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Mathematics 2022 Paper II 50 marks Solve

Paper II — Q6

(a) Solve the heat equation ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x)…

(a)

Solve the heat equation ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x), 0 ≤ x ≤ l. 20 marks

(b)

Find a combinatorial circuit corresponding to the Boolean function f(x, y, z) = [x · (ȳ + z)] + y and write the input/output table for the circuit. 15 marks

(c)

Find the moment of inertia of a right circular solid cone about one of its slant sides (generator) in terms of its mass M, height h and the radius of base as a. 15 marks

हिंदी में प्रश्न पढ़ें
(a)

उष्मा समीकरण ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 का शर्तों u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x), 0 ≤ x ≤ l से प्रतिबंधित हल ज्ञात कीजिये। (20 अंक)

(b)

बूलिय फलन f(x, y, z) = [x · (ȳ + z)] + y का संगत संयोजीव्यास परिपथ (कॉम्बिनेटोरियल सर्किट) ज्ञात कीजिये तथा परिपथ के लिये निवेश/निर्गत (इनपुट/आउटपुट) सारणी लिखिये। (15 अंक)

(c)

एक लम्ब वृत्तीय ठोस शंकु का उसकी संहति M, ऊँचाई h तथा आधार की त्रिज्या a के रूप में उसकी एक तिर्यक रेखा (जनक रेखा) के सापेक्ष जड़त्व-आघूर्ण ज्ञात कीजिये। (15 अंक)

Q6 of the 2022 UPSC Mains Mathematics Paper II, as printed
The question as printed in the 2022 Mathematics paper
Evaluate your answer to this question → See all 2022 Mathematics questions

Model answer

Written 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.

(a) Using separation of variables, write u(x,t)=X(x)T(t). Then

T′/T = X″/X = -λ.

The boundary conditions u(0,t)=u(l,t)=0 give X(0)=X(l)=0. Hence λₙ=(nπ/l)² and Xₙ(x)=sin(nπx/l), n=1,2,3,… . Therefore

u(x,t)=Σₙ₌₁^∞ Bₙ sin(nπx/l) exp(-n²π²t/l²).

At t=0,

x(l-x)=Σₙ₌₁^∞ Bₙ sin(nπx/l),

so by the Fourier sine series,

Bₙ=(2/l)∫₀^l x(l-x) sin(nπx/l) dx.

Let k=nπ/l. Then

Bₙ=(2/l)[l∫₀^l x sin(kx) dx − ∫₀^l x² sin(kx) dx].

Now,

∫₀^l x sin(kx) dx = −l(−1)ⁿ/k,

and

∫₀^l x² sin(kx) dx = −l²(−1)ⁿ/k + 2((−1)ⁿ−1)/k³.

Thus

Bₙ=(2/l)·2(1−(−1)ⁿ)/k³ = 4l²(1−(−1)ⁿ)/(n³π³).

So Bₙ=0 for even n, and for odd n,

Bₙ=8l²/(n³π³).

Hence

u(x,t)=8l²/π³ Σₘ₌₀^∞ sin((2m+1)πx/l) exp(−(2m+1)²π²t/l²)/(2m+1)³, 0≤x≤l, t≥0.

(b) The Boolean function is

f(x,y,z)=[x·(¬y+z)]+y.

Construct the circuit by these gates:

  • Take input y through a NOT gate to get ¬y.
  • Feed ¬y and z into an OR gate. Its output is A=¬y+z.
  • Feed x and A into an AND gate. Its output is B=x·(¬y+z).
  • Feed B and y into a final OR gate. Its output is f=B+y.

Thus the gate sequence is: y→NOT→OR with z→AND with x→OR with y→f.

Input/output table:

  • x=0, y=0, z=0 → f=0
  • x=0, y=0, z=1 → f=0
  • x=1, y=0, z=0 → f=1
  • x=1, y=0, z=1 → f=1
  • x=0, y=1, z=0 → f=1
  • x=0, y=1, z=1 → f=1
  • x=1, y=1, z=0 → f=1
  • x=1, y=1, z=1 → f=1

Equivalently, when y=1 the output is always 1; when y=0 the output equals x. Thus the function simplifies to f=x+y (OR).

(c) Let the cone have apex at the origin, axis along z, height h, base radius a. Its uniform density is

ρ=M/(⅓πa²h)=3M/(πa²h).

