Q2
(a) Find all solutions to the following system of equations by row-reduced method : x₁ + 2x₂ − x₃ = 2, 2x₁ + 3x₂ + 5x₃ = 5, − x₁ − 3x₂ + 8x₃ = − 1. 15 (b) A wire of length l is cut into two parts which are bent in the form of a square and a circle respectively. Using Lagrange's method of undetermined multipliers, find the least value of the sum of the areas so formed. 15 (c) If P, Q, R; P', Q', R' are feet of the six normals drawn from a point to the ellipsoid x²/a² + y²/b² + z²/c² = 1, and the plane PQR is represented by lx + my + nz = p, show that the plane P'Q'R' is given by x/a²l + y/b²m + z/c²n + 1/p = 0. 20
हिंदी में प्रश्न पढ़ें
(a) निम्नलिखित समीकरण निकाय के सभी हलों को पंक्ति-समानित विधि से ज्ञात कीजिए : x₁ + 2x₂ − x₃ = 2, 2x₁ + 3x₂ + 5x₃ = 5, − x₁ − 3x₂ + 8x₃ = − 1. 15 (b) एक l लम्बाई के तार को दो भागों में काटकर क्रमशः एक वर्ग तथा एक वृत्त के रूप में मोड़ा गया है । लग्रांज की अनिर्धारित गुणक विधि का प्रयोग करके, इस तरह से प्राप्त किए गए क्षेत्रफलों के योगफल का न्यूनतम मान ज्ञात कीजिए । 15 (c) यदि P, Q, R; P', Q', R', एक बिंदु से दीर्घवृत्तज x²/a² + y²/b² + z²/c² = 1 पर छः (सिक्स) अभिलंब पाद हैं तथा lx + my + nz = p से समतल PQR निरूपित है, दर्शाइए कि x/a²l + y/b²m + z/c²n + 1/p = 0, समतल P'Q'R' को निरूपित करता है । 20
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How this answer will be evaluated
Approach
Solve all three parts systematically, allocating approximately 30% time to part (a) [15 marks], 30% to part (b) [15 marks], and 40% to part (c) [20 marks]. Begin with clear statement of given data for each part, show complete working with row operations for (a), Lagrangian formulation and optimization for (b), and coordinate geometry with normal properties for (c). Conclude with boxed final answers for each sub-part.
Key points expected
- Part (a): Correct augmented matrix formation and systematic row reduction to echelon form, identification of rank and consistency, complete solution set with free variable parameterization
- Part (b): Proper constraint equation (4x + 2πr = l), correct Lagrangian L = x² + πr² + λ(4x + 2πr - l), accurate partial derivatives and solving for optimal x, r in terms of l
- Part (c): Equation of normal to ellipsoid at point (x₁,y₁,z₁) as a²(x-x₁)/x₁ = b²(y-y₁)/y₁ = c²(z-z₁)/z₁, condition that normal passes through given point (α,β,γ), feet of normals satisfying the sextic equation
- Part (c): Use of Joachimsthal's notation or equivalent for plane equations, reciprocal property relating PQR and P'Q'R' planes through the ellipsoid center, verification of the given plane equation
- Clear geometric interpretation: for (a) intersection of three planes; for (b) optimal allocation between perimeter and circumference; for (c) conjugate diametral planes and polar reciprocity
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 15% | 7.5 | For (a): correct 3×4 augmented matrix with RHS values; for (b): accurate perimeter constraint 4x + 2πr = l and area expression; for (c): correct normal line equations and identification of six feet of normals | Minor errors in matrix entries or constraint setup that don't fundamentally derail solution; partially correct normal equations | Fundamentally wrong setup: incorrect matrix dimensions, wrong constraint (e.g., area instead of perimeter), or misunderstanding normal geometry |
| Method choice | 20% | 10 | Explicit use of Gauss-Jordan elimination for (a); proper Lagrange multiplier technique with clear λ introduction for (b); elegant use of reciprocal directions or polar plane properties for (c) | Correct but inefficient methods (e.g., Cramer's rule for (a)); incomplete Lagrangian setup; brute force coordinate approach for (c) | Wrong method entirely (e.g., Cramer's rule for singular system in (a)), or complete omission of Lagrange multipliers in (b) |
| Computation accuracy | 25% | 12.5 | Flawless arithmetic: correct RREF with precise pivot operations, exact optimal values x = l/(4+π), r = l/(2(4+π)), and exact algebraic verification of plane equation in (c) | Minor arithmetic slips (sign errors, fraction mistakes) that propagate but method remains clear; correct final answers despite intermediate errors | Major computational errors: wrong rank determination, incorrect optimal values, or algebraic mistakes preventing verification of plane equation |
| Step justification | 25% | 12.5 | Every row operation stated explicitly; clear reasoning for λ elimination and second derivative/physical argument for minimum in (b); geometric justification for plane P'Q'R' using conjugate points or reciprocal radii | Some steps shown but gaps in justification (e.g., assuming minimum without verification, skipping row operation details) | Missing crucial justifications: no check for consistency in (a), no verification of minimum in (b), or purely computational approach to (c) without geometric insight |
| Final answer & units | 15% | 7.5 | Complete parametric solution for (a) showing infinite solutions; exact minimum area l²/(4(4+π)) with proper units; fully verified plane equation for P'Q'R' with clear substitution showing satisfaction | Correct answers but incomplete form (e.g., particular solution only for (a), numerical approximation instead of exact form) | Missing final answers, wrong answers due to earlier errors, or answers without required form (e.g., no parameterization for dependent system) |
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