Q4
(a) Find a linear map T : $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ which rotates each vector of $\mathbb{R}^2$ by an angle $\theta$. Also, prove that for $\theta = \frac{\pi}{2}$, T has no eigenvalue in $\mathbb{R}$. 15 (b) Trace the curve y²x² = x² – a², where a is a real constant. 20 (c) If the plane ux + vy + wz = 0 cuts the cone ax² + by² + cz² = 0 in perpendicular generators, then prove that (b + c) u² + (c + a) v² + (a + b) w² = 0. 15
हिंदी में प्रश्न पढ़ें
(a) एक रैखिक प्रतिचित्र T : $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ ज्ञात कीजिए जो कि $\mathbb{R}^2$ के प्रत्येक सदिश को $\theta$ कोण से घुमा देता है । यह भी सिद्ध कीजिए कि $\theta = \frac{\pi}{2}$ के लिए, T का कोई भी अभिलक्षणिक मान (आइगेनमान) $\mathbb{R}$ में नहीं है । 15 (b) वक्र y²x² = x² – a² का अनुरेख (ट्रेस) कीजिए, जहाँ a एक वास्तविक अचर है । 20 (c) यदि समतल ux + vy + wz = 0, शंकु ax² + by² + cz² = 0 को लंब जनकों में काटता है, तो सिद्ध कीजिए कि (b + c) u² + (c + a) v² + (a + b) w² = 0. 15
Directive word: Trace
This question asks you to trace. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.
See our UPSC directive words guide for a full breakdown of how to respond to each command word.
How this answer will be evaluated
Approach
The directive 'trace' in part (b) demands a systematic graphical analysis of the curve's behavior. Structure your answer with: (a) rotation matrix derivation and eigenvalue proof (~15 min, 30%); (b) complete curve tracing with symmetry, asymptotes, tangents, and sketch (~25 min, 40%); (c) condition for perpendicular generators using direction cosines and orthogonality condition (~15 min, 30%). Begin each part with clear setup, proceed through rigorous derivation, and conclude with boxed final results.
Key points expected
- Part (a): Derivation of rotation matrix T = [[cos θ, -sin θ], [sin θ, cos θ]] and verification of linearity
- Part (a): Characteristic equation λ² + 1 = 0 for θ = π/2, showing no real eigenvalues exist
- Part (b): Domain analysis (|x| ≥ a), symmetry about both axes, and asymptotes y = ±1
- Part (b): Behavior at x → ±∞, points of intersection with axes, and rough sketch of four branches
- Part (c): Parametric representation of generators and condition for perpendicularity using direction cosines
- Part (c): Derivation that (b+c)u² + (c+a)v² + (a+b)w² = 0 from the orthogonality condition of generator directions
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly defines rotation transformation using standard basis images; for (b): properly identifies domain restrictions |x| ≥ a and coordinate transformations; for (c): correctly sets up cone equation and plane intersection geometry with proper identification of generator concept | Sets up most parts correctly but misses domain restriction in (b) or confuses cone generators with simple lines; some notational inconsistencies | Major setup errors: wrong rotation matrix direction, ignores |x| ≥ a condition entirely, or fails to identify what constitutes a generator of the cone |
| Method choice | 20% | 10 | For (a): uses matrix representation approach; for (b): applies systematic curve tracing protocol (symmetry, asymptotes, special points, sketch); for (c): employs direction cosine method or parametric generator approach rather than ad hoc elimination | Uses acceptable methods but misses optimal approaches—e.g., uses coordinate rotation unnecessarily in (b), or brute-force algebra in (c) without geometric insight | Inappropriate methods: attempts coordinate geometry without matrices in (a), ignores asymptote analysis in (b), or tries to solve (c) by eliminating variables without using generator properties |
| Computation accuracy | 20% | 10 | Flawless computations: correct characteristic polynomial λ² + 1 in (a), accurate asymptote derivation y² → 1 in (b), and error-free algebraic manipulation leading to the symmetric condition in (c) | Minor computational slips: sign errors in rotation matrix, incorrect limit evaluation for asymptotes, or arithmetic errors in expanding the perpendicularity condition that still preserve answer structure | Serious computational errors: wrong determinant for eigenvalue problem, incorrect asymptote equations, or failure to obtain the required symmetric form in (c) due to algebraic mistakes |
| Step justification | 20% | 10 | For (a): proves linearity via T(αu+βv) = αT(u)+βT(v); for (b): justifies each curve feature with limit arguments or derivative tests; for (c): explicitly states why perpendicular generators require l₁l₂ + m₁m₂ + n₁n₂ = 0 and traces this to the final condition | States key results but skips some justifications—e.g., asserts linearity without proof, claims asymptotes without limits, or states the final condition without showing the orthogonality link | Bare assertions without reasoning: presents rotation matrix without derivation, sketches curve without analytical support, or writes final answer in (c) with no derivation from perpendicularity |
| Final answer & units | 20% | 10 | Clear boxed/presented answers: explicit rotation matrix with θ parameter, complete curve sketch with all features labeled, and clean derivation of (b+c)u² + (c+a)v² + (a+b)w² = 0 with concluding statement; proper handling of 'a' as arbitrary positive constant | Answers present but incomplete: matrix given without θ generalization, sketch missing some branches or asymptotes, final condition stated but with minor notational issues | Missing or garbled final answers: no explicit matrix form, no sketch in (b), or failure to reach the required symmetric condition in (c); treats 'a' as specific value rather than parameter |
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