(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

Question

(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)

UPSC Answer Check · Accepted Answer

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

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	For (a), correctly identifies arguments ξ = x-kt+iαy, η = x-kt-iαy and applies chain rule properly; for (b), sets up correct 3×4 augmented matrix; for (c), recognizes sign-magnitude format with MSB as sign; for (d), draws correct free-body diagram and identifies system boundaries; for (e), correctly computes ∇φ for velocity field	Sets up most parts correctly but has minor errors in matrix dimensions, sign interpretation, or partial derivative notation; missing one setup element	Major setup errors: wrong augmented matrix size, confuses sign-magnitude with 2's complement, incorrect identification of independent variables in PDE, or wrong velocity-potential relationship
Method choice	20%	10	Selects optimal methods: chain rule for PDE verification, Gauss-Jordan over Gaussian elimination for complete reduction, positional weight method for hex conversion, conservation of linear momentum for mechanics, and streamline integration from velocity components	Uses correct but suboptimal methods (e.g., Gaussian elimination without reduction to RREF, or Cramer's rule for part b); some inefficiency in approach	Wrong method selection: uses direct substitution without chain rule for (a), matrix inversion instead of row reduction for (b), or force analysis instead of momentum conservation for (d)
Computation accuracy	20%	10	All calculations precise: correct partial derivatives with proper handling of iα factors, exact row operations yielding integer solutions, accurate hex-to-decimal conversions with fractional parts, correct algebraic manipulation for displacement formula, and exact streamline integration	Minor arithmetic slips: sign errors in derivatives, one row operation error corrected later, or rounding in hex conversion; final answers slightly off but method evident	Major computational errors: incorrect derivative calculations, wrong pivot selections leading to incorrect solutions, fundamental errors in base conversion, or integration mistakes in streamline derivation
Step justification	20%	10	Every non-trivial step explained: explicit chain rule application with intermediate variables shown, row operations labeled (R₂ → R₂ - 2R₁), positional weights written out for hex digits, conservation principle stated before application, and streamline parameterization justified	Most steps shown but some shortcuts taken: skips obvious chain rule steps, omits some row operation labels, or assumes streamline form without derivation	Minimal working shown: jumps from problem statement to answer with inadequate intermediate steps, making error tracing impossible; missing justification for key substitutions
Final answer & units	20%	10	All answers boxed/clearly stated with appropriate units: displacement in terms of a, M, m; streamlines in parametric or implicit form; hex conversions exact; PDE verification concluded explicitly; all five parts answered completely	Answers present but format inconsistent: missing units on displacement, decimal approximations instead of exact fractions, or incomplete streamline description; one part may be unanswered	Missing or wrong answers: no conclusion for PDE verification, unsolved system, incorrect final hex values, displacement without direction, or no streamline equations derived

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