Q1 50M Compulsory solve Number theory, analysis, complex analysis, linear programming
(a) Let $m_1, m_2, \cdots, m_k$ be positive integers and $d > 0$ the greatest common divisor of $m_1, m_2, \cdots, m_k$. Show that there exist integers $x_1, x_2, \cdots, x_k$ such that $$d = x_1m_1 + x_2m_2 + \cdots + x_km_k$$ (10 marks)
(b) Test the uniform convergence of the series $$x^4 + \frac{x^4}{1+x^4} + \frac{x^4}{(1+x^4)^2} + \frac{x^4}{(1+x^4)^3} + \cdots$$ on $[0, 1]$. (10 marks)
(c) If a function $f$ is monotonic in the interval $[a, b]$, then prove that $f$ is Riemann integrable in $[a, b]$. (10 marks)
(d) Let $c : [0, 1] \to \mathbb{C}$ be the curve, where $c(t) = e^{4\pi it}$, $0 \leq t \leq 1$. Evaluate the contour integral $\displaystyle\int_c \frac{dz}{2z^2 - 5z + 2}$. (10 marks)
(e) A department of a company has five employees with five jobs to be performed. The time (in hours) that each man takes to perform each job is given in the effectiveness matrix. Assign all the jobs to these five employees to minimize the total processing time:
Employees
I II III IV V
A 10 5 13 15 16
B 3 9 18 13 6
Jobs C 10 7 2 2 2
D 7 11 9 7 12
E 7 9 10 4 12
(10 marks)
हिंदी में पढ़ें
(a) मान लीजिए कि $m_1, m_2, \cdots, m_k$ धनात्मक पूर्णांक हैं तथा $d > 0$, $m_1, m_2, \cdots, m_k$ का महत्तम समापवर्तक है। दर्शाइए कि ऐसे पूर्णांक $x_1, x_2, \cdots, x_k$ अस्तित्व में हैं ताकि $$d = x_1m_1 + x_2m_2 + \cdots + x_km_k$$ (10 अंक)
(b) श्रेणी $$x^4 + \frac{x^4}{1+x^4} + \frac{x^4}{(1+x^4)^2} + \frac{x^4}{(1+x^4)^3} + \cdots$$ के $[0, 1]$ पर एकसमान अभिसरण की जाँच कीजिए। (10 अंक)
(c) यदि एक फलन $f$, अन्तराल $[a, b]$ में एकदिशे है, तब सिद्ध कीजिए कि $f$, $[a, b]$ में रीमान समाकलनीय है। (10 अंक)
(d) मान लीजिए कि $c : [0, 1] \to \mathbb{C}$, $c(t) = e^{4\pi it}$, $0 \leq t \leq 1$ के द्वारा परिभाषित एक वक्र है। कन्टूर समाकल $$\int_c \frac{dz}{2z^2 - 5z + 2}$$ का मान निकालिए। (10 अंक)
(e) एक कम्पनी के एक विभाग के पाँच कर्मचारियों को पाँच कार्य सम्पन्न करने हैं। जितना समय (घंटों में) एक व्यक्ति एक कार्य को सम्पन्न करने के लिए लेता है, वह प्रभाविता आव्यूह में दिया गया है। इन पाँच कर्मचारियों को इन सभी कार्यों को इस तरह निर्धारित कीजिए जिससे कि समस्त कार्य सम्पन्न करने का समय न्यूनतम हो :
कर्मचारी
| | I | II | III | IV | V |
|---|---|---|---|---|---|
| A | 10 | 5 | 13 | 15 | 16 |
| B | 3 | 9 | 18 | 13 | 6 |
| C | 10 | 7 | 2 | 2 | 2 |
| D | 7 | 11 | 9 | 7 | 12 |
| E | 7 | 9 | 10 | 4 | 12 |
कार्य
(10 अंक)
Answer approach & key points
Solve all five sub-parts systematically, allocating approximately 20% time each since all carry equal marks. For (a), apply the Extended Euclidean Algorithm or ideal-theoretic proof; for (b), identify the geometric series and test convergence at x=0; for (c), use Darboux's theorem or direct ε-δ argument with partition refinement; for (d), apply residue theorem after factorizing the denominator and checking pole locations relative to the curve (unit circle traversed twice); for (e), execute the Hungarian algorithm with row/column reductions. Present each solution with clear theorem citations and boxed final answers.
