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Mathematics 2025 Paper II 50 marks Solve

Paper II — Q6

(a) Solve (∂² u)/(∂ x²) + (∂² u)/(∂ y²) = 0 for a rectangular plate subject to the boundary conditions u(0,y) = 0, u(a,y) =…

(a)

Solve (∂² u)/(∂ x²) + (∂² u)/(∂ y²) = 0 for a rectangular plate subject to the boundary conditions u(0,y) = 0, u(a,y) = 0 u(x,0) = 0, u(x,b) = f(x) 20 marks

(b)

Simplify the Boolean function F(x,y,z) = xyz + x'yz + xy'z + xyz' and draw the corresponding GATE network. 15 marks

(c)

Calculate the moment of inertia of a uniform solid cylinder of mass M, radius R and length L with respect to a set of axes passing through the centre of the cylinder, where z-axis is the axis of the cylinder and ρ is the constant density at any point of the cylinder. Also find L/R for which the moment of inertia about x- or y-axis will be minimum for a given mass of the cylinder. 15 marks

हिंदी में प्रश्न पढ़ें
(a)

एक आयताकार प्लेट के लिए (∂² u)/(∂ x²) + (∂² u)/(∂ y²) = 0 को परिसीमा प्रतिबंधों u(0,y) = 0, u(a,y) = 0 u(x,0) = 0, u(x,b) = f(x) के अधीन हल कीजिए। (20 अंक)

(b)

बूलिय फलन F(x,y,z) = xyz + x'yz + xy'z + xyz' का सरलीकरण कीजिए और संगत GATE परिपथ को रेखांकित कीजिए। (15 अंक)

(c)

द्रव्यमान M, त्रिज्या R और लंबाई L के एक एकसमान ठोस बेलन का, बेलन के केंद्र से होकर जाने वाले अक्षों के एक समुच्चय के सापेक्ष जड़त्व आघूर्ण की गणना कीजिए, जहाँ z-अक्ष, बेलन का अक्ष है और ρ, बेलन के किसी भी बिंदु पर अचर घनत्व है। बेलन के एक दिए गए द्रव्यमान के लिए L/R, जिसके लिए x- या y-अक्ष के सापेक्ष जड़त्व आघूर्ण न्यूनतम होगा, भी ज्ञात कीजिए। (15 अंक)

Q6 of the 2025 UPSC Mains Mathematics Paper II, as printed
The question as printed in the 2025 Mathematics paper
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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) Let u(x,y)=X(x)Y(y). Substituting into ∂²u/∂x² + ∂²u/∂y² = 0 gives

X″/X = -Y″/Y = -k².

Thus X″ + k²X = 0. The boundary conditions u(0,y)=0 and u(a,y)=0 give X(0)=X(a)=0. Therefore the non-trivial solutions are

Xₙ(x)=sin(nπx/a), k=nπ/a, n=1,2,3,...

For Y, Y″ - (nπ/a)²Y = 0. Since u(x,0)=0, we have Y(0)=0, so

Yₙ(y)=sinh(nπy/a).

Hence

u(x,y)=Σ_n=1^∞ Aₙ sin(nπx/a) sinh(nπy/a).

Using u(x,b)=f(x),

f(x)=Σ_n=1^∞ Aₙ sin(nπx/a) sinh(nπb/a).

This is a Fourier sine series. Therefore

Aₙ sinh(nπb/a) = (2/a)∫_0^a f(x) sin(nπx/a) dx,

so

Aₙ = [2/(a sinh(nπb/a))]∫_0^a f(x) sin(nπx/a) dx.

Final answer:

u(x,y)=Σ_n=1^∞ Aₙ sin(nπx/a) sinh(nπy/a),

where

Aₙ = [2/(a sinh(nπb/a))]∫_0^a f(x) sin(nπx/a) dx.

This holds for a piecewise smooth f(x) satisfying f(0)=f(a)=0, the necessary compatibility from the side boundaries.

(b) Use Boolean algebra. The given function is

F = xyz + x'yz + xy'z + xyz'.

Now

xy + yz + zx = xy(z+z') + yz(x+x') + zx(y+y') = xyz + xyz' + xyz + x'yz + xyz + xy'z = xyz + xyz' + x'yz + xy'z = F.

Thus

Final answer:

F = xy + yz + zx.

This is the majority function of x, y, z.

Gate network: use three 2-input AND gates to obtain xy, yz and zx. Feed these three outputs into a 3-input OR gate. The output of the OR gate is F.

