(a) Consider a linear operator T on R³ over R defined by T(x, y, z) = (2x, 4x – y, 2x + 3y – z). Is T invertible? If yes, justify your answer and find T⁻¹. (15 marks)

(b) If u = (x+y)/(1-xy) and v = tan⁻¹x + tan⁻¹y, then find ∂(u, v)/∂(x, y). Are u

Question

(a) Consider a linear operator T on R³ over R defined by T(x, y, z) = (2x, 4x – y, 2x + 3y – z). Is T invertible? If yes, justify your answer and find T⁻¹. (15 marks)

(b) If u = (x+y)/(1-xy) and v = tan⁻¹x + tan⁻¹y, then find ∂(u, v)/∂(x, y). Are u and v functionally related? If yes, find the relationship. (15 marks)

(c) Find the image of the line x = 3-6t, y = 2t, z = 3+2t in the plane 3x+4y-5z+26 = 0. (20 marks)

UPSC Answer Check · Accepted Answer

Solve all three parts systematically, allocating approximately 30% time to part (a) on linear operator invertibility, 30% to part (b) on Jacobian and functional dependence, and 40% to part (c) on finding image of a line in a plane. Begin each part with clear statement of the mathematical approach, show complete working with logical flow, and conclude with verified final answers. For part (c), explicitly verify that the given line intersects the plane before finding the image line.
- For (a): Construct matrix representation of T, compute determinant to establish invertibility (det = 2 ≠ 0), and find inverse using adjugate or row reduction: T⁻¹(x,y,z) = (x/2, 2x-y, 7x-3y-z)/2
- For (b): Compute Jacobian ∂(u,v)/∂(x,y) = 0, establishing functional dependence; derive relationship u = tan(v) using tan⁻¹ addition formula
- For (c): Verify line intersects plane at point P(3,0,3), find direction vector of image line using reflection formula or two-point method with foot of perpendicular
- For (c): Correctly apply image line formula: find reflected point or use symmetric point method to determine image line equation
- All parts: Explicit verification of results (TT⁻¹ = I for (a), direct substitution for (b), checking image line lies in plane for (c))

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies matrix of T for (a), recognizes u = tan(v) identity connection for (b), and verifies line-plane intersection before proceeding in (c); all initial conditions properly interpreted	Matrix representation has minor errors or missing verification of intersection in (c); some setup steps implied rather than explicit	Major setup errors: wrong matrix for T, fails to recognize tan⁻¹ addition formula relevance, or assumes line is parallel to plane without checking
Method choice	20%	10	Uses determinant for invertibility (a), Jacobian determinant for functional dependence (b), and either reflection formula or symmetric point method for image line (c); methods are optimal and clearly stated	Correct methods chosen but inefficient alternatives used (e.g., solving 9 equations for inverse); image line found by two arbitrary points without systematic approach	Attempts to find inverse before checking invertibility, computes wrong partial derivatives, or uses rotation instead of reflection for image
Computation accuracy	20%	10	All calculations error-free: det(T) = 2, T⁻¹ matrix correct, Jacobian determinant = 0, and image line equation precisely determined with correct direction ratios	Minor arithmetic slips (e.g., sign error in cofactor, partial derivative miscalculation) that don't fundamentally derail the solution	Major computational errors: incorrect determinant, wrong inverse formula, or image line that doesn't satisfy plane equation
Step justification	20%	10	Each step rigorously justified: why det ≠ 0 implies invertibility, explicit derivation of tan(A+B) formula for functional relation, geometric reasoning for image construction; cites relevant theorems	Steps shown but with gaps in logical connection; some justifications like 'by calculation' or 'it can be shown' without elaboration	Missing crucial justifications: no explanation why Jacobian zero implies dependence, or asserts image line without geometric reasoning
Final answer & units	20%	10	All answers precisely stated: explicit T⁻¹ formula, clear u = tan(v) relationship with domain restrictions, and image line in symmetric/cartesian form; includes verification for each part	Correct answers but poorly formatted (e.g., inverse as system of equations rather than single formula, image line as two points rather than equation)	Missing final answers, or answers that fail basic verification (TT⁻¹ ≠ I, u ≠ tan(v), or image line not lying in given plane)

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