Paper I — Q3
(a)(i) If u = x² + y², v = x² - y², where x = rcosθ, y = rsinθ, then find (∂(u,v))/(∂(r,θ)). (7 marks) (a)(ii) If ∫limits₀^x…
If u = x² + y², v = x² - y², where x = rcosθ, y = rsinθ, then find (∂(u,v))/(∂(r,θ)). 7 marks
If ∫limits₀^x f(t) dt = x + ∫limitsₓ¹ tf(t) dt, then find the value of f(1). 5 marks
Express ∫limitsₐ^b (x-a)^m (b-x)ⁿ dx in terms of Beta function. 8 marks
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
Prove that the eigen vectors, corresponding to two distinct eigen values of a real symmetric matrix, are orthogonal. 8 marks
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
यदि u = x² + y², v = x² - y², जहाँ पर x = rcosθ, y = rsinθ है, तब (∂(u,v))/(∂(r,θ)) ज्ञात कीजिए। (7 अंक)
यदि ∫limits₀^x f(t) dt = x + ∫limitsₓ¹ tf(t) dt है, तो f(1) का मान ज्ञात कीजिए। (5 अंक)
∫limitsₐ^b (x-a)^m (b-x)ⁿ dx को बीटा-फलन के रूप में व्यक्त कीजिए। (8 अंक)
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)
सिद्ध कीजिए कि एक वास्तविक सममित आव्यूह के दो भिन्न अभिलक्षणिक मानों के संगत अभिलक्षणिक सदिश, लंबिक हैं। (8 अंक)
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)
Model answer
Drafted with AI by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key. How we build rubrics and answers.
(a)(i) Jacobian ∂(u, v)/∂(r, θ)
u = r², v = r²cos 2θ. So u_r = 2r, u_θ = 0, v_r = 2r cos 2θ, v_θ = −2r² sin 2θ.
∂(u, v)/∂(r, θ) = (2r)(−2r² sin 2θ) − 0 = −4r³ sin 2θ.
(a)(ii) Value of f(1)
Differentiating ∫₀ˣ f(t)dt = x + ∫ₓ¹ t f(t)dt with respect to x (f continuous): f(x) = 1 − x f(x). So f(x) = 1/(1 + x) and f(1) = 1/2.
(a)(iii) Beta function form
Put x = a + (b − a)t, so dx = (b − a)dt, x − a = (b − a)t and b − x = (b − a)(1 − t), with t from 0 to 1.
∫ₐᵇ (x − a)ᵐ(b − x)ⁿ dx = (b − a)^(m+n+1) ∫₀¹ tᵐ(1 − t)ⁿ dt = (b − a)^(m+n+1) B(m + 1, n + 1), for m, n > −1.
(b) Locus of the foot of the perpendicular
Let the centre be (α, β, γ), with α² + β² + γ² = r² because the sphere passes through O. Its equation is x² + y² + z² − 2αx − 2βy − 2γz = 0, which meets the axes at A(2α, 0, 0), B(0, 2β, 0), C(0, 0, 2γ).
Plane ABC: X/2α + Y/2β + Z/2γ = 1, with normal N = (1/2α, 1/2β, 1/2γ). The foot of the perpendicular from O is P = (X, Y, Z) = N/|N|². Since |P|² = 1/|N|², we get 1/(2α) = X/(X² + Y² + Z²), so α = (X² + Y² + Z²)/(2X), and likewise β, γ with Y, Z.
Substituting in α² + β² + γ² = r²:
(X² + Y² + Z²)²(1/X² + 1/Y² + 1/Z²) = 4r².
(c)(i) Eigenvectors of a real symmetric matrix
Let Ax = λx and Ay = μy with λ ≠ μ. Using Aᵀ = A: λ(x·y) = (Ax)·y = x·(Ay) = μ(x·y). So (λ − μ)(x·y) = 0, and since λ ≠ μ, x·y = 0. The eigenvectors are orthogonal.
(c)(ii) trace(AB) = trace(BA)
For 2 × 2 matrices, trace(AB) = Σᵢⱼ aᵢⱼbⱼᵢ = a₁₁b₁₁ + a₁₂b₂₁ + a₂₁b₁₂ + a₂₂b₂₂, and trace(BA) = Σᵢⱼ bᵢⱼaⱼᵢ is the same sum. Hence trace(AB) = trace(BA).
If AB − BA = I₂, taking traces gives 0 on the left and 2 on the right, a contradiction. So AB − BA ≠ I₂.
How this answer will be evaluated
What earns full marks: Complete stepwise derivations with all justifications, correct final results, and verification steps.
Evaluation rubric
Each sub-part is marked on its own, against the marks and word limit printed on the paper.
- (a)(i) Find the Jacobian ∂(u,v)/∂(r,θ). 7 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Express u = r² and v = r² cos 2θ
- Compute the four partial derivatives
- Evaluate the determinant −4r³ sin 2θ
Loses marks
- Sign error
- Computing ∂(u,v)/∂(x,y) instead
Earns more
- Equivalent form −8r³ sinθ cosθ
- (a)(ii) Find f(1) from the integral equation. 5 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Differentiate both sides using the fundamental theorem
- Obtain f(x) = 1 − x f(x)
- State f(1) = 1/2
Loses marks
- Sign error in derivative of ∫ₓ¹
Earns more
- Note continuity of f for differentiation
- (a)(iii) Express the integral in terms of the Beta function. 8 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- Substitute x = a + (b−a)t
- Change limits to 0 and 1
- Result (b−a)^(m+n+1) B(m+1, n+1)
Loses marks
- Missing factor (b−a)^(m+n+1)
- Wrong Beta arguments
Earns more
- State validity m, n > −1
- (b) Find the locus of the foot of the perpendicular from O to plane ABC. 15 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- Equation of the sphere and intercepts 2α, 2β, 2γ
- Foot of the perpendicular and expressions for α, β, γ
- Locus (X²+Y²+Z²)²(1/X²+1/Y²+1/Z²) = 4r²
Loses marks
- Wrong intercepts
- Failing to eliminate α, β, γ
Earns more
- Use α²+β²+γ² = r²
- (c)(i) Prove eigenvectors of distinct eigenvalues of a real symmetric matrix are orthogonal. 8 marks
justify— claim → 3-4 reasons → evidence → conclusion
Must cover
- Set Ax = λx, Ay = μy with λ ≠ μ
- Use symmetry to get λ(x·y) = μ(x·y)
- Conclude x·y = 0
Loses marks
- Assuming orthogonality
- Missing use of λ ≠ μ
Earns more
- Explicit use of Aᵀ = A
- (c)(ii) Show trace(AB) = trace(BA) and deduce AB − BA ≠ I₂. 7 marks
justify— claim → 3-4 reasons → evidence → conclusion
Must cover
- Write both traces as the same sum
- Take trace of AB − BA = I₂
- Reach the contradiction 0 = 2
Loses marks
- Assuming AB = BA
Earns more
- Explicit 2 × 2 entries
What "Solve" is asking you to do
Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.
Structure that answers it
Given data and what is required → method chosen, with the reason it applies → set-up (equation, circuit, free body, trial balance) → working, step by step → answer with units and any condition of validity
Where marks are lost
Doing the middle steps mentally and writing only the result. In mathematics papers, a further loss comes from giving a decimal where the exact value in surds or fractions was wanted, or from skipping the justification a part explicitly asks for.
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