(a) Show that there is no tangent plane to the sphere
x² + y² + z² - 4x + 2y - 4z + 4 = 0
that can be passed through the straight line
(x+6)/2 = y + 3 = z + 1. (15 marks)

(b) If f(x, y) = { xy(x²-y²)/(x²+y²), when (x,y) ≠ (0,0)
{ 0,

Question

(a) Show that there is no tangent plane to the sphere
x² + y² + z² - 4x + 2y - 4z + 4 = 0
that can be passed through the straight line
(x+6)/2 = y + 3 = z + 1. (15 marks)

(b) If f(x, y) = { xy(x²-y²)/(x²+y²), when (x,y) ≠ (0,0)
             { 0,                  when (x,y) = (0,0),
then find f_xy(0,0) and f_yx(0,0). (15 marks)

(c) (i) Find the eigenvalues and the corresponding eigenvectors of the matrix
A = [1  2   0]
    [2  1  -6]
    [2 -2   3] (12 marks)

(ii) Let P_n denote the vector space of all polynomials of degree ≤ n over R. Verify that
dim(P_4/P_2) = dim P_4 - dim P_2. (8 marks)

UPSC Answer Check · Accepted Answer

Prove the geometric impossibility in (a), calculate mixed partial derivatives in (b), and solve the eigenvalue problem with quotient space verification in (c). Allocate approximately 30% time to part (a) as it requires conceptual rigour, 30% to part (b) for careful limit analysis, 25% to (c)(i) for eigenvalue computation, and 15% to (c)(ii) for the dimension theorem verification. Begin with sphere-line analysis, proceed to partial derivative limits, then matrix characteristic polynomial, and conclude with basis construction for the quotient space.
- For (a): Complete the square to find sphere centre (2,-1,2) and radius 3; verify the given line does not intersect the sphere or lies entirely outside; show the distance from centre to line exceeds radius, making tangent plane impossible
- For (b): Compute f_x(0,y) using limit definition, then f_xy(0,0); compute f_y(x,0) then f_yx(0,0); demonstrate f_xy(0,0) = -1 and f_yx(0,0) = 1, showing inequality of mixed partials
- For (c)(i): Find characteristic equation det(A-λI)=0 yielding eigenvalues λ=3,3,-2; find eigenvectors for λ=3 (geometric multiplicity 1) and λ=-2; handle the defective case for repeated eigenvalue appropriately
- For (c)(ii): Identify basis {1,x,x²,x³,x⁴} for P₄ and {1,x,x²} for P₂; construct coset basis {x³+P₂, x⁴+P₂} for P₄/P₂; verify dim(P₄/P₂)=2=5-3=dim P₄-dim P₂
- Cross-part rigour: Use ε-δ arguments or explicit limit calculations in (b); justify why tangent plane condition fails via distance formula or system inconsistency in (a)

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies sphere centre (2,-1,2), radius 3 in (a); properly defines difference quotients for partial derivatives at origin in (b); accurately writes characteristic polynomial det(A-λI)=0 in (c)(i); correctly identifies standard bases for P₄ and P₂ in (c)(ii)	Minor errors in completing squares or radius calculation; partially correct limit setup for partial derivatives; characteristic polynomial with sign errors; basis identification with minor omissions	Incorrect sphere centre/radius; fails to use limit definition for partial derivatives; wrong characteristic polynomial; confuses P₄/P₂ with subspace rather than quotient space
Method choice	20%	10	Uses distance from point to line formula or plane pencil method in (a); applies limit definition before differentiation in (b); employs row reduction for eigenvectors and dimension theorem for quotient space in (c); selects most efficient path for each sub-part	Correct but inefficient methods; some reliance on formulas without derivation; acceptable eigenvalue computation but messy eigenvector finding; adequate but not optimal basis construction	Attempts geometric arguments without analytic verification; uses direct differentiation ignoring (0,0) discontinuity; tries to diagonalize defective matrix; confuses direct sum with quotient space
Computation accuracy	20%	10	Precise distance calculation showing d=√35>3 in (a); exact values f_xy(0,0)=-1, f_yx(0,0)=1 in (b); correct eigenvalues 3,3,-2 with accurate eigenvectors in (c)(i); exact dimension count 2=5-3 in (c)(ii)	Minor arithmetic slips in distance or limit evaluation; sign errors in partial derivatives; eigenvalue calculation with one error; dimension verification with computational gaps	Major calculation errors in distance formula; incorrect limit values; wrong eigenvalues; fundamental misunderstanding of dimension formula application
Step justification	20%	10	Explicitly justifies why distance>radius implies no tangent plane; shows limit existence for each partial derivative step; proves linear independence of eigenvectors and geometric multiplicity; rigorously proves cosets form basis for quotient space	Some justification present but gaps in logical flow; asserts limits exist without showing ε-δ; eigenvector verification incomplete; basis claim without full linear independence proof	Unjustified assertions throughout; no explanation for limit existence; eigenvectors stated without verification; dimension equality asserted without basis construction
Final answer & units	20%	10	Clear conclusion: no tangent plane exists with geometric interpretation; explicit statement f_xy(0,0)≠f_yx(0,0) noting continuity failure; complete eigenvalue-eigenvector table; precise verification dim(P₄/P₂)=2 with explicit basis	Correct answers but poorly formatted; missing explicit inequality statement in (b); eigenvectors without normalization or clear presentation; dimension verification without explicit basis	Missing or wrong final answers; no conclusion drawn; incomplete eigenvalue information; failure to verify the dimension formula explicitly

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