Mathematics 2023 Paper I 50 marks Compulsory Solve

Q5

(a) Obtain the solution of the initial-value problem dy/dx - 2xy = 2, y(0) = 1 in the form y = eˣ²[1 + √π erf(x)]. (10 marks) (b) Given that L{f(t); p} = F(p). Show that ∫₀^∞ f(t)/t dt = ∫₀^∞ F(x)dx. Hence evaluate the integral ∫₀^∞ (e⁻ᵗ - e⁻³ᵗ)/t dt. (10 marks) (c) A cylinder of radius 'a' touches a vertical wall along a generating line. Axis of the cylinder is fixed horizontally. A uniform flat beam of length 'l' and weight 'W' rests with its extremities in contact with the wall and the cylinder, making an angle of 45° with the vertical. If frictional forces are neglected, then show that a/l = (√5 + 5)/(4√2). Also, find the reactions of the cylinder and wall. (10 marks) (d) A particle is moving under Simple Harmonic Motion of period T about a centre O. It passes through the point P with velocity v along the direction OP and OP = p. Find the time that elapses before the particle returns to the point P. What will be the value of p when the elapsed time is T/2 ? (10 marks) (e) If ā = sinθî + cosθĵ + θk̂, b̄ = cosθî - sinθĵ - 3k̂, c̄ = 2î + 3ĵ - 3k̂, then find the values of the derivative of the vector function ā × (b̄ × c̄) w.r.t. θ at θ = π/2 and θ = π. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) प्रारंभिक-मान समस्या : dy/dx - 2xy = 2, y(0) = 1 का हल y = eˣ²[1 + √π erf(x)] के रूप में प्राप्त कीजिए। (10 अंक) (b) दिया गया है L{f(t); p} = F(p). दर्शाइए कि ∫₀^∞ f(t)/t dt = ∫₀^∞ F(x)dx. अतः समाकल ∫₀^∞ (e⁻ᵗ - e⁻³ᵗ)/t dt का मान ज्ञात कीजिए। (10 अंक) (c) अर्ध्व्यास 'a' का एक बेलन (सिलिंडर) एक जनक रेखा के अनुदिश एक उद्वाधर दीवार को स्पर्श किया हुआ है। बेलन का अक्ष क्षैतिजतः स्थिर है। लम्बाई 'l' तथा भार 'W' का एक एक समान समतल दंड उद्वाधर से 45° का कोण बनाते हुए अपने सिरों को दीवार के सहारे तथा बेलन पर टिकाए है। अगर घर्षण बल नगण्य हैं, तब दर्शाइए कि a/l = (√5 + 5)/(4√2). दीवार और बेलन की प्रतिक्रियाएँ भी ज्ञात कीजिए। (10 अंक) (d) कोई कण केन्द्र 'O' के सापेक्ष आवर्त काल T के साथ सरल आवर्त गति में गतिशील है। कण बिन्दु P से OP के अनुदिश दिशा में v वेग से गुजरता है तथा OP = p है। कण का बिन्दु P पर पुनः लौटने में लगा समय ज्ञात कीजिए। यदि लगा समय T/2 हो, तो p का मान क्या होगा ? (10 अंक) (e) यदि ā = sinθî + cosθĵ + θk̂, b̄ = cosθî - sinθĵ - 3k̂, c̄ = 2î + 3ĵ - 3k̂, तो सदिश फलन ā × (b̄ × c̄) के θ के सापेक्ष अवकलज के मान, θ = π/2 और θ = π पर ज्ञात कीजिए। (10 अंक)

Evaluate your answer to this question → See all 2023 Mathematics questions

Directive word: Solve

This question asks you to solve. 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

Solve all five sub-parts systematically, allocating approximately 20% time to each part given equal 10-mark weighting. For (a), apply integrating factor method for linear ODE; for (b), use Laplace transform properties with Fubini's theorem; for (c), draw free-body diagram and apply equilibrium conditions; for (d), use SHM equations with phase analysis; for (e), apply vector triple product identity before differentiation. Present solutions with clear intermediate steps and boxed final answers.

