Mathematics 2021 Paper I 50 marks Solve

Q3

(a)(i) If $u = x^2 + y^2$, $v = x^2 - y^2$, where $x = r\cos\theta$, $y = r\sin\theta$, then find $\frac{\partial(u,v)}{\partial(r,\theta)}$. (7 marks) (a)(ii) If $\int\limits_{0}^{x} f(t)\,dt = x + \int\limits_{x}^{1} tf(t)\,dt$, then find the value of $f(1)$. (5 marks) (a)(iii) Express $\int\limits_{a}^{b} (x-a)^m (b-x)^n\,dx$ in terms of Beta function. (8 marks) (b) A sphere of constant radius $r$ passes through the origin $O$ and cuts the axes at the points $A, B$ and $C$. Find, the locus of the foot of the perpendicular drawn from $O$ to the plane $ABC$. (15 marks) (c)(i) Prove that the eigen vectors, corresponding to two distinct eigen values of a real symmetric matrix, are orthogonal. (8 marks) (c)(ii) For two square matrices A and B of order 2, show that trace (AB) = trace (BA). Hence show that AB - BA ≠ I₂, where I₂ is an identity matrix of order 2. (7 marks)

हिंदी में प्रश्न पढ़ें

(a)(i) यदि $u = x^2 + y^2$, $v = x^2 - y^2$, जहाँ पर $x = r\cos\theta$, $y = r\sin\theta$ है, तब $\frac{\partial(u,v)}{\partial(r,\theta)}$ ज्ञात कीजिए। (7 अंक) (a)(ii) यदि $\int\limits_{0}^{x} f(t)\,dt = x + \int\limits_{x}^{1} tf(t)\,dt$ है, तो $f(1)$ का मान ज्ञात कीजिए। (5 अंक) (a)(iii) $\int\limits_{a}^{b} (x-a)^m (b-x)^n\,dx$ को बीटा-फलन के रूप में व्यक्त कीजिए। (8 अंक) (b) अचर त्रिज्या $r$ का एक गोला मूल-बिंदु $O$ से गुजरता है तथा अक्षों को $A, B, C$ बिंदुओं पर काटता है। $O$ से समतल $ABC$ पर खींचे गए लंब-पाद का बिंदुपथ ज्ञात कीजिए। (15 अंक) (c)(i) सिद्ध कीजिए कि एक वास्तविक सममित आव्यूह के दो भिन्न अभिलक्षणिक मानों के संगत अभिलक्षणिक सदिश, लंबिक हैं। (8 अंक) (c)(ii) दो वर्ग आव्यूह A तथा B जिनकी कोटि, 2 है के लिए दर्शाइए कि अनुरेख (AB) = अनुरेख (BA)। अतैव दर्शाइए कि AB - BA ≠ I₂ जहाँ I₂ एक 2-कोटि का तत्समक आव्यूह है। (7 अंक)

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Approach

Solve all six sub-parts systematically, allocating approximately 35% time to part (b) (15 marks) as the highest-weighted component, 30% to part (a) (20 marks across three items), and 35% to part (c) (15 marks across two items). Begin each sub-part with clear statement of given conditions, show complete working with proper mathematical notation, and conclude with boxed final answers. For (a)(iii) and (c)(i), explicitly state theorems being applied (Beta function definition, spectral theorem for symmetric matrices).

Key points expected

  • For (a)(i): Correct application of chain rule for Jacobians, computing ∂(u,v)/∂(x,y) and ∂(x,y)/∂(r,θ) separately, then multiplying to get final result 8r³sinθcosθ or equivalent simplified form
  • For (a)(ii): Differentiation under integral sign using Leibniz rule, establishing f(x) = 1 - xf(x), solving to get f(x) = 1/(1+x), hence f(1) = 1/2
  • For (a)(iii): Substitution x = a + (b-a)t to transform limits to 0 and 1, identifying parameters p = m+1, q = n+1, final answer as (b-a)^(m+n+1) B(m+1, n+1)
  • For (b): Setting up sphere equation x²+y²+z²-2ux-2vy-2wz=0 with center (u,v,w), using |OA|=|OB|=|OC|=2r condition, finding plane ABC as x/u + y/v + z/w = 2, deriving foot of perpendicular coordinates and eliminating parameters to get locus x⁻² + y⁻² + z⁻² = r⁻²
  • For (c)(i): Using definition of eigenvectors AX=λX, AY=μY with λ≠μ, exploiting symmetry A=Aᵀ to show λXᵀY = μXᵀY, hence XᵀY=0 proving orthogonality
  • For (c)(ii): Direct computation of trace(AB) and trace(BA) showing equality via ΣΣaᵢⱼbⱼᵢ, then using trace(AB-BA)=0 while trace(I₂)=2 to establish contradiction

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%12Correctly identifies all given conditions for each sub-part: proper Jacobian chain rule setup for (a)(i), correct integral equation interpretation for (a)(ii), appropriate substitution identification for (a)(iii), accurate sphere equation with center coordinates for (b), precise eigenvalue-eigenvector definitions for (c)(i), and correct trace definition for (c)(ii)Minor errors in initial setup such as wrong Jacobian multiplication order, incorrect limits handling, or misstated sphere equation, but recoverable with partial creditFundamental misunderstanding of problem requirements: treating Jacobian as single partial derivative, ignoring integral limits, or using wrong coordinate system for sphere
Method choice20%12Selects optimal methods: chain rule decomposition for Jacobians, Leibniz differentiation for integral equations, standard Beta substitution, parametric elimination for locus, inner product approach for orthogonality proof, and direct element-wise computation for trace equalityUses correct but inefficient methods, or makes suboptimal choices like direct computation of ∂(u,v)/∂(r,θ) without chain rule simplification, or coordinate geometry instead of vector approach for (b)Applies completely wrong methods: attempting numerical integration for (a)(ii), using spherical coordinates unnecessarily, or algebraic expansion instead of trace properties for (c)(ii)
Computation accuracy20%12Flawless arithmetic and algebraic manipulation: correct partial derivatives (4x, 4y, -4y, 4x for u,v w.r.t x,y), accurate determinant calculations, precise integration results, correct elimination of three parameters in (b), and valid algebraic steps in trace proofMinor computational slips: sign errors in Jacobian, arithmetic mistakes in Beta function exponents, or algebraic errors in locus derivation that don't fundamentally derail the solutionMajor computational errors: incorrect determinant evaluation, wrong differentiation under integral sign, or invalid algebraic steps leading to wrong final forms
Step justification20%12Clear justification at each step: explicit statement of Jacobian multiplication theorem, Leibniz rule application with conditions, Beta function definition reference, geometric reasoning for sphere properties, and logical flow in orthogonality proof with explicit use of symmetrySome steps justified but others assumed without comment, or missing explicit theorem citations while maintaining correct logical flowMinimal or no justification: jumps between steps without explanation, omits critical reasoning for orthogonality conclusion, or presents unexplained algebraic manipulations
Final answer & units20%12All six sub-parts with correct final answers in proper form: 8r³sinθcosθ or 4r³sin2θ for (a)(i), f(1)=1/2 for (a)(ii), (b-a)^(m+n+1)B(m+1,n+1) for (a)(iii), x⁻²+y⁻²+z⁻²=r⁻² for (b), complete orthogonality proof for (c)(i), and clear contradiction demonstration for (c)(ii)Most answers correct but some incomplete: missing simplified forms, wrong numerical values in one or two parts, or incomplete final statement for (b)Multiple incorrect final answers, missing answers for sub-parts, or answers without supporting working that appear guessed

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