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Mathematics 2021 Paper I 50 marks Solve

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…

(a)
(i)

If u = x² + y², v = x² - y², where x = rcosθ, y = rsinθ, then find (∂(u,v))/(∂(r,θ)). 7 marks

(ii)

If ∫limits₀^x f(t) dt = x + ∫limitsₓ¹ tf(t) dt, then find the value of f(1). 5 marks

(iii)

Express ∫limitsₐ^b (x-a)^m (b-x)ⁿ dx in terms of Beta function. 8 marks

(b)

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

(c)
(i)

Prove that the eigen vectors, corresponding to two distinct eigen values of a real symmetric matrix, are orthogonal. 8 marks

(ii)

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
(a)
(i)

यदि u = x² + y², v = x² - y², जहाँ पर x = rcosθ, y = rsinθ है, तब (∂(u,v))/(∂(r,θ)) ज्ञात कीजिए। (7 अंक)

(ii)

यदि ∫limits₀^x f(t) dt = x + ∫limitsₓ¹ tf(t) dt है, तो f(1) का मान ज्ञात कीजिए। (5 अंक)

(iii)

∫limitsₐ^b (x-a)^m (b-x)ⁿ dx को बीटा-फलन के रूप में व्यक्त कीजिए। (8 अंक)

(b)

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)

(c)
(i)

सिद्ध कीजिए कि एक वास्तविक सममित आव्यूह के दो भिन्न अभिलक्षणिक मानों के संगत अभिलक्षणिक सदिश, लंबिक हैं। (8 अंक)

(ii)

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)

Q3 of the 2021 UPSC Mains Mathematics Paper I, as printed
The question as printed in the 2021 Mathematics paper
Evaluate your answer to this question → See all 2021 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)(i) Put x = r cosθ, y = r sinθ. Then

u = x² + y² = r²,

v = x² − y² = r²(cos²θ − sin²θ) = r² cos2θ.

By the Jacobian formula,

∂u/∂r = 2r, ∂u/∂θ = 0,

∂v/∂r = 2r cos2θ, ∂v/∂θ = −2r² sin2θ.

Hence

∂(u,v)/∂(r,θ) =

| 2r 0 | | 2r cos2θ −2r² sin2θ |

= (2r)(−2r² sin2θ) − 0

= −4r³ sin2θ = −8r³ sinθ cosθ.

Answer: ∂(u,v)/∂(r,θ) = −4r³ sin2θ = −8r³ sinθ cosθ.

(a)(ii) Assume f is continuous so that Leibniz rule applies. Differentiate both sides with respect to x:

d/dx ∫₀ˣ f(t)dt = f(x).

Also,

d/dx [x + ∫ₓ¹ t f(t)dt] = 1 − x f(x).

Therefore,

f(x) = 1 − x f(x),

so

(1 + x)f(x) = 1.

Putting x = 1,

2f(1) = 1, hence f(1) = 1/2.

Answer: f(1) = 1/2.

(a)(iii) Use the substitution

x = a + (b − a)t, so dx = (b − a)dt.

When x = a, t = 0; when x = b, t = 1. Then

x − a = (b − a)t, b − x = (b − a)(1 − t).

Thus

∫ₐᵇ (x − a)^m (b − x)^n dx

= ∫₀¹ [(b − a)t]^m [(b − a)(1 − t)]^n (b − a)dt

= (b − a)^(m+n+1) ∫₀¹ t^m (1 − t)^n dt.

By definition of the Beta function,

∫₀¹ t^m (1 − t)^n dt = B(m + 1, n + 1).

Therefore,

Answer: ∫ₐᵇ (x − a)^m (b − x)^n dx = (b − a)^(m+n+1) B(m + 1, n + 1), valid for m > −1, n > −1.

(b) Let the centre of the sphere be (α, β, γ). Since the sphere has radius r and passes through the origin O,

α² + β² + γ² = r².

Its equation is

x² + y² + z² − 2αx − 2βy − 2γz = 0.

On the x-axis, y = z = 0, so

x² − 2αx = 0, hence x = 0 or x = 2α.

Thus A = (2α, 0, 0). Similarly,

B = (0, 2β, 0), C = (0, 0, 2γ).

The plane ABC has intercept form

X/(2α) + Y/(2β) + Z/(2γ) = 1.

Its normal vector is

N = (1/(2α), 1/(2β), 1/(2γ)).

