Mathematics 2023 Paper I 50 marks Solve

Q3

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ (i) Verify the Cayley-Hamilton theorem for the matrix $A$. (ii) Show that $A^n = A^{n-2} + A^2 - I$ for $n \geq 3$, where $I$ is the identity matrix of order 3. Hence, find $A^{40}$. 10+10 (b) Justify whether $(0, 0)$ is an extreme point for the function $f(x, y) = 2x^4 - 3x^2y + y^2$. 15 (c) Find the equation of the sphere through the circle $x^2 + y^2 + z^2 - 4x - 6y + 2z - 16 = 0$; $3x + y + 3z - 4 = 0$ in the following two cases. (i) the point $(1, 0, -3)$ lies on the sphere. (ii) the given circle is a great circle of the sphere. 15

हिंदी में प्रश्न पढ़ें

दिया गया है $A=\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ (i) आव्यूह A के लिये कैले-हैमिल्टन प्रमेय को सत्यापित कीजिए। (ii) दर्शाइए कि n ≥ 3 के लिये Aⁿ = Aⁿ⁻² + A² – I; जहाँ I कोटि 3 का तत्समक आव्यूह है। अतः A⁴⁰ ज्ञात कीजिए। 10+10 (b) तर्क सहित दर्शाइये कि $(0, 0)$, फलन $f(x, y) = 2x^4 - 3x^2y + y^2$ का चरम-बिन्दु है अथवा नहीं। 15 (c) वृत्त $x^2 + y^2 + z^2 - 4x - 6y + 2z - 16 = 0$; $3x + y + 3z - 4 = 0$ से होकर गुजरने वाले गोले का समीकरण निम्न दो स्थितियों में ज्ञात कीजिए। (i) बिन्दु $(1, 0, -3)$ गोले पर हो। (ii) दिया गया वृत्त गोले का एक बृहत् वृत्त हो। 15

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How this answer will be evaluated

Approach

Solve this multi-part problem by allocating approximately 40% time to part (a) covering Cayley-Hamilton verification and recurrence relation (20 marks), 30% to part (b) on extreme point analysis using Hessian and higher-order tests (15 marks), and 30% to part (c) on sphere equations through given circle with two conditions (15 marks). Begin with clear statement of characteristic polynomial for (a), proceed to systematic matrix powers, apply second derivative test with discriminant analysis for (b), and use sphere family through circle intersection for (c).

Key points expected

  • For (a)(i): Correct computation of characteristic polynomial det(A - λI) = -λ³ + λ² + λ - 1 and verification that A³ = A² + A - I
  • For (a)(ii): Proof of recurrence Aⁿ = Aⁿ⁻² + A² - I using Cayley-Hamilton, and efficient computation of A⁴⁰ via pattern or binary exponentiation
  • For (b): Computation of first partials fx = 8x³ - 6xy, fy = -3x² + 2y, verification that (0,0) is critical point, and application of discriminant D = fxx·fyy - (fxy)² with higher-order analysis showing saddle point
  • For (c)(i): Formation of sphere family S: x²+y²+z²-4x-6y+2z-16 + λ(3x+y+3z-4) = 0 and substitution of (1,0,-3) to find λ
  • For (c)(ii): Condition that given circle is great circle requires sphere center to lie on plane 3x+y+3z-4=0, yielding center (-4+3λ)/2, (-6+λ)/2, (2+3λ)/2 satisfying plane equation
  • Explicit final answers: A⁴⁰ expression, conclusion that (0,0) is not extreme point (saddle), and two sphere equations with specific λ values

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness18%9Correctly identifies characteristic polynomial coefficients for (a), sets up proper critical point conditions for (b), and forms valid sphere family equation S + λP = 0 for (c) with correct geometric interpretationMinor errors in polynomial setup or sphere family form, but recognizable attempt at correct structure; critical point conditions partially correctFundamental errors: wrong characteristic polynomial, incorrect partial derivatives, or invalid sphere equation setup
Method choice22%11Uses Cayley-Hamilton for recurrence proof in (a), applies discriminant test with higher-order verification for (b), and correctly distinguishes point-substitution vs. great-circle condition for (c)Appropriate methods chosen but inefficient computation of A⁴⁰ or incomplete analysis of critical point nature; correct approach to sphere but algebraic errorsWrong methods: direct computation of A⁴⁰ without recurrence, using only second derivative test without checking D=0 case, or confusing sphere conditions
Computation accuracy24%12Accurate determinant calculation, correct matrix powers, precise discriminant computation showing D(0,0)=0 with proper follow-up, and exact λ values -2/3 and 2 for (c)(i) and (c)(ii)Minor arithmetic errors in matrix entries or λ calculation, but salvageable structure; sign errors in partial derivativesMajor computational errors: incorrect characteristic roots, wrong A² or A³, algebraic mistakes in discriminant, or incorrect sphere equations
Step justification20%10Clear proof that A satisfies its characteristic equation, explicit induction or direct verification for recurrence, justification of higher-order test when D=0, and geometric reasoning for great circle condition (center on plane, radius condition)Steps shown but gaps in logical flow; assumes recurrence without proof, or states saddle point without examining f(x,√(3/2)x²) behaviorMissing crucial justifications: no verification of Cayley-Hamilton, asserts A⁴⁰ without showing pattern, or claims extreme point without analysis
Final answer & units16%8Complete answers: explicit A⁴⁰ in terms of A², A, I; clear statement that (0,0) is saddle point not extreme; simplified sphere equations x²+y²+z²-6x-6y-z-8=0 for (c)(i) and x²+y²+z²-2x-8y+8z+4=0 for (c)(ii)Correct forms but unsimplified or missing one component; conclusion present but poorly expressedMissing final answers, wrong conclusions, or incomplete sphere equations without numerical values

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