Mathematics 2024 Paper I 50 marks Solve

Q4

(a) Let A = $\begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{pmatrix}$ be a 3×3 matrix. Find the eigenvalues and the corresponding eigenvectors of A. Hence find the eigenvalues and the corresponding eigenvectors of A⁻¹⁵, where A⁻¹⁵ = (A⁻¹)¹⁵. (20 marks) (b) Using double integration, find the area lying inside the cardioid r = a(1+cos θ) and outside the circle r = a. (15 marks) (c) Find the equation of the sphere which touches the plane 3x+2y-z+2=0 at the point (1, -2, 1) and cuts orthogonally the sphere x²+y²+z²-4x+6y+4=0. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) माना A = $\begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{pmatrix}$ एक 3×3 आव्यूह है। A का अभिलक्षणिक मान तथा संगत अभिलक्षणिक सदिश ज्ञात कीजिए। अतः A⁻¹⁵ का अभिलक्षणिक मान तथा संगत अभिलक्षणिक सदिश ज्ञात कीजिए, जहाँ A⁻¹⁵ = (A⁻¹)¹⁵ है। (20 अंक) (b) दिशा: समाकलन का प्रयोग करते हुए हृदयाक्ष (कार्डिओइड) r = a(1+cos θ) के अंदर तथा वृत्त r = a के बाह्य स्थित क्षेत्र का क्षेत्रफल ज्ञात कीजिए। (15 अंक) (c) उस गोले का समीकरण ज्ञात कीजिए, जो समतल 3x+2y-z+2=0 को बिंदु (1, -2, 1) पर स्पर्श करता है और गोले x²+y²+z²-4x+6y+4=0 को लंबिकतः काटता है। (15 अंक)

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Approach

Solve all three parts systematically, allocating approximately 40% of time to part (a) due to its 20 marks weightage, and 30% each to parts (b) and (c). Begin with setting up the characteristic equation for eigenvalues in (a), then proceed to polar integration for the cardioid area in (b), and finally apply the orthogonal sphere condition in (c). Present each part with clear headings and show all computational steps explicitly.

Key points expected

  • Part (a): Correct characteristic equation det(A-λI)=0 yielding λ³-6λ²-3λ+18=0, eigenvalues λ=6,-1,-1, corresponding eigenvectors for distinct and repeated eigenvalues, and application of eigenvalue power property for A⁻¹⁵
  • Part (a): Proper handling of inverse matrix eigenvalues (reciprocals) raised to power 15, giving eigenvalues (1/6)¹⁵, (-1)¹⁵=-1, (-1)¹⁵=-1 with same eigenvectors
  • Part (b): Correct identification of intersection points at θ=±π/2, proper setup of double integral ∫∫ r dr dθ with limits r from a to a(1+cos θ) and θ from -π/2 to π/2
  • Part (b): Accurate evaluation yielding area = a²(8+π)/2 or equivalent simplified form, with correct polar area element r dr dθ
  • Part (c): Application of sphere touching plane condition: center lies on normal line through (1,-2,1), giving center as (1+3t, -2+2t, 1-t)
  • Part (c): Use of orthogonal spheres condition 2u₁u₂+2v₁v₂+2w₁w₂=d₁+d₂ for spheres x²+y²+z²+2ux+2vy+2wz+d=0, leading to correct radius and final equation

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly sets up characteristic polynomial for (a), identifies proper integration limits and region for (b), and establishes both geometric constraints (touching plane and orthogonal condition) for (c); all initial equations error-freeSets up most equations correctly but has minor errors in characteristic polynomial coefficients, integration limits, or sphere condition applicationMajor errors in setup: wrong characteristic equation, incorrect region identification for cardioid, or failure to apply orthogonal sphere condition
Method choice20%10Uses efficient eigenvalue power property for A⁻¹⁵ rather than computing inverse directly; employs polar coordinates optimally for (b); applies standard sphere-family method with proper parameter elimination for (c)Uses correct but inefficient methods, such as computing A⁻¹ explicitly; acceptable coordinate choice but suboptimal integration order; correct sphere method with algebraic inefficiencyInappropriate methods: attempting direct matrix power computation, Cartesian coordinates for cardioid, or ignoring standard sphere equation forms
Computation accuracy20%10All arithmetic accurate: eigenvalues computed correctly, integration evaluated without error, sphere radius and center coordinates exact; algebraic simplifications precise throughoutMinor computational slips: sign errors in eigenvector calculation, arithmetic mistakes in integration that don't affect final structure, or small errors in sphere parameter determinationMajor computational errors: incorrect eigenvalues, wrong integration result, or fundamentally incorrect sphere equation due to calculation mistakes
Step justification20%10Explicitly states why eigenvectors of A and A⁻¹⁵ coincide, justifies polar area element and symmetry usage in (b), and proves why orthogonal condition reduces to 2uu'+2vv'+2ww'=d+d' for (c)Shows key steps with minimal justification; states results without explaining why eigenvalue power property holds, or presents integration without geometric reasoningMissing critical justifications: no explanation for eigenvector preservation, unexplained integration limits, or asserted sphere conditions without derivation
Final answer & units20%10All three parts with complete final answers: eigenvalues and eigenvectors clearly paired for A and A⁻¹⁵, area expressed in terms of a², sphere equation in standard form x²+y²+z²+2ux+2vy+2wz+d=0 with numerical coefficientsCorrect final forms but incomplete: missing eigenvector for repeated eigenvalue, unsimplified area expression, or sphere equation not fully reducedMissing final answers, incorrect pairing of eigenvalues/eigenvectors, wrong units or no units where expected, or incomplete sphere equation

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