(a) Let V₁ = (2, -1, 3, 2), V₂ = (-1, 1, 1, -3) and V₃ = (1, 1, 9, -5) be three vectors of the space ℝ⁴. Does (3, -1, 0, -1) ∈ span {V₁, V₂, V₃} ? Justify your answer. (10 marks)

(b) Find the rank and nullity of the linear transformation : T : ℝ³ →

Question

(a) Let V₁ = (2, -1, 3, 2), V₂ = (-1, 1, 1, -3) and V₃ = (1, 1, 9, -5) be three vectors of the space ℝ⁴. Does (3, -1, 0, -1) ∈ span {V₁, V₂, V₃} ? Justify your answer. (10 marks)

(b) Find the rank and nullity of the linear transformation : T : ℝ³ → ℝ³ given by T(x, y, z) = (x + z, x + y + 2z, 2x + y + 3z) (10 marks)

(c) Find the values of p and q for which limₓ→₀ [x(1 + p cos x) - q sin x]/x³ exists and equals 1. (10 marks)

(d) Examine the convergence of the integral ∫₀¹ (log x)/(1+x) dx (10 marks)

(e) A variable plane which is at a constant distance 3p from the origin O cuts the axes in the points A, B, C respectively. Show that the locus of the centroid of the tetrahedron OABC is 9(1/x² + 1/y² + 1/z²) = 16/p². (10 marks)

UPSC Answer Check · Accepted Answer

Solve each sub-part systematically with equal time allocation (~20% per part) since all carry 10 marks each. For (a), set up and solve the linear system for span membership; for (b), construct the matrix of T and apply rank-nullity theorem; for (c), use Taylor series expansion or L'Hôpital's rule to find p and q; for (d), apply comparison tests for improper integrals; for (e), use the distance formula and centroid coordinates to derive the locus. Present solutions clearly with proper mathematical notation and concluding statements for each part.
- Part (a): Set up the augmented matrix [V₁ V₂ V₃ | (3,-1,0,-1)] and determine consistency via row reduction; conclude whether the target vector lies in the span
- Part (b): Construct the 3×3 matrix of T, compute its rank via determinant or row reduction, then apply rank-nullity theorem (rank + nullity = 3)
- Part (c): Expand cos x and sin x using Taylor series up to x³ terms, equate coefficients to ensure limit exists and equals 1, solving for p = -5/2 and q = -3/2
- Part (d): Identify the singularity at x=0, use limit comparison test with x^(-1/2) or direct evaluation showing |log x|/(1+x) is integrable, proving convergence
- Part (e): Use the plane equation x/a + y/b + z/c = 1 with distance condition 1/√(1/a²+1/b²+1/c²) = 3p, find centroid (a/4, b/4, c/4), eliminate parameters to obtain 9(1/x²+1/y²+1/z²) = 16/p²

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies the mathematical framework for each part: augmented matrix for span in (a), matrix representation of linear transformation in (b), Taylor expansion setup for limit in (c), improper integral classification in (d), and intercept form of plane with distance formula in (e)	Sets up most parts correctly but has minor errors in one part, such as wrong matrix dimensions or incorrect plane equation form	Fundamental setup errors in multiple parts, such as confusing span with linear independence, wrong transformation matrix, or incorrect integral bounds
Method choice	20%	10	Selects optimal methods: Gaussian elimination for (a), determinant/rank-nullity for (b), Taylor series (superior to repeated L'Hôpital) for (c), comparison test or direct integration for (d), and parameter elimination for (e); avoids computationally intensive approaches	Uses acceptable but suboptimal methods, such as repeated L'Hôpital instead of Taylor series in (c), or cofactor expansion instead of row reduction in (b)	Chooses inappropriate methods, such as trying to find inverse for rank in (b), or failing to recognize the need for series expansion in (c)
Computation accuracy	20%	10	All calculations error-free: correct row reduction in (a), accurate determinant and rank calculation in (b), precise coefficient matching yielding p = -5/2, q = -3/2 in (c), correct limit evaluation in (d), and exact algebraic manipulation in (e)	Minor arithmetic errors in one or two parts, such as sign errors in row operations or slight coefficient miscalculations, but method remains sound	Significant computational errors in multiple parts, such as wrong rank, incorrect p/q values, or failure to simplify the locus equation properly
Step justification	20%	10	Every logical step justified: explains why consistency implies span membership in (a), cites rank-nullity theorem explicitly in (b), shows why higher order terms vanish in (c), proves convergence rigorously in (d), and clearly tracks parameter elimination in (e)	Most steps justified but some gaps, such as assuming convergence without proof in (d), or stating rank-nullity without verification of conditions	Missing crucial justifications, such as no explanation for why limit exists, unverified claims about convergence, or logical gaps in the locus derivation
Final answer & units	20%	10	Clear definitive answers for all parts: explicit yes/no with justification for (a), numerical rank and nullity for (b), exact values p = -5/2, q = -3/2 for (c), 'convergent' with value or bound for (d), and fully derived locus equation for (e); proper mathematical formatting throughout	Correct answers but poorly presented, such as missing the final locus equation simplification, or unclear statement of rank/nullity values	Missing or incorrect final answers, such as wrong conclusion about span membership, incorrect rank/nullity, unsolved for p and q, or incomplete locus derivation

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