Q4
Find the rank of the matrix $A = \begin{bmatrix} 1 & 2 & -1 & 0 \\ -1 & 3 & 0 & -4 \\ 2 & 1 & 3 & -2 \\ 1 & 1 & 1 & -1 \end{bmatrix}$ by reducing it to row-reduced echelon form. 15 (b) Trace the curve $y^2(x^2 - 1) = 2x - 1$. 20 (c) Prove that the locus of a line which meets the lines y = mx, z = c; y = -mx, z = -c and the circle x² + y² = a², z = 0 is c²m²(cy - mzx)² + c²(yz - cmx)² = a²m²(z² - c²)². 15
हिंदी में प्रश्न पढ़ें
आव्यूह $A = \begin{bmatrix} 1 & 2 & -1 & 0 \\ -1 & 3 & 0 & -4 \\ 2 & 1 & 3 & -2 \\ 1 & 1 & 1 & -1 \end{bmatrix}$ का पंक्ति समानतित सोपानक रूप में समान्यन करके उसकी कोटि ज्ञात कीजिए। 15 (b) वक्र $y^2(x^2 - 1) = 2x - 1$ को अनुरेखित कीजिए। 20 (c) सिद्ध कीजिए कि रेखाओं y = mx, z = c; y = -mx, z = -c और वृत्त x² + y² = a², z = 0 से मिलने वाली रेखा का विद्यु-पथ c²m²(cy - mzx)² + c²(yz - cmx)² = a²m²(z² - c²)² है। 15
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How this answer will be evaluated
Approach
Solve all three parts systematically, allocating approximately 30% time to part (a) matrix rank reduction, 40% to part (b) curve tracing as it carries highest marks, and 30% to part (c) 3D geometry proof. Begin with clear identification of each part, show complete row operations for (a), detailed curve analysis with asymptotes and intercepts for (b), and rigorous parametric derivation for (c).
Key points expected
- Part (a): Correct reduction of 4×4 matrix to row-reduced echelon form using elementary row operations, identification of pivot positions, and accurate rank determination
- Part (b): Complete curve tracing of y²(x²-1)=2x-1 including symmetry analysis, asymptotes (vertical at x=±1), intercepts, domain restrictions, and sketch of two branches
- Part (c): Proper parameterization of lines meeting given skew lines and circle, elimination of parameters to derive the stated quartic locus surface
- Verification of rank by checking linear dependence of rows/columns or using determinant test for part (a)
- Analysis of singular points and behavior near asymptotes for part (b) curve
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 15% | 7.5 | Correctly sets up augmented matrix for part (a), identifies curve type and restrictions for part (b), and establishes proper parametric equations for lines in part (c) with correct geometric interpretation of given constraints | Basic setup present but minor errors in initial matrix writing, incomplete domain analysis for curve, or partially correct parameterization | Major setup errors such as wrong matrix entries, incorrect curve equation interpretation, or fundamentally wrong approach to 3D geometry |
| Method choice | 20% | 10 | Selects optimal row reduction strategy for (a), applies systematic curve tracing methodology (asymptotes, symmetry, derivatives) for (b), and chooses efficient elimination method for (c) using appropriate geometric insight | Acceptable methods chosen but suboptimal (e.g., using cofactor expansion for rank), incomplete curve analysis, or longer elimination route | Inappropriate methods such as determinant calculation instead of row reduction when specifically asked, or trial-and-error approach without methodology |
| Computation accuracy | 25% | 12.5 | Flawless arithmetic in all row operations with correct pivot selection, accurate algebraic manipulation in curve analysis, and error-free elimination leading to exact final locus equation matching the required form | Minor computational slips (sign errors, arithmetic mistakes) that don't fundamentally alter the answer, or partially correct final form | Persistent calculation errors leading to wrong rank, incorrect curve shape, or failure to reach the stated locus equation |
| Step justification | 25% | 12.5 | Every row operation explicitly stated with clear notation (R1→R1+R2 etc.), curve properties rigorously justified with limits and derivatives, and geometric reasoning clearly explained for the locus derivation with intermediate steps shown | Some steps shown but gaps in justification, missing explanations for key transitions, or assumed results without verification | Bare answers with no working, or unjustified jumps that make verification impossible |
| Final answer & units | 15% | 7.5 | Clear statement of rank=3 for part (a), accurate labeled sketch with key features for part (b), and exact reproduction of target locus equation for part (c) with proper interpretation of the surface | Correct final answers but poorly presented, incomplete sketch, or correct equation but without geometric interpretation | Missing final answers, wrong conclusions, or failure to present the required equation form in part (c) |
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