(a) Prove that any set of n linearly independent vectors in a vector space V of dimension n constitutes a basis for V. 10

(b) Let T : ℝ² → ℝ³ be a linear transformation such that T\begin{pmatrix} 1 \ 0 \end{pmatrix} = \begin{pmatrix} 1 \ 2 \ 3 \e

Question

(a) Prove that any set of n linearly independent vectors in a vector space V of dimension n constitutes a basis for V. 10

(b) Let T : ℝ² → ℝ³ be a linear transformation such that T\begin{pmatrix} 1 \ 0 \end{pmatrix} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} and T\begin{pmatrix} 1 \ 1 \end{pmatrix} = \begin{pmatrix} -3 \ 2 \ 8 \end{pmatrix}. Find T\begin{pmatrix} 2 \ 4 \end{pmatrix}. 10

(c) Evaluate $\lim\limits_{x 	o \infty} (e^x + x)^{\frac{1}{x}}$. 10

(d) Examine the convergence of $\int\limits_{0}^{2} \frac{dx}{(2x - x^2)}$. 10

(e) A variable plane passes through a fixed point (a, b, c) and meets the axes at points A, B and C respectively. Find the locus of the centre of the sphere passing through the points O, A, B and C, O being the origin. 10

UPSC Answer Check · Accepted Answer

Begin with a clear statement of definitions for part (a), then proceed systematically through all five sub-parts. Allocate approximately 20% time to (a) for rigorous proof construction, 15% each to (b), (c), (d), and (e) for calculations and analysis. For (b), express the target vector as a linear combination first; for (c), use logarithmic transformation; for (d), analyze improper integral behavior near singularities at x=0 and x=2; for (e), use intercept form of plane and sphere properties. Conclude each part with explicit final answers boxed or clearly stated.
- Part (a): Correct definition of basis, dimension, and linear independence; proof that n linearly independent vectors in an n-dimensional space must span V (using exchange lemma or dimension argument)
- Part (b): Express (2,4) as linear combination of (1,0) and (1,1); apply linearity of T to obtain T(2,4) = 2T(1,0) + 4[T(1,1) - T(1,0)] = (-14, 2, 22)
- Part (c): Take logarithm, evaluate limit using L'Hôpital's rule or asymptotic comparison; final answer e^1 = e
- Part (d): Factor denominator as x(2-x), identify singularities at 0 and 2; show divergence by comparison with ∫dx/x near each endpoint or partial fractions
- Part (e): Use intercept form x/α + y/β + z/γ = 1 with constraint a/α + b/β + c/γ = 1; find sphere center at (α/2, β/2, γ/2); eliminate parameters to get locus a/x + b/y + c/z = 2
- Part (e) alternative: Recognize sphere through origin with diameter from O to opposite point on plane; use diametric sphere equation

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly states all definitions (basis, dimension, linear transformation properties, improper integral criteria, intercept form of plane); properly identifies singular points in (d) and geometric constraints in (e); sets up coordinate systems and notation consistently across all parts	States most definitions correctly but may miss subtle points like the spanning requirement in (a) or both singularities in (d); minor errors in geometric setup for (e)	Fundamental definitional errors (confuses basis with linear independence only, misses singularities in improper integral, wrong plane equation form); inconsistent or missing notation
Method choice	20%	10	Selects optimal methods: exchange lemma or dimension theorem for (a), linear combination approach for (b), logarithmic limit for (c), partial fraction decomposition with limit comparison for (d), intercept elimination for (e); avoids unnecessary computation	Uses correct but suboptimal methods (e.g., matrix inversion for (b), direct expansion without log for (c)); completes tasks but with extra steps; partial fraction errors in (d)	Inappropriate methods (attempts matrix inverse for non-square in (b), ignores indeterminate form in (c), treats (d) as proper integral, uses general sphere equation without geometric insight in (e))
Computation accuracy	20%	10	All arithmetic flawless: correct coefficients in linear combination for (b) yielding (-14, 2, 22), accurate limit evaluation giving e, correct partial fraction decomposition 1/(2x-x²) = 1/(2x) + 1/(2(2-x)), clean elimination in (e) producing a/x + b/y + c/z = 2	Minor arithmetic slips (sign errors in vector components, coefficient mistakes in partial fractions, algebraic errors in elimination) that don't fundamentally derail the solution	Major computational errors (wrong vector values, incorrect limit evaluation, failure to decompose integrand, algebraic mistakes preventing locus determination); answers numerically unreasonable
Step justification	20%	10	Rigorous justification at each step: explicit appeal to dimension theorem in (a), clear linearity justification in (b), proper limit theorems and L'Hôpital justification in (c), explicit divergence proof via limit comparison test in (d), geometric reasoning for sphere center location in (e)	Most steps justified but some gaps (assumes spanning without proof in (a), skips continuity justification for limit interchange in (c), asserts divergence without explicit test in (d))	Missing critical justifications (no proof that vectors span in (a), unjustified limit manipulations in (c), no convergence test in (d), purely computational with no geometric reasoning in (e))
Final answer & units	20%	10	All five answers clearly stated: (a) complete proof with concluding statement, (b) vector (-14, 2, 22), (c) e, (d) 'divergent' with reason, (e) locus equation a/x + b/y + c/z = 2; proper mathematical formatting, boxed or emphasized final answers	Most answers present but poorly formatted or with minor errors; incomplete concluding statements; answers buried in working	Missing or incorrect final answers; wrong format (scalars instead of vectors, missing divergence conclusion, no locus equation); illegible presentation; failure to answer specific question asked

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