Q1 50M Compulsory prove Group 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
$$(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) \log |f'(z)| = 0.$$ 10 marks
(c) Test the convergence of $$\int\limits_{0}^{2} \frac{\log x}{\sqrt{(2-x)}} dx$$ . 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 = $$\frac{\partial \phi}{\partial y} - \frac{\partial \psi}{\partial x}$$ and q = $$\frac{\partial \phi}{\partial x} + \frac{\partial \psi}{\partial y}$$ . 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
Answer approach & key points
Prove the five mathematical statements systematically, allocating approximately 20% time to each sub-part since all carry equal marks. For (a), apply Sylow's theorems or Lagrange's theorem with counting arguments; for (b), use Cauchy-Riemann equations and harmonic function properties; for (c), apply limit comparison test or Dirichlet's test for improper integrals; for (d), verify Cauchy-Riemann equations using the given harmonic functions; for (e), execute Phase I (artificial variables) and Phase II (simplex method) with clear tableau presentations. Structure as five distinct proofs with clear headings, showing all logical steps without skipping to conclusions.
- For (a): Apply Sylow's third theorem or direct counting argument using Lagrange's theorem to show uniqueness of subgroup of order m, noting that number of such subgroups divides n and is ≡ 1 (mod m), forcing exactly one subgroup
- For (b): Express log|f'(z)| in terms of real and imaginary parts, apply Laplacian operator, and use analyticity of f(z) (Cauchy-Riemann equations) to show the sum of second partial derivatives vanishes identically
- For (c): Identify singularities at x=0 and x=2, split integral, apply limit comparison test with appropriate standard integrals (e.g., 1/x^α near 0 and 1/(2-x)^β near 2) to establish convergence
- For (d): Verify that p and q satisfy Cauchy-Riemann equations (∂p/∂x = ∂q/∂y and ∂p/∂y = -∂q/∂x) using the given Laplace equations for φ and ψ, and equality of mixed partials
- For (e): Phase I: Introduce surplus and slack variables, add artificial variables for ≥ constraint, minimize sum of artificials; Phase II: Remove artificials, use final Phase I tableau to maximize original objective, obtaining optimal solution x₁=13/3, x₂=4/3, z=7
Q2 50M prove Convergence, 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
Answer approach & key points
Prove the three mathematical statements systematically: for (a) apply Cauchy's general principle by showing |f_{n+p} - f_n| < ε for n ≥ m; for (b) prove the homomorphism property for abelian groups and provide a concrete counterexample for the converse; for (c) use Poisson's integral formula or Schwarz's formula to construct the analytic function from boundary values. Allocate approximately 30% time to (a), 30% to (b), and 40% to (c) given their mark distribution and complexity.
- For (a): Correct application of Cauchy's general principle showing |f_{n+p} - f_n| = Σ_{k=n+1}^{n+p} 1/k! < 1/n!·(e-1) → 0 as n → ∞, establishing convergence to e
- For (b): Proof that φ(ab) = φ(a)φ(b) = φ(b)φ(a) = φ(ba) for homomorphism φ: G → H when G is abelian; counterexample using S₃ (non-abelian) with abelian quotient S₃/A₃ ≅ ℤ₂
- For (c): Recognition that boundary data represents real part of analytic function; application of Schwarz's formula or Poisson integral to find harmonic conjugate
- For (c): Simplification of denominator using a⁴ - 2a²cos2θ + 1 = |a² - e^{2iθ}|² and identification with Re[(a²+z²)/(a²-z²)] or similar standard form
- For (c): Final analytic function f(z) = (a²+z²)/(a²-z²) or equivalent verified on |z|=1
- Clear logical flow with proper mathematical notation and statement of theorems used
Q3 50M solve Complex 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)
Answer approach & key points
Solve this multi-part numerical problem by allocating approximately 30% time to part (a) on complex analysis, 40% to part (b) on series differentiation, and 30% to part (c) on linear programming via duality. Begin each part with clear identification of the mathematical technique required, proceed through systematic computation with explicit formula citations, and conclude with verified numerical answers for residues, convergence justification, and optimal primal/dual values.
- For (a): Identify all poles at z = 0, z = -1/2, and z = n (n ∈ ℤ, n ≠ 0) with correct orders; calculate residues using Laurent series or limit formulas, especially handling the simple pole at z = -1/2 and double pole at z = 0
- For (a): Correctly classify z = 0 as a double pole (order 2) due to the combined effect of z and sin(πz), and simple poles at z = n for non-zero integers
- For (b): Derive U_n(x) = S_n(x) - S_{n-1}(x), show term-by-term differentiation is valid by verifying uniform convergence of ΣU_n(x) and conditions for differentiation under the series
- For (b): Demonstrate that ΣU'_n(x) does not converge uniformly on [0,1] by analyzing the behavior at x = 0 versus x > 0, particularly showing sup-norm of partial sums does not tend to zero
- For (c): Formulate the dual LP correctly (maximize w = 12y₁ + 8y₂ subject to y₁ + 3y₂ ≤ 4, 2y₁ + 2y₂ ≤ 3, 4y₁ + y₂ ≤ 1, y₁, y₂ ≥ 0), solve using graphical or simplex method, and apply complementary slackness to recover primal optimal solution
- For (c): Verify strong duality by confirming equal optimal values and interpret shadow prices for the primal constraints
Q4 50M prove Ring 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)
Answer approach & key points
Prove the three mathematical claims systematically: for (a) establish primality via quotient ring isomorphism to Z and non-maximality by embedding in (x,2); for (b) compute upper and lower Darboux sums using density of rationals and irrationals, showing unequal integrals; for (c) solve the unbalanced assignment problem by adding a dummy officer with zero costs, then apply Hungarian algorithm. Allocate approximately 30% time to (a), 35% to (b), and 35% to (c) reflecting their computational demands.
- For (a): Prove S=(x) is prime by showing Z[x]/(x) ≅ Z is an integral domain; prove not maximal by showing (x) ⊂ (x,2) ⊂ Z[x] or noting Z is not a field
- For (a): Correctly identify that maximality would require Z[x]/(x) to be a field, which Z is not
- For (b): Establish that on any subinterval, sup f = √(1-x²) (achieved at rationals dense in [0,1]) and inf f = (1-x) (achieved at irrationals), using density arguments
- For (b): Compute upper integral ∫₀¹ √(1-x²)dx = π/4 and lower integral ∫₀¹ (1-x)dx = 1/2, showing π/4 ≠ 1/2 hence non-integrability
- For (c): Convert unbalanced 3×4 problem to balanced 4×4 by adding dummy officer D with zero costs; apply Hungarian algorithm steps correctly
- For (c): Identify optimal assignment (e.g., A-Chennai, B-Delhi, C-Mumbai or equivalent) with minimum cost calculation in thousand Rupees
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)
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)
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)
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)
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