- (a) Correctly states and applies Bézout's identity/Extended Euclidean Algorithm for k integers, showing d generates the ideal (m₁,...,mₖ)
- (b) Identifies geometric series with ratio 1/(1+x⁴), finds pointwise limit function (0 for x>0, 1 for x=0), and proves non-uniform convergence via supremum norm or discontinuity of limit
- (c) Proves monotonic functions have at most countably many discontinuities (or uses Darboux integrability criterion), establishes upper and lower sums converge
- (d) Factorizes denominator as (2z-1)(z-2), identifies poles at z=½ and z=2, determines only z=½ lies inside |z|=1 (traversed twice), computes residue correctly
- (e) Correctly applies Hungarian algorithm: row reduction, column reduction, minimum lines covering zeros, optimal assignment with minimum total time calculation
Q2 50M solve Calculus, field theory, complex analysis
(a) Find the maximum and minimum values of f(x) = x³ - 9x² + 26x - 24 for 0 ≤ x ≤ 1. (15 marks)
(b) Let F be a field and f(x) ∈ F[x] a polynomial of degree > 0 over F. Show that there is a field F' and an imbedding q : F → F' s.t. the polynomial f^q ∈ F'[x] has a root in F', where f^q is obtained by replacing each coefficient a of f by q(a). (15 marks)
(c) Find the Laurent series expansion of f(z) = (z² - z + 1)/[z(z² - 3z + 2)] in the powers of (z+1) in the region |z+1| > 3. (20 marks)
हिंदी में पढ़ें
(a) f(x) = x³ - 9x² + 26x - 24 का, 0 ≤ x ≤ 1 के लिए, अधिकतम तथा न्यूनतम मान निकालिए। (15 अंक)
(b) मान लीजिए कि F एक क्षेत्र है तथा f(x) ∈ F[x], क्षेत्र F के ऊपर घात > 0 का एक बहुपद है। दर्शाइए कि एक क्षेत्र F' तथा एक अंतःस्थापन q : F → F' इस प्रकार से अस्तित्व में है कि बहुपद f^q ∈ F'[x] का एक मूल F' में है, जहाँ f^q, f के प्रत्येक गुणांक a को q(a) द्वारा प्रतिस्थापित करने से प्राप्त होता है। (15 अंक)
(c) क्षेत्र |z+1| > 3 में f(z) = (z² - z + 1)/[z(z² - 3z + 2)] का लॉरें श्रेणी प्रसार, (z+1) की घातों में ज्ञात कीजिए। (20 अंक)
Answer approach & key points
Solve this three-part problem by allocating time proportionally to marks: approximately 15 minutes for part (a) on cubic optimization, 15 minutes for part (b) on field extension theory, and 20 minutes for part (c) on Laurent series expansion. Begin each part with clear statement of the mathematical approach, show all working steps with proper justification, and conclude with boxed final answers. For part (c), explicitly note the substitution w = z+1 and verify convergence in the specified annular region.
- Part (a): Correctly find f'(x) = 3x² - 18x + 26, determine no critical points in [0,1] since discriminant < 0 and f'(x) > 0 throughout, hence extrema occur at endpoints with f(0) = -24 (minimum) and f(1) = -6 (maximum)
- Part (b): Construct the field extension F' = F[x]/(p(x)) where p(x) is an irreducible factor of f(x), define the natural embedding q: F → F', and prove that the coset α = x + (p(x)) is a root of f^q in F' using the evaluation homomorphism
- Part (c): Substitute z = w - 1 where w = z + 1, rewrite f(z) in terms of w, perform partial fraction decomposition, and expand each term as geometric series valid for |w| > 3 (i.e., |z+1| > 3)
- Part (c): Identify singularities at z = 0, 1, 2 which correspond to w = 1, 2, 3, confirming |w| > 3 excludes all singularities and ensures convergence
- Part (c): Obtain Laurent series with only negative powers of w (analytic part vanishes), presenting coefficients explicitly as rational numbers
Q3 50M prove Complex analysis, optimization and Lagrange multipliers
(a) Let f be an entire function whose Taylor series expansion with centre z = 0 has infinitely many terms. Show that z = 0 is an essential singularity of f(1/z). (15 marks)
(b) Find the stationary values of $x^2 + y^2 + z^2$ subject to the conditions $ax^2 + by^2 + cz^2 = 1$ and $lx + my + nz = 0$. Interpret the result geometrically. (20 marks)
(c) Convert the following LPP into dual LPP : Minimize $Z = x_1 - 3x_2 - 2x_3$ subject to $$3x_1 - x_2 + 2x_3 \leq 7$$ $$2x_1 - 4x_2 \geq 12$$ $$-4x_1 + 3x_2 + 8x_3 = 10$$ where $x_1, x_2 \geq 0$ and $x_3$ is unrestricted in sign. (15 marks)
हिंदी में पढ़ें
(a) मान लीजिए कि f एक सर्वत्र वैश्लेषिक फलन है जिसके केन्द्र z = 0 पर टेलर श्रेणी प्रसार में अपरिमित रूप से अनेक पद हैं। दर्शाइए कि f(1/z) की z = 0 एक अनिवार्य विचित्रता है। (15 अंक)
(b) शर्तों $ax^2 + by^2 + cz^2 = 1$ तथा $lx + my + nz = 0$ से प्रतिबंधित $x^2 + y^2 + z^2$ के स्थाय (अचर) मान निकालिए। परिणाम की ज्यामितीय व्याख्या कीजिए। (20 अंक)
(c) निम्न रैखिक प्रोग्राम समस्या को द्वैती रैखिक प्रोग्राम समस्या में परिवर्तित कीजिए : न्यूनतमीकरण कीजिए $Z = x_1 - 3x_2 - 2x_3$ बशर्ते कि $$3x_1 - x_2 + 2x_3 \leq 7$$ $$2x_1 - 4x_2 \geq 12$$ $$-4x_1 + 3x_2 + 8x_3 = 10$$ जहाँ $x_1, x_2 \geq 0$ तथा $x_3$ का चिह्न अप्रतिबंधित है। (15 अंक)
Answer approach & key points
The directive 'prove' for part (a) and 'find' for parts (b)-(c) demand rigorous mathematical demonstration. Allocate approximately 30% time to part (a) establishing the essential singularity via Laurent series analysis, 40% to part (b) solving the constrained optimization with Lagrange multipliers and geometric interpretation, and 30% to part (c) systematically converting the primal LPP to dual form. Structure as: brief statement of key theorems → step-by-step derivation for each part → concluding verification of results.