Key points expected

  • Part (a): Identify integrating factor e^(-x²), solve using standard formula, apply initial condition y(0)=1, and manipulate to express using error function erf(x)
  • Part (b): Prove the identity using Laplace definition and changing order of integration (Fubini/Tonelli), then apply to f(t)=e^(-t)-e^(-3t) with F(p)=1/(p+1)-1/(p+3) to get ln(3)
  • Part (c): Construct geometry with 45° beam, locate contact points, write three equilibrium equations (ΣFx=0, ΣFy=0, ΣM=0), eliminate reactions to find a/l ratio, then back-substitute for reaction magnitudes
  • Part (d): Use SHM equation x = Asin(ωt+φ), apply conditions at point P to find phase, determine return time using periodicity and symmetry, evaluate p when elapsed time is T/2
  • Part (e): Apply vector identity ā×(b̄×c̄) = (ā·c̄)b̄ - (ā·b̄)c̄, compute scalar products, differentiate resultant vector component-wise, evaluate at θ=π/2 and θ=π

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies integrating factor for (a), states Laplace definition and domain conditions for (b), draws accurate free-body diagram with geometry for (c), writes standard SHM equation with ω=2π/T for (d), applies correct vector triple product identity for (e)Most setups correct but minor errors in initial conditions for (a), missing convergence justification for (b), incomplete geometry for (c), wrong angular frequency for (d), or direct expansion instead of identity for (e)Fundamental setup errors: wrong integrating factor, incorrect Laplace definition, missing forces in diagram, linear motion equations for SHM, or incorrect vector operations
Method choice20%10Uses optimal methods: integrating factor for linear ODE, Fubini's theorem for integral swap, moment equilibrium about strategic point for (c), phase-angle method for SHM, vector identity before differentiationCorrect but suboptimal methods: variation of parameters for (a), integration by parts without Fubini for (b), force resolution without moment optimization for (c), energy method missing time for (d), component-wise cross product for (e)Inappropriate methods: separation of variables for (a), residue calculus without justification for (b), virtual work without setup for (c), kinematic equations for (d), numerical differentiation for (e)
Computation accuracy20%10Flawless calculations: correct integration ∫e^(-x²)dx to erf form, proper evaluation of ∫[1/x - 1/(x+2)]dx = ln(3), exact algebraic simplification to (√5+5)/(4√2), correct return time formula, accurate vector derivatives at both pointsMinor computational slips: sign errors in integration constants, arithmetic errors in fraction simplification, one incorrect reaction value, wrong phase calculation, or one incorrect derivative componentMajor computational failures: incorrect Gaussian integral handling, divergent integral claimed convergent, wrong trigonometric substitution in (c), SHM period errors, or completely wrong vector derivatives
Step justification20%10Explicitly justifies: integrating factor derivation, Fubini/Tonelli conditions for swapping integrals, equilibrium condition selection, SHM phase determination uniqueness, vector identity proof or citationSome steps justified but gaps: assumes integrating factor form, states integral swap without conditions, asserts equilibrium without showing force directions, states phase without derivation, applies identity without namingMinimal or no justification: jumps to solution, claims results without derivation, missing free-body diagram explanation, asserts SHM properties without proof, computes derivatives without showing steps
Final answer & units20%10All answers in required forms: y=e^(x²)[1+√π erf(x)] for (a), ln(3) for (b), exact ratio and reaction forces in terms of W for (c), time expression and p=0 for (d), specific numerical vectors for (e)Correct answers but not in specified form: equivalent but unsimplified expressions, decimal approximations instead of exact values, missing one reaction component, incomplete time expression, one wrong evaluation pointMissing or wrong final answers: incorrect functional form, wrong numerical value, incomplete reaction set, physically impossible time, or no evaluation at specified points

Practice this exact question

Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.

Evaluate my answer →

More from Mathematics 2023 Paper I