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Mathematics 2021 Paper II 50 marks Solve

Paper II — Q6

(a) Solve the wave equation a²∂²u/∂x² = ∂²u/∂t², 0<x<L, t>0 subject to the conditions u(0,t)=0, u(L,t)=0 u(x,0)=(1/4)x(L-x)…

(a)

Solve the wave equation

a²∂²u/∂x² = ∂²u/∂t², 0<x<L, t>0

subject to the conditions

u(0,t)=0, u(L,t)=0 u(x,0)=(1/4)x(L-x), ∂u/∂t|ₜ₌₀=0 20 marks

(b)

Obtain the Boolean function F(x, y, z) based on the table given below. Then simplify F(x, y, z) and draw the corresponding GATE network:

xyzF(x, y, z)
1111
1101
1011
1000
0111
0100
0010
0000(15 marks)
(c)

Obtain the Lagrangian equation for the motion of a system of two particles of unequal masses connected by an inextensible string passing over a small smooth pulley. 15 marks

हिंदी में प्रश्न पढ़ें
(a)

तरंग समीकरण

a²∂²u/∂x² = ∂²u/∂t², 0<x<L, t>0

का शर्तों

u(0,t)=0, u(L,t)=0 u(x,0)=(1/4)x(L-x), ∂u/∂t|ₜ₌₀=0

से प्रतिबंधित हल प्राप्त कीजिए। (20 अंक)

(b)

नीचे दी गई सारणी पर आधारित बूलियन फलन F(x,y,z) को निकालिए और तब F(x,y,z) को सरल कीजिए तथा उसके अनुरूप GATE परिपथ खींचिए :

xyzF(x,y,z)
1111
1101
1011
1000
0111
0100
0010
0000(15 अंक)
(c)

एक छोटी चिकनी घिरनी के ऊपर से गुजरने वाली एक अवितान्य डोरी के सिरों से बंधे असमान संहति वाले दो कणों के निकाय की गति के लिए लग्रांजी समीकरण प्राप्त कीजिए। (15 अंक)

Q6 of the 2021 UPSC Mains Mathematics Paper II, as printed
The question as printed in the 2021 Mathematics paper
Evaluate your answer to this question → See all 2021 Mathematics questions

The figure this question refers to, in words

The question paper is a scan and the diagram did not survive as text. This is the figure as read from the original page — every component, value and label — so the question can be worked from the text below.

(b) Table with four columns: x, y, z, F(x, y, z) Row 1: 1, 1, 1, 1 Row 2: 1, 1, 0, 1 Row 3: 1, 0, 1, 1 Row 4: 1, 0, 0, 0 Row 5: 0, 1, 1, 1 Row 6: 0, 1, 0, 0 Row 7: 0, 0, 1, 0 Row 8: 0, 0, 0, 0

(b) Table with 4 columns and 8 data rows: x, y, z, F(x, y, z) 1, 1, 1, 1 1, 1, 0, 1 1, 0, 1, 1 1, 0, 0, 0 0, 1, 1, 1 0, 1, 0, 0 0, 0, 1, 0 0, 0, 0, 0

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) Wave equation with fixed ends

Put u = X(x)T(t). Then X″/X = T″/(a²T) = −λ². The conditions u(0,t) = u(L,t) = 0 give X(0) = X(L) = 0, so λₙ = nπ/L and Xₙ = sin(nπx/L). From T″ + a²λₙ²T = 0 and u_t(x,0) = 0 we get Tₙ = cos(nπat/L). Hence

u(x,t) = Σ Aₙ sin(nπx/L) cos(nπat/L).

The condition u(x,0) = x(L−x)/4 makes Aₙ the Fourier sine coefficient:

Aₙ = (2/L)∫₀ᴸ [x(L−x)/4] sin(nπx/L) dx = (1/2L)∫₀ᴸ x(L−x) sin(nπx/L) dx.