- Part (a): Taylor series of f(z) with infinitely many terms implies Laurent series of f(1/z) has infinitely many negative powers, proving z=0 is essential singularity via Casorati-Weierstrass or definition
- Part (b): Correct formulation of Lagrangian with two multipliers λ and μ; derivation of 5 equations from partial derivatives; elimination leading to characteristic equation for stationary values
- Part (b): Geometric interpretation as finding extremal distances from origin to intersection of ellipsoid and plane (ellipse), yielding maximum and minimum distance squared values
- Part (c): Conversion of unrestricted x₃ to x₃⁺ - x₃⁻; transformation of ≥ constraint to ≤ by sign reversal; proper assignment of dual variables (y₁ ≥ 0, y₂ ≤ 0, y₃ unrestricted)
- Part (c): Correct dual objective (maximize) with right-hand side coefficients and proper constraint coefficients from primal transpose, verifying weak duality structure
Q4 50M solve Group theory, complex integration and linear programming
(a) Show that there are infinitely many subgroups of the additive group $\mathbb{Q}$ of rational numbers. (15 marks)
(b) Using contour integration, evaluate the integral $\int_{-\infty}^{\infty} \frac{\sin x \, dx}{x(x^2+a^2)}$, $a > 0$. (20 marks)
(c) Solve the following linear programming problem using Big M method : Maximize Z = 4x₁ + 5x₂ + 2x₃ subject to 2x₁ + x₂ + x₃ ≥ 10, x₁ + 3x₂ + x₃ ≤ 12, x₁ + x₂ + x₃ = 6, x₁, x₂, x₃ ≥ 0. (15 marks)
हिंदी में पढ़ें
(a) दर्शाइए कि परिमेय संख्याओं के योज्य समूह $\mathbb{Q}$ के अपरिमित रूप से अनेक उपसमूह हैं। (15 अंक)
(b) कंटूर समाकलन का उपयोग कर समाकलन $\int_{-\infty}^{\infty} \frac{\sin x \, dx}{x(x^2+a^2)}$, $a > 0$ का मान ज्ञात कीजिए। (20 अंक)
(c) बड़ा M (बिग M) विधि का उपयोग करके निम्नलिखित रैखिक प्रोग्राम समस्या को हल कीजिए : अधिकतमीकरण कीजिए $Z = 4x_1 + 5x_2 + 2x_3$ बशर्ते कि $2x_1 + x_2 + x_3 \geq 10$, $x_1 + 3x_2 + x_3 \leq 12$, $x_1 + x_2 + x_3 = 6$, $x_1, x_2, x_3 \geq 0$। (15 अंक)
Answer approach & key points
Solve this multi-part problem by allocating time proportionally to marks: approximately 30% (15 minutes) for part (a) on infinite subgroups of (Q,+), 40% (20 minutes) for part (b) on contour integration, and 30% (15 minutes) for part (c) on Big M method. Begin each part with clear statement of approach, show complete working with proper mathematical justification, and conclude with explicit final answers. For (b), explicitly state contour choice and residue calculations; for (c), present the simplex tableaux clearly.
- Part (a): Construct explicit infinite family of subgroups, such as H_n = {m/n^k : m ∈ Z, k ≥ 0} for fixed n > 1, or Z[1/p] for varying primes p, proving closure under addition and inverses
- Part (a): Prove distinctness of infinitely many subgroups by showing H_n ≠ H_m for n ≠ m, or using prime-based constructions
- Part (b): Identify integrand has simple pole at z=0 and simple poles at z=±ia, choose semicircular contour in upper half-plane, handle pole on real axis via principal value
- Part (b): Apply residue theorem correctly: compute Res(f, ia) and half-residue at z=0, combine to get π(1-e^{-a})/a² for the sine integral
- Part (c): Convert to standard form using surplus, slack, and artificial variables with Big M penalty: minimize W = -4x₁-5x₂-2x₃ + M(a₁+a₂) or equivalent
- Part (c): Execute simplex iterations showing entering and leaving variables, pivot operations, until optimality reached with x₁=3, x₂=0, x₃=3, Z=18