(a) Can the set {(0, 0, 0, 3), (1, 1, 0, 0), (0, 1, –1, 0)} be extended to form a basis of the vector space ℝ⁴? Justify your answer. 10 marks

(b) Find the range, rank, kernel and nullity of the linear transformation T : ℝ⁴ → ℝ³ given by T(x, y, z, w

Question

(a) Can the set {(0, 0, 0, 3), (1, 1, 0, 0), (0, 1, –1, 0)} be extended to form a basis of the vector space ℝ⁴? Justify your answer. 10 marks

(b) Find the range, rank, kernel and nullity of the linear transformation T : ℝ⁴ → ℝ³ given by T(x, y, z, w) = (x – w, y + z, z – w). 10 marks

(c) A rectangular sheet of metal of length 6 meters and width 2 meters is given. Four equal squares are removed from the four corners. The sides of this sheet are now folded up to form an open rectangular box. Find approximately the height of the box, such that the volume of the box is maximum. 10 marks

(d) Given that f(x + y) = f(x) f(y) for all real x, y, f(x) ≠ 0 for any real x and f'(0) = 2. Show that for all real x, f'(x) = 2f(x). Hence find f(x). 10 marks

(e) Find the equation of the cone whose vertex is the point (1, 1, 0) and whose guiding curve is y = 0, x² + z² = 4. 10 marks

UPSC Answer Check · Accepted Answer

Solve each sub-part systematically with clear mathematical reasoning. For (a), verify linear independence and extend to basis; for (b), construct the matrix representation and apply rank-nullity; for (c), set up the volume function and optimize using calculus; for (d), use the functional equation to derive the differential equation; for (e), use the cone generator method. Allocate approximately 20% time per sub-part given equal marks distribution, ensuring each part receives complete treatment with proper justification.
- (a) Verify linear independence of the three vectors by checking no non-trivial combination equals zero; extend to basis by adding a fourth vector not in their span, such as (0,0,1,0) or (1,0,0,0)
- (b) Construct the 3×4 matrix representation of T; find rank by row reduction or determinant of 3×3 minors; determine nullity using rank-nullity theorem (nullity = 4 - rank = 2)
- (c) Express volume V = (6-2h)(2-2h)h = 4h(3-h)(1-h); find dV/dh = 0 yielding 3h² - 8h + 3 = 0; select valid root h = (4-√7)/3 ≈ 0.45m within domain 0 < h < 1
- (d) Use f(x+y) = f(x)f(y) to show [f(x+h)-f(x)]/h = f(x)[f(h)-1]/h; as h→0, f'(x) = f(x)f'(0) = 2f(x); solve to get f(x) = e^(2x)
- (e) For cone with vertex (1,1,0) and guiding curve y=0, x²+z²=4: parametrize generator lines (1+(x₀-1)t, 1-t, z₀t) where x₀²+z₀²=4; eliminate parameters to get (x-1)² + z² = 4(y-1)² with y ≤ 1

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies vector spaces, dimensions, and domains for all parts: recognizes ℝ⁴ needs 4 basis vectors in (a), proper matrix dimensions in (b), valid height constraints 0<h<1 in (c), functional equation properties in (d), and cone generator geometry in (e)	Sets up most parts correctly but has minor errors like wrong matrix dimensions, incorrect domain restrictions, or misidentified vector space properties in one or two parts	Major setup errors: confuses domain/codomain, uses wrong dimensions, invalid constraints, or fundamentally misunderstands the mathematical structures involved
Method choice	20%	10	Selects optimal methods: linear independence test and extension for (a), matrix rank-nullity for (b), calculus optimization with second derivative test for (c), limit definition of derivative for (d), and parametric elimination for cone equation in (e)	Uses acceptable but suboptimal methods, or correct methods with unnecessary complications; may use trial-and-error for basis extension or skip second derivative test in optimization	Inappropriate methods: attempts to use determinant for 3×4 matrix, uses Lagrange multipliers unnecessarily, or fails to recognize the exponential function characterization in (d)
Computation accuracy	20%	10	All calculations precise: correct row reduction yielding rank 2 and nullity 2 in (b), accurate quadratic solution h = (4-√7)/3 in (c), correct integration yielding f(x) = e^(2x) in (d), and error-free algebraic elimination for cone equation	Minor computational slips: arithmetic errors in row operations, approximate rather than exact value for height, or small algebraic mistakes in final cone equation that preserve correct structure	Serious computational errors: wrong rank calculation, incorrect quadratic formula application, integration errors, or fundamentally wrong final equations
Step justification	20%	10	Every non-trivial step justified: explicit linear independence verification, citation of rank-nullity theorem, verification of maximum via second derivative or sign analysis, rigorous limit argument for f'(x), and clear geometric reasoning for cone generators	Most key steps justified but some gaps: assumes linear independence without check, states rank-nullity without verification, or skips justification for why critical point gives maximum	Unjustified leaps: asserts results without calculation, omits critical verification steps, or presents answers without showing how they were obtained
Final answer & units	20%	10	Complete precise answers: explicit basis extension vector, complete description of range/kernel with basis vectors, exact height h = (4-√7)/3 meters with approximate value ~0.45m, f(x) = e^(2x), and standard cone equation (x-1)² - 4(y-1)² + z² = 0	Correct answers but incomplete: missing basis for kernel, approximate height without exact form, or correct equation without specifying domain restriction y ≤ 1 for the cone	Missing or wrong final answers: no basis extension provided, incorrect rank/nullity values, wrong height, wrong functional form, or incorrect cone equation

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