कोटि mn का एक परिमित समूह G है, जहाँ m और n, (m > n) अभाज्य संख्याएँ हैं । दर्शाइए कि G का कोटि m का अधिक-से-अधिक एक उपसमूह है । 10 अंक
(b) यदि w = f(z), z का एक विसलेषिक फलन है, तब दर्शाइए कि
((∂²)/(∂ x²) + (∂²)/(∂ y²)) log |f'(z)| = 0 है । 10 अंक
(c) ∫limits₀² (log x)/(√((2-x))) dx के अभिसरण का परीक्षण कीजिए । 10 अंक
(d) यदि x तथा y के फलन φ और ψ लाप्लास समीकरण को सन्तुष्ट करते हैं, तो दर्शाइए कि f(z) = p + iq, i = √−1 एक विसलेषिक फलन है, जहाँ p = (∂ φ)/(∂ y) - (∂ ψ)/(∂ x) तथा q = (∂ φ)/(∂ x) + (∂ ψ)/(∂ y) हैं । 10 अंक
(e) निम्नलिखित रैखिक प्रोग्रामन समस्या को हल करने के लिए द्विचरण विधि का उपयोग कीजिए :
अधिकतमीकरण कीजिए z = x₁ + 2x₂
बशर्ते कि x₁ - x₂ ≥ 3
2x₁ + x₂ ≤ 10
x₁, x₂ ≥ 0
10 अंक
Answer approach & key points
(a) justify: claim > 3-4 reasons > evidence > conclusion | (b) derive: given > assumptions > stepwise derivation > result > check | (c) calculate: given > formula > substitution > result with units > interpretation | (d) justify: claim > 3-4 reasons > evidence > conclusion | (e) calculate: given > formula > substitution > result with units > interpretation Full marks: All parts solved with clear, step-by-step derivations and correct application of theorems.
Subgroup of order m is a Sylow m-subgroup
Number s ≡ 1 (mod m) and s | n
s = n impossible since m ∤ (n − 1) as n − 1 < m
Hence s = 1: unique (normal) subgroup of order m
f′ analytic; assume f′(z) ≠ 0
log f′ = log|f′| + i arg f′ locally analytic
Real part of analytic function is harmonic (via CR equations)
Conclusion
Identify singular points x = 0 and x = 2
Near 0: compare with ∫|log x| (or x^(−1/2)) — convergent
Near 2: compare with (2 − x)^(−1/2) — convergent
Conclusion: convergent
Compute p_x, q_y, p_y, q_x
Use Laplace equations and equality of mixed partials
Verify CR equations p_x = q_y, p_y = −q_x
Continuity of partials ⇒ analytic
Standard form with surplus, slack and artificial variables
Phase I: minimise a₁; x₁ enters, a₁ leaves, W = 0
Phase II: x₂ enters, s₂ leaves
Optimum x₁ = 13/3, x₂ = 4/3, z = 7 with optimality check
50MproveConvergence, group theory, complex analysis
(a) Using Cauchy's general principle of convergence, examine the convergence of the sequence < fₙ >, where fₙ = 1 + 1/1! + 1/2! + ... + 1/n!.
15 marks
(b) Show that every homomorphic image of an abelian group is abelian, but the converse is not necessarily true.
15 marks
(c) Find the function which is analytic inside and on the circle C : z = e^(iθ), 0 ≤ θ ≤ 2π and has the value
(a² - 1) cos θ + i(a² + 1) sin θ
──────────────────────────────
a⁴ - 2a² cos 2θ + 1
on the circumference of C, where a² > 1.
20 marks
हिंदी में पढ़ें
(a) कोशी के अभिसरण के व्यापक (जनरल) सिद्धांत का उपयोग करते हुए, अनुक्रम < fₙ > के अभिसरण की जाँच कीजिए, जहाँ fₙ = 1 + 1/1! + 1/2! + ... + 1/n! ।
15 अंक
(b) दर्शाइए कि एक अबेली समूह का प्रत्येक समाकारी प्रतिबिंब अबेली है, लेकिन इसका विपरीत आवश्यक रूप से सत्य नहीं है ।
15 अंक
(c) वह फलन ज्ञात कीजिए जो कि वृत्त C : z = e^(iθ), 0 ≤ θ ≤ 2π, के अंदर तथा उसके ऊपर विश्लेषिक है और C की परिधि पर जिसका मान
(a² - 1) cos θ + i(a² + 1) sin θ
──────────────────────────────
a⁴ - 2a² cos 2θ + 1
है, जहाँ a² > 1 है ।
20 अंक
Answer approach & key points
(a) examine: intro > how/why with reasoning > evidence > conclusion | (b) justify: claim > 3-4 reasons > evidence > conclusion | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Rigorous application of theorems with complete derivations and clear counterexamples.