Text diagram:

x ──┐ y ──┴── AND1 ── xy ──┐ y ──┐ │ z ──┴── AND2 ── yz ── OR ── F z ──┐ │ x ──┴── AND3 ── zx ──┘

(c) Let the cylinder have uniform density ρ. Its volume is

V = πR²L,

so

M = ρπR²L.

Take the z-axis along the cylinder axis and the x- and y-axes perpendicular to it through the centre. Use cylindrical coordinates (r,θ,z), with dV = r dr dθ dz.

For the z-axis, the perpendicular distance from the axis is r. Therefore

I_z = ∫ ρ r² dV = ρ ∫_-L/2^L/2 dz ∫_0^2π dθ ∫_0^R r² · r dr = ρ L · 2π · R⁴/4 = (1/2)ρπR⁴L = (1/2)MR².

For the x-axis, the squared distance from the x-axis is y²+z² = r² sin²θ + z². Hence

I_x = ρ∫(r² sin²θ + z²) dV.

The cross-sectional part is

ρ∫ r² sin²θ dV = ρL∫_0^2π sin²θ dθ ∫_0^R r³ dr = ρL · π · R⁴/4 = MR²/4.

The longitudinal part is

ρ∫ z² dV = ρ · 2π ∫_0^R r dr ∫_-L/2^L/2 z² dz = ρ · 2π · R²/2 · L³/12 = ML²/12.

Therefore

I_x = MR²/4 + ML²/12.

By symmetry, I_y = I_x.

Final answers:

I_z = (1/2)MR²,

I_x = I_y = MR²/4 + ML²/12.

All moments are in kg m² if M is in kg and R, L are in m.

For fixed mass M and fixed density ρ, the volume πR²L is constant, so R²L is constant. Let C = R²L. Then

I_x = M C/(4L) + M L²/12.

Differentiate with respect to L:

dI_x/dL = -M C/(4L²) + M L/6.

For minimum, dI_x/dL = 0, so

L/6 = C/(4L²)

⇒ 4L³ = 6C

⇒ L³ = (3/2)C.

The second derivative is

d²I_x/dL² = M C/(2L³) + M/6 > 0,

so this stationary point gives a minimum. But C = R²L, hence

L³ = (3/2)R²L

⇒ L² = (3/2)R²

⇒ L/R = √(3/2).

Final answer:

L/R = √(3/2).

At this ratio, the moment of inertia about the x- or y-axis is minimum for a given mass.

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) derive: given > assumptions > stepwise derivation > result > check | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Complete derivations with all steps, correct results, and clear diagrams.

Key points expected

  • State separation of variables method
  • Solve X''/X = -Y''/Y = -λ²
  • Apply u(0,y)=0 and u(a,y)=0 to find λ
  • Expand f(x) in Fourier sine series
  • Apply Boolean algebra laws to simplify F
  • Identify redundant terms
  • Draw gate network for simplified F
  • Label inputs and output clearly

Evaluation rubric

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

  1. (a) General solution of Laplace's equation for a rectangular plate with given BCs. 20 marks

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

    Must cover

    • State separation of variables method
    • Solve X''/X = -Y''/Y = -λ²
    • Apply u(0,y)=0 and u(a,y)=0 to find λ
    • Expand f(x) in Fourier sine series

    Loses marks

    • Missing eigenvalue determination
    • Incorrect Fourier coefficients

    Earns more

    • Explicit form of eigenfunctions
    • General series solution for u(x,y)
    • Verification of boundary conditions

    Extra mark

    • Sketch of first mode shape
  2. (b) Simplified Boolean expression and corresponding gate network diagram. 15 marks

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

    Must cover

    • Apply Boolean algebra laws to simplify F
    • Identify redundant terms
    • Draw gate network for simplified F
    • Label inputs and output clearly

    Loses marks

    • Incorrect gate symbols
    • Missing input/output labels

    Earns more

    • K-map verification of simplification
    • Alternative simplification path noted

    Extra mark

    • Truth table for verification
  3. (c) Moments of inertia I_z, I_x, I_y and optimal L/R for minimum I_x. 15 marks

    calculate— given → formula → substitution → result with units → interpretation

    Must cover

    • Set up volume integral for I_z
    • Calculate I_z = (1/2)MR²
    • Calculate I_x = I_y = (1/12)M(3R²+L²)
    • Minimize I_x to find L/R

    Loses marks

    • Missing integration limits
    • Incorrect minimization derivative

    Earns more

    • Use of perpendicular axis theorem
    • Explicit integration steps shown
    • Differentiation for minimization

    Extra mark

    • Physical interpretation of result

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