Q4
(a)(i) Reduce the following matrix to a row-reduced echelon form and hence also, find its rank: A = [1 3 2 4 1 0 0 2 2 0 2 6 2 6 2 3 9 1 10 6] (10 marks) (a)(ii) Find the eigen values and the corresponding eigen vectors of the matrix A = (0 -i i 0), over the complex-number field. (10 marks) (b) Show that the entire area of the Astroid : x^(2/3) + y^(2/3) = a^(2/3) is (3/8)πa². (15 marks) (c) Find equation of the plane containing the lines $$\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7},$$ $$\frac{x-2}{1} = \frac{y-4}{3} = \frac{z-6}{5}.$$ Also find the point of intersection of the given lines. (15 marks)
हिंदी में प्रश्न पढ़ें
(a)(i) निम्नलिखित आव्यूह का पंक्ति-समानीत सोपानक रूप में समान्यन कीजिए एवं अतैव इसकी कोटि भी ज्ञात कीजिए। A = [1 3 2 4 1 0 0 2 2 0 2 6 2 6 2 3 9 1 10 6] (10 अंक) (a)(ii) सम्मिश्र संख्या क्षेत्र पर आव्यूह A = (0 -i i 0) के अभिलक्षणिक मान तथा संगत अभिलक्षणिक सदिशों को ज्ञात कीजिए। (10 अंक) (b) दर्शाइए कि ऐस्ट्रॉइड : x^(2/3) + y^(2/3) = a^(2/3) का पूरा क्षेत्रफल (3/8)πa² है। (15 अंक) (c) रेखाओं (x+1)/3 = (y+3)/5 = (z+5)/7, (x-2)/1 = (y-4)/3 = (z-6)/5 को अंतर्विष्ट करने वाले समतल का समीकरण ज्ञात कीजिए। दी गई रेखाओं के प्रतिच्छेद बिंदु को भी ज्ञात कीजिए। (15 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve this multi-part numerical problem by allocating approximately 40% time to part (a) covering matrix reduction and eigenvalue computation (20 marks), 30% to part (b) deriving the astroid area using parametric integration (15 marks), and 30% to part (c) finding the plane equation and intersection point of skew lines (15 marks). Begin with clear statement of given data, proceed with systematic computational steps showing all row operations/integrations/vector calculations, and conclude with boxed final answers for each sub-part.
Key points expected
- For (a)(i): Correct application of elementary row operations to reduce 4×5 matrix to row-reduced echelon form; identification of pivot positions and correct rank determination (rank = 3)
- For (a)(ii): Setting up and solving characteristic equation det(A - λI) = 0; obtaining eigenvalues λ = ±1; finding normalized eigenvectors (1, i) and (1, -i) or equivalent scalar multiples
- For (b): Parametric representation x = a cos³θ, y = a sin³θ; correct area integral setup using symmetry and Jacobian; evaluation yielding (3/8)πa²
- For (c): Verification that lines are coplanar via condition [a₂-a₁, b₁, b₂] = 0; calculation of plane normal via cross product of direction vectors; point of intersection by solving parametric equations simultaneously
- Correct handling of complex numbers in (a)(ii) and proper geometric interpretation of astroid as hypocycloid with four cusps
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies initial matrix dimensions, writes proper characteristic equation for Hermitian matrix, sets correct parametric equations for astroid with appropriate limits, and verifies coplanarity condition before proceeding in (c) | Minor errors in initial setup such as wrong parametric form or missing coplanarity check but recovers later; characteristic equation has sign errors but correct degree | Fundamental setup errors: treats matrix as 5×4, uses wrong parametric form (e.g., x = a cos θ), or assumes lines intersect without verification |
| Method choice | 20% | 10 | Uses Gaussian elimination with partial pivoting for (a)(i), exploits Hermitian property for real eigenvalues in (a)(ii), employs parametric integration with symmetry for (b), and uses vector triple product method for plane equation in (c) | Correct but inefficient methods: computes determinant by Laplace expansion unnecessarily, uses Cartesian integration without parametric substitution, or solves plane equation by point substitution without cross product | Inappropriate methods: attempts cofactor expansion for 4×4 RREF, ignores complex field in eigenvalue problem, or uses incorrect area formula |
| Computation accuracy | 20% | 10 | Flawless arithmetic: correct RREF with leading 1s and zeros above/below pivots, accurate eigenvalue ±1 with proper complex eigenvectors, exact integration yielding (3/8)πa², and precise intersection point coordinates | Minor computational slips: arithmetic errors in row operations (recovered), sign errors in integration, or algebraic mistakes in solving for intersection that partially affect final answers | Major computational errors: wrong rank due to arithmetic, incorrect eigenvalues, wrong area result, or impossible intersection point indicating lines are actually skew |
| Step justification | 20% | 10 | Explicitly states each elementary row operation (R₂ → R₂ - 2R₁), justifies why eigenvalues of Hermitian matrix are real, explains symmetry argument reducing astroid integral to first quadrant, and proves coplanarity before finding plane | Shows most steps but omits key justifications: row operations shown without naming, eigenvector calculation without normalization discussion, or integration limits stated without explanation | Minimal working shown: jumps to RREF without operations, states eigenvalues without characteristic equation, or presents final answers with no derivation |
| Final answer & units | 20% | 10 | All four final answers clearly boxed/labeled: RREF matrix with rank = 3, eigenvalues ±1 with corresponding eigenvectors, area = (3/8)πa² with proper dimensional analysis, and plane equation in standard form with exact intersection point coordinates | Correct answers but poorly presented: missing units on area, unnormalized eigenvectors, or intersection point given as parametric values rather than Cartesian coordinates | Missing or wrong final answers: no rank stated, eigenvectors not found, incorrect area formula, or claims lines are parallel/skew without plane equation |
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