Statement of Cauchy's criterion
|f_m − f_n| = Σ_{k=n+1}^{m} 1/k!
Bound k! ≥ 2^(k−1) giving |f_m − f_n| < 1/2^(n−1)
Choice of N for given ε; conclusion convergent (limit e)
Take u = φ(x), v = φ(y) in φ(G)
uv = φ(xy) = φ(yx) = vu
Counter-example: non-abelian group with abelian image (e.g., S₃ → {±1} via sign, or G → G/G′)
Conclusion
Substitutions cos θ = (z + z⁻¹)/2, sin θ = (z − z⁻¹)/2i, cos 2θ = (z² + z⁻²)/2
50MsolveComplex analysis, series differentiation, linear programming
(a) Locate the poles and their order for the function f(z) = 1/[z(sin πz)(z + 1/2)]. Also, find the residue of f(z) at these poles. (15 marks)
(b) Consider the series Σ(n=1 to ∞) U_n(x), 0 ≤ x ≤ 1, the sum of whose first n terms is given by S_n(x) = (1/2n²)log(1 + n⁴x²), x ∈ [0,1]. Show that the given series can be differentiated term-by-term, though Σ(n=1 to ∞) U'_n(x), does not converge uniformly on [0,1]. (20 marks)
(c) Using duality principle, solve the following linear programming problem: Minimize z = 4x₁ + 3x₂ + x₃ subject to x₁ + 2x₂ + 4x₃ ≥ 12, 3x₁ + 2x₂ + x₃ ≥ 8, x₁, x₂, x₃ ≥ 0. (15 marks)
हिंदी में पढ़ें
(a) फलन f(z) = 1/[z(sin πz)(z + 1/2)] के अनंतक तथा उनकी घात (ऑर्डर) का पता लगाइए। इन अनंतकों पर f(z) के अवशेष भी ज्ञात कीजिए। (15 अंक)
(b) श्रेणी Σ(n=1 to ∞) U_n(x), 0 ≤ x ≤ 1 का विचार कीजिए, जिसके पहले n पदों का योगफल S_n(x) = (1/2n²)log(1 + n⁴x²), x ∈ [0,1] के द्वारा दिया गया है। दर्शाइए कि दी गई श्रेणी को पद-दर-पद अवकलित किया जा सकता है, यद्यपि Σ(n=1 to ∞) U'_n(x), [0,1] पर एकसमान अभिसरित नहीं होती है। (20 अंक)
(c) द्वैत सिद्धांत का उपयोग करते हुए, निम्नलिखित रैखिक प्रोग्रामन समस्या को हल कीजिए: न्यूनतमीकरण कीजिए z = 4x₁ + 3x₂ + x₃ बशर्ते कि x₁ + 2x₂ + 4x₃ ≥ 12, 3x₁ + 2x₂ + x₃ ≥ 8, x₁, x₂, x₃ ≥ 0। (15 अंक)
Answer approach & key points
(a) calculate: given > formula > substitution > result with units > interpretation | (b) justify: claim > 3-4 reasons > evidence > conclusion | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Complete derivations, correct theorems cited, all parts solved with verification.