Integrating by parts twice, ∫₀ᴸ x(L−x) sin(nπx/L) dx = 2L³[1 − (−1)ⁿ]/(n³π³). So

Aₙ = L²[1 − (−1)ⁿ]/(n³π³),

which is 0 for even n and 2L²/(n³π³) for odd n. Therefore

u(x,t) = Σ (n = 1, 3, 5, …) [2L²/(n³π³)] sin(nπx/L) cos(nπat/L).

Check: at t = 0, x = L/2 the series gives (2L²/π³)·(π³/32) = L²/16, which equals x(L−x)/4 there.

(b) Boolean function and gate network

F = 1 for the rows 111, 110, 101 and 011, so

F = xyz + xyz′ + xy′z + x′yz.

Pairing adjacent minterms (as on a K-map): xyz + xyz′ = xy, xyz + xy′z = xz, xyz + x′yz = yz. Hence

F(x,y,z) = xy + xz + yz.

This is the majority function: F = 1 when at least two inputs are 1.

Gate network: three 2-input AND gates form xy, xz and yz; their outputs feed one 3-input OR gate, whose output is F.

Diagram: AND-OR gate network for F = xy + xz + yz

(c) Lagrangian for two masses over a smooth pulley

Because the string is inextensible, one coordinate suffices. Let x be the downward displacement of m₁; then m₂ rises by x and both move with speed ẋ.

T = ½(m₁ + m₂)ẋ².

V = −m₁gx + m₂gx + constant = −(m₁ − m₂)gx + constant.

L = T − V = ½(m₁ + m₂)ẋ² + (m₁ − m₂)gx.

Lagrange's equation d/dt(∂L/∂ẋ) − ∂L/∂x = 0 gives

(m₁ + m₂)ẍ = (m₁ − m₂)g,

so ẍ = [(m₁ − m₂)/(m₁ + m₂)]g. This is downward for m₁ if m₁ > m₂, and zero when m₁ = m₂.

How this answer will be evaluated

What earns full marks: Complete derivations with all steps shown, correct final results, and clear diagrams where required.

Evaluation rubric

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

  1. (a) Solve the wave equation by separation of variables with the given boundary and initial conditions 20 marks

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

    Must cover

    • Separation of variables with Xₙ = sin(nπx/L) from X(0) = X(L) = 0
    • Tₙ = cos(nπat/L) from u_t(x,0) = 0
    • Aₙ = 2L²/(n³π³) for odd n, 0 for even n, with the final series

    Loses marks

    • Using cosine instead of sine for the spatial part
    • Missing the zero even-n coefficients
    • Ignoring u_t(x,0) = 0

    Earns more

    • Correct separated ODEs with λₙ = nπ/L
    • Fourier sine coefficient integral set up and evaluated
    • Final series stated

    Extra mark

    • Checks that the series at t = 0, x = L/2 gives L²/16
  2. (b) Obtain F from the truth table, simplify it and draw the gate network 15 marks

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

    Must cover

    • Minterms 111, 110, 101, 011 identified
    • F simplified to xy + xz + yz
    • Network of three AND gates feeding one OR gate

    Loses marks

    • Including minterms where F = 0
    • Leaving F unsimplified
    • Wrong gate type in the network

    Earns more

    • Canonical sum of products written correctly
    • Pairing of minterms (K-map or Boolean laws) shown
    • Network inputs and gates correctly connected

    Extra mark

    • Notes that F is the majority function
  3. (c) Obtain the Lagrangian equation of motion for two unequal masses over a smooth pulley 15 marks

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

    Must cover

    • Single coordinate x using the inextensible-string constraint
    • L = ½(m₁ + m₂)ẋ² + (m₁ − m₂)gx
    • (m₁ + m₂)ẍ = (m₁ − m₂)g

    Loses marks

    • Using two independent coordinates without the constraint
    • Wrong sign of potential energy
    • Writing Newton's laws instead of Lagrange's equation

    Earns more

    • Kinetic energy with equal speeds for both masses
    • Potential energy with the correct sign
    • Acceleration (m₁ − m₂)g/(m₁ + m₂) stated

    Extra mark

    • Checks the case m₁ = m₂
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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