Choose one generator, the line from the apex to the base point (a,0,h). Put s²=a²+h². At height z, the cone satisfies 0≤r≤az/h, with x=r cosθ, y=r sinθ. For a point P(x,y,z), the square of its perpendicular distance from the generator is

d²=x²+y²+z²−(ax+hz)²/s².

Substituting x=r cosθ, y=r sinθ gives

d²=r²(1−a²cos²θ/s²)+a²z²/s²−2ah r z cosθ/s².

Integrating over θ from 0 to 2π, the cosθ term vanishes and ∫cos²θ dθ=π, so

∫₀^(2π) d² dθ=πr²(a²+2h²)/s²+2πa²z²/s².

Therefore the moment of inertia about the generator is

I=ρ∫₀^h ∫₀^(az/h) ∫₀^(2π) d²·r dθ dr dz

=ρπ/s²∫₀^h ∫₀^(az/h)[(a²+2h²)r³+2a²z²r] dr dz.

The inner integral equals

(a²+2h²)(az/h)⁴/4+a²z²(az/h)²

=a⁴z⁴(a²+6h²)/(4h⁴).

Hence

I=ρπ/s²·a⁴(a²+6h²)/(4h⁴)·h⁵/5

=ρπa⁴h(a²+6h²)/(20s²).

Using ρ=3M/(πa²h), this becomes

I=3Ma²(a²+6h²)/(20(a²+h²)).

This is valid for a uniform right circular solid cone about a generator, with mass M, height h and base radius a.

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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How this answer will be evaluated

Approach

(a) derive: given > assumptions > stepwise derivation > result > check | (b) describe: define > structure or process in order > labelled diagram > significance | (c) derive: given > assumptions > stepwise derivation > result > check Full marks: Complete derivations with all steps, correct results, and verification

Key points expected

  • Apply separation of variables to find general solution
  • Satisfy boundary conditions u(0,t)=u(l,t)=0
  • Expand initial condition x(l-x) in Fourier sine series
  • Calculate Fourier coefficients explicitly
  • Draw circuit with correct logic gates
  • Implement function f(x,y,z) = [x·(ȳ+z)]+y
  • Provide complete truth table for all 8 inputs
  • Label inputs and outputs clearly

Evaluation rubric

Each sub-part is marked on its own, against the marks and word limit printed on the paper.

  1. (a) Complete solution to the heat equation satisfying boundary and initial conditions. 20 marks

    derive— given → assumptions → stepwise derivation → result → check

    Must cover

    • Apply separation of variables to find general solution
    • Satisfy boundary conditions u(0,t)=u(l,t)=0
    • Expand initial condition x(l-x) in Fourier sine series
    • Calculate Fourier coefficients explicitly

    Loses marks

    • Skipping intermediate steps in derivation
    • Answer without working
    • Incorrect Fourier coefficient calculation

    Earns more

    • State separation of variables method by name
    • Show eigenvalue problem for spatial part
    • Verify solution satisfies PDE and conditions
    • Check special case or limit behavior

    Extra mark

    • Neat figure showing initial condition
    • Alternative method noted briefly
  2. (b) Combinatorial circuit diagram and complete input/output table for given Boolean function. 15 marks

    describe— define → structure or process in order → labelled diagram → significance

    Must cover

    • Draw circuit with correct logic gates
    • Implement function f(x,y,z) = [x·(ȳ+z)]+y
    • Provide complete truth table for all 8 inputs
    • Label inputs and outputs clearly

    Loses marks

    • Missing or incorrect truth table entries
    • Circuit not matching Boolean function
    • Unlabelled or unclear diagram

    Earns more

    • Show intermediate gate outputs
    • Verify circuit against truth table
    • Use standard logic gate symbols
    • Simplify or optimize circuit if possible

    Extra mark

    • Neat labelled diagram
    • Alternative circuit implementation noted
  3. (c) Moment of inertia of solid cone about slant side in terms of M, h, a. 15 marks

    derive— given → assumptions → stepwise derivation → result → check

    Must cover

    • Set up coordinate system with slant side as axis
    • Use integration to find moment of inertia
    • Express result in terms of M, h, a
    • Show all integration steps clearly

    Loses marks

    • Skipping integration steps
    • Incorrect coordinate setup
    • Final answer without derivation

    Earns more

    • State parallel axis theorem if used
    • Define all variables before use
    • Verify dimensions of final result
    • Check special case (e.g., h→0 or a→0)

    Extra mark

    • Neat figure showing cone and axis
    • Alternative method noted briefly

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