Double pole at z = 0; simple poles at z = n (n ≠ 0 integer) and z = −1/2
Res at n: (−1)ⁿ/[πn(n + 1/2)] = 2(−1)ⁿ/[πn(2n + 1)]
Res at −1/2 = 2
Res at 0 = −4/π via Laurent expansion or derivative formula
50MproveRing theory, Riemann integration, assignment problem
(a) Consider the polynomial ring Z[x] over the ring Z of integers. Let S be an ideal of Z[x] generated by x. Show that S is prime but not a maximal ideal of Z[x]. (15 marks)
(b) Find the upper and lower Riemann integrals for the function f defined on [0, 1] as follows: f(x) = (1-x²)^½, if x is rational; f(x) = (1-x), if x is irrational. Hence, show that f is not Riemann integrable on [0, 1]. (15 marks)
(c) The personnel manager of a company wants to assign officers A, B and C to the regional offices at Delhi, Mumbai, Kolkata and Chennai. The cost of relocation (in thousand Rupees) of the three officers at the four regional offices are given below:
| Officer | Delhi | Mumbai | Kolkata | Chennai |
|---------|-------|--------|---------|---------|
| A | 16 | 22 | 24 | 20 |
| B | 10 | 32 | 26 | 16 |
| C | 10 | 20 | 46 | 30 |
Find the assignment which minimizes the total cost of relocation and also determine the minimum cost. (20 marks)
हिंदी में पढ़ें
(a) पूर्णांकों के वलय Z पर बहुपद वलय Z[x] का विचार कीजिए। मान लीजिए x द्वारा जनित Z[x] की एक गुणजावली S है। दर्शाइए कि S, Z[x] की एक अभाज्य गुणजावली है लेकिन उच्चिष्ट गुणजावली नहीं है। (15 अंक)
(b) [0, 1] पर परिभाषित निम्नलिखित फलन f के लिए ऊपर तथा निम्न रीमान समाकल ज्ञात कीजिए: f(x) = (1-x²)^½, यदि x परिमेय है; f(x) = (1-x), यदि x अपरिमेय है। अतः दर्शाइए कि [0, 1] पर f रीमान समाकलनीय नहीं है। (15 अंक)
(c) एक कंपनी का कर्मिक प्रबंधक, अधिकारियों A, B और C को क्षेत्रीय कार्यालयों दिल्ली, मुंबई, कोलकाता और चेन्नई में नियुक्त करना चाहता है। चार क्षेत्रीय कार्यालयों में इन तीन अधिकारियों के स्थानांतरण की लागत (हजार रुपयों में) नीचे दी गई है: [तालिका दी गई है] वह नियतन (असाइनमेंट) ज्ञात कीजिए, जो स्थानांतरण की कुल लागत को न्यूनतम करता है और न्यूनतम लागत भी निर्धारित कीजिए। (20 अंक)
Answer approach & key points
(a) justify: claim > 3-4 reasons > evidence > conclusion | (b) calculate: given > formula > substitution > result with units > interpretation | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Rigorous proofs with named theorems; complete Hungarian method steps.
Evaluation map p ↦ p(0) is a surjective homomorphism Z[x] → Z with kernel (x)
Z[x]/(x) ≅ Z, an integral domain ⇒ (x) prime
Z not a field ⇒ (x) not maximal
Explicit chain (x) ⊊ (2, x) ⊊ Z[x]
Density of rationals and irrationals in each subinterval
√(1 − x²) ≥ 1 − x on [0, 1], so M_i and m_i come from the respective functions
Upper integral = ∫₀¹ √(1 − x²) dx = π/4
Lower integral = ∫₀¹ (1 − x) dx = 1/2; unequal ⇒ not integrable
Dummy row with zero costs to balance 4 × 4
Hungarian method: row/column reduction and covering lines
Optimal assignment A → Mumbai, B → Chennai, C → Delhi (Kolkata vacant)
50MCompulsorysolvePDE, 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
(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) calculate: given > formula > substitution > result with units > interpretation | (e) calculate: given > formula > substitution > result with units > interpretation Full marks: Complete derivations with all steps, correct results, and verification.
Chain rule for ξ, η with ξ_y = iα, η_y = −iα, ξ_t = η_t = −k
u_xx + u_yy = (1 − α²)(f″ + g″)
u_tt = k²(f″ + g″)
Use 1 − α² = k²/C² to conclude
Augmented matrix
Row operations to echelon form (z = 3)
Back to reduced row-echelon form
x = 1, y = 2, z = 3 with check
Binary 1000 1101 and 1111 1111; MSB as sign
(8D)₁₆ = −13, (FF)₁₆ = −127
Positional expansion of 9B2.1A
(9B2.1A)₁₆ = 2482.1015625
No external horizontal force ⇒ centre of mass fixed
Initial and final positions of man and board centre
M(−a) = M(s + a) + ms
Distance 2Ma/(M + m), opposite to the man's walk
Velocity components (x, y, −2z) (sign convention stated)
50MderiveLaplace 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
(a) derive: given > assumptions > stepwise derivation > result > check | (b) describe: define > structure or process in order > labelled diagram > significance | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Complete derivations with all steps justified and verified.
