(a) It is given that the equation of any cone with vertex at (a, b, c) is f((x-a)/(z-c), (y-b)/(z-c)) = 0. Find the differential equation of the cone. (10 marks)

(b) Solve, by Gauss elimination method, the system of equations
2x + 2y + 4z = 18
x + 3

Question

(a) It is given that the equation of any cone with vertex at (a, b, c) is f((x-a)/(z-c), (y-b)/(z-c)) = 0. Find the differential equation of the cone. (10 marks)

(b) Solve, by Gauss elimination method, the system of equations
2x + 2y + 4z = 18
x + 3y + 2z = 13
3x + y + 3z = 14 (10 marks)

(c) (i) Convert the number (1093·21875)₁₀ into octal and the number (1693·0628)₁₀ into hexadecimal systems.
(ii) Express the Boolean function F(x, y, z) = xy + x'z in a product of maxterms form. (10 marks)

(d) A particle at a distance r from the centre of force moves under the influence of the central force F = -k/r², where k is a constant. Obtain the Lagrangian and derive the equations of motion. (10 marks)

(e) The velocity components of an incompressible fluid in spherical polar coordinates (r, θ, ψ) are (2Mr⁻³cosθ, Mr⁻²sinθ, 0), where M is a constant. Show that the velocity is of the potential kind. Find the velocity potential and the equations of the streamlines. (10 marks)

UPSC Answer Check · Accepted Answer

Solve each sub-part systematically with clear section headings. For (a), set up the cone equation and eliminate the arbitrary function f to get the PDE; for (b), perform Gauss elimination with back substitution; for (c)(i), convert decimal to octal/hexadecimal using successive division/multiplication, and for (c)(ii), expand to canonical POS form using Boolean algebra or K-map; for (d), construct Lagrangian in polar coordinates and derive Euler-Lagrange equations; for (e), verify irrotational flow condition, integrate for velocity potential, and solve streamline equations. Allocate approximately 15% time each to (a), (b), (c)(i), (c)(ii), and 20% each to (d) and (e) due to their derivational complexity.
- For (a): Eliminate arbitrary function f by differentiating the cone equation with respect to x, y, z and eliminating f_u, f_v to obtain the PDE (z-c)p + (z-c)q = (x-a) + (y-b) where p=∂z/∂x, q=∂z/∂y
- For (b): Form augmented matrix, perform row operations to achieve upper triangular form, back-substitute to get x=1, y=2, z=3 with verification
- For (c)(i): Convert (1093.21875)₁₀ = (2105.16)₈ and (1693.0628)₁₀ = (69D.101)₁₆ showing integer and fractional part conversions separately
- For (c)(ii): Express F(x,y,z) = xy + x'z in product of maxterms as ΠM(0,2,4,6) or (x+y+z)(x+y'+z)(x'+y+z)(x'+y'+z) using truth table or algebraic expansion
- For (d): Construct Lagrangian L = ½m(ṙ² + r²θ̇²) + k/r in polar coordinates, derive equations: m(r̈ - rθ̇²) = -k/r² and d/dt(mr²θ̇) = 0 (conservation of angular momentum)
- For (e): Verify ∇×v = 0, obtain velocity potential φ = -Mr⁻²cosθ, and derive streamline equations as r²sin²θ = constant, ψ = constant

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies the mathematical setup for each sub-part: proper elimination strategy for (a), correct augmented matrix for (b), accurate place-value decomposition for (c)(i), correct truth table/K-map setup for (c)(ii), appropriate coordinate system and kinetic/potential energy terms for (d), and correct spherical polar curl expression for (e)	Sets up most parts correctly but has minor errors in one sub-part, such as wrong matrix dimensions, incorrect place values in conversion, or missing terms in Lagrangian	Fundamental setup errors in multiple sub-parts, such as treating cone equation incorrectly, wrong elimination method, or incorrect coordinate identification in (d)-(e)
Method choice	20%	10	Selects optimal methods: Lagrange's method of eliminating arbitrary functions for (a), systematic Gauss elimination with partial pivoting awareness for (b), standard base conversion algorithms for (c)(i), canonical POS conversion using maxterm identification for (c)(ii), Euler-Lagrange formalism for (d), and curl verification with potential integration for (e)	Uses acceptable but suboptimal methods, such as Cramer's rule instead of Gauss elimination, or algebraic expansion without K-map for Boolean function	Chooses inappropriate methods, such as direct integration for (a), matrix inversion for (b), or fails to recognize central force nature in (d)
Computation accuracy	20%	10	All calculations error-free: correct partial derivatives and elimination in (a), accurate row operations yielding x=1, y=2, z=3 in (b), precise octal/hexadecimal conversions with correct fractional parts in (c)(i), accurate maxterm identification ΠM(0,2,4,6) in (c)(ii), correct Lagrangian derivatives and equations of motion in (d), verified curl zero and integrated potential φ = -Mr⁻²cosθ in (e)	Minor arithmetic slips in one or two sub-parts, such as sign errors in row operations, rounding errors in base conversion, or coefficient errors in partial derivatives	Major computational errors in multiple sub-parts, such as wrong solutions to linear system, completely wrong base conversions, or incorrect equations of motion
Step justification	20%	10	Every non-trivial step is justified: explains why f can be eliminated via Jacobian condition in (a), shows each row operation explicitly in (b), explains why integer and fractional parts are treated separately in (c)(i), justifies maxterm selection via truth table or De Morgan's laws in (c)(ii), derives generalized momenta and forces in (d), and proves irrotational condition before asserting potential existence in (e)	Shows key steps but skips some justifications, such as omitting verification of solution in (b) or not explaining the zero-curl condition in (e)	Minimal working shown with large logical gaps, or presents final answers without intermediate steps in most sub-parts
Final answer & units	20%	10	All final answers clearly stated with appropriate notation: PDE for cone in standard form, explicit solution vector (1,2,3) with verification, octal and hexadecimal numbers with base subscripts, complete POS expression with all maxterms, explicit Lagrangian and both equations of motion, velocity potential with streamline equations and physical interpretation	Most answers correct but poorly formatted, missing subscripts for number bases, or incomplete streamline equations in (e)	Missing or incorrect final answers for multiple sub-parts, no verification of solutions, or answers without proper mathematical notation

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