Fourier transform in x: Φ″ − k²Φ = 0
Bounded solution Φ = F(k)e^(−|k|y)
Inverse transform and ∫e^(−|k|y)e^(iks)dk = 2y/(y² + s²)
φ = (y/π)∫ f(ξ)dξ/[y² + (x − ξ)²]; boundary check as y → 0⁺
Simplification Y = B(C̄ + Ā) = B·(AC)̄ (or circuit of unsimplified form)
Circuit drawn with NAND/AND (or NOT/OR/AND) gates
Bit-by-bit evaluation of the eight positions
Output Y = 00111000
Elliptic polar substitution x = ar cos θ, y = br sin θ, J = abr
(a) Find the integral surface of the following quasi-linear equation (y - φ) (∂ φ)/(∂ x) + (φ - x) (∂ φ)/(∂ y) = x - y, which passes through the curve φ = 0, xy = 1 and through the circle x + y + φ = 0, x² + y² + φ² = a². (15 marks)
(b) Integrate f(x) = 5x³ - 3x² + 2x + 1 from x = -2 to x = 4 using (i) Simpson's 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) = (B(x² - y²))/((x² + y²)²), v(x, y) = 2Bxy/((x² + y²)²), 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 - φ) (∂ φ)/(∂ x) + (φ - x) (∂ φ)/(∂ y) = x - y, का वह समाकल पृष्ठ ज्ञात कीजिए, जो कि वक्र φ = 0, xy = 1 और वृत्त x + y + φ = 0, x² + y² + φ² = a² से होकर गुजरता है। (15 अंक)
(b) (i) चौड़ाई h = 1 के साथ सिम्पसन के 3/8 नियम, और (ii) चौड़ाई h = 1 के साथ समलंबी (ट्रेपिजॉइडल) नियम का उपयोग करके f(x) = 5x³ - 3x² + 2x + 1 का x = -2 से x = 4 तक समाकलन कीजिए। (15 अंक)
(c) मान लीजिए कि वेग क्षेत्र u(x, y) = (B(x² - y²))/((x² + y²)²), v(x, y) = 2Bxy/((x² + y²)²), 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
(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: Complete derivations with all steps shown, correct application of theorems, verification of results, and clear presentation.
Lagrange auxiliary equations and integrals c₁ = x + y + φ, c₂ = x² + y² + φ²
Circle has c₁ = 0, c₂ = a² constant ⇒ it is a characteristic; infinitely many surfaces (e.g., sphere x² + y² + φ² = a², plane x + y + φ = 0, or F(c₁, c₂) = 0 with F(0, a²) = 0)
Verification of the surface or clear statement of non-uniqueness
(a) Solve the partial differential equation (∂)/(∂ y)((∂ φ)/(∂ x) + φ) + 2x^2y((∂ φ)/(∂ x) + φ) = 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 (x² + (y-a)²)/(x² + (y+a)²) + 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 (∂)/(∂ y)((∂ φ)/(∂ x) + φ) + 2x^2y((∂ φ)/(∂ x) + φ) = 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 (x² + (y-a)²)/(x² + (y+a)²) + 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
(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: Complete derivations with all steps justified and verified.
Classification: B² − 4AC = 1 > 0, hyperbolic; characteristics x = const, y = const
Canonical form in ξ = x, η = y
Substitute v = φ_ξ + φ, solve v = F(ξ)e^(−ξ²η²)
φ = e^(−x)[∫ F(s)e^(s − s²y²) ds + G(y)]
Difference table from the data
x₀ = 2 (or 1), p chosen correctly
Δf₀ = 7, Δ²f₀ = 12, Δ³f₀ = 6 (from x₀ = 2)
f(2.5) = 3.375
Vortices ±Γ at (0, ±a); ψ = (Γ/2π) ln(r₂/r₁)
Velocity of the pair U = Γ/(4πa) along x-axis
Relative stream function ψ − Uy = const
Result log[(x² + (y − a)²)/(x² + (y + a)²)] + y/a = C