Let P = (X, Y, Z) be the foot of the perpendicular from O to the plane ABC. For the plane N·R = 1, the foot from the origin is

P = N/|N|².

Let

D = |N|² = 1/(4α²) + 1/(4β²) + 1/(4γ²).

Then

X = (1/(2α))/D, Y = (1/(2β))/D, Z = (1/(2γ))/D.

Hence

α = 1/(2XD), β = 1/(2YD), γ = 1/(2ZD).

Also |P|² = 1/D, so

D = 1/(X² + Y² + Z²).

Therefore,

α = (X² + Y² + Z²)/(2X),

β = (X² + Y² + Z²)/(2Y),

γ = (X² + Y² + Z²)/(2Z).

Substitute into α² + β² + γ² = r²:

[(X² + Y² + Z²)/(2X)]² + [(X² + Y² + Z²)/(2Y)]² + [(X² + Y² + Z²)/(2Z)]² = r².

So

(X² + Y² + Z²)²/4 · (1/X² + 1/Y² + 1/Z²) = r².

Thus

Answer: locus is (X² + Y² + Z²)²(1/X² + 1/Y² + 1/Z²) = 4r², or equivalently (X² + Y² + Z²)²(Y²Z² + Z²X² + X²Y²) = 4r²X²Y²Z².

(c)(i) Let A be a real symmetric matrix. Let x and y be eigenvectors corresponding to distinct eigenvalues λ and μ:

A x = λx, A y = μy, λ ≠ μ.

Since A is symmetric,

(Ax)·y = x·(Ay).

But (Ax)·y = (λx)·y = λ(x·y), and x·(Ay) = x·(μy) = μ(x·y).

Hence

λ(x·y) = μ(x·y),

so

(λ − μ)(x·y) = 0.

Since λ ≠ μ, we get x·y = 0.

Answer: the eigenvectors corresponding to distinct eigenvalues are orthogonal.

(c)(ii) Let A = (aᵢⱼ), B = (bᵢⱼ), i, j = 1, 2. Then

trace(AB) = a₁₁b₁₁ + a₁₂b₂₁ + a₂₁b₁₂ + a₂₂b₂₂.

Similarly,

trace(BA) = b₁₁a₁₁ + b₁₂a₂₁ + b₂₁a₁₂ + b₂₂a₂₂.

The two sums are identical because scalar products commute. Hence

trace(AB) = trace(BA).

Now suppose, if possible, AB − BA = I₂. Taking trace,

trace(AB − BA) = trace(I₂).

But

trace(AB − BA) = trace(AB) − trace(BA) = 0,

while trace(I₂) = 1 + 1 = 2.

Thus 0 = 2, impossible.

Answer: trace(AB) = trace(BA); hence AB − BA ≠ I₂.

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(i)) calculate: given > formula > substitution > result with units > interpretation | (a(ii)) calculate: given > formula > substitution > result with units > interpretation | (a(iii)) derive: given > assumptions > stepwise derivation > result > check | (b) derive: given > assumptions > stepwise derivation > result > check | (c(i)) justify: claim > 3-4 reasons > evidence > conclusion | (c(ii)) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete stepwise derivations with all justifications, correct final results, and verification steps.

Key points expected

  • Compute partial derivatives ∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y
  • Compute partial derivatives ∂x/∂r, ∂x/∂θ, ∂y/∂r, ∂y/∂θ
  • Apply chain rule to find ∂(u,v)/∂(x,y) and ∂(x,y)/∂(r,θ)
  • Multiply determinants to obtain final Jacobian value
  • Differentiate both sides with respect to x
  • Apply Leibniz rule for variable limits of integration
  • Substitute x=1 into the resulting differential equation
  • Solve for f(1) using the boundary condition

Evaluation rubric

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

  1. (a(i)) Compute the Jacobian determinant ∂(u,v)/∂(r,θ) using the chain rule. 7 marks

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

    Must cover

    • Compute partial derivatives ∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y
    • Compute partial derivatives ∂x/∂r, ∂x/∂θ, ∂y/∂r, ∂y/∂θ
    • Apply chain rule to find ∂(u,v)/∂(x,y) and ∂(x,y)/∂(r,θ)
    • Multiply determinants to obtain final Jacobian value

    Loses marks

    • Skipping intermediate partial derivative steps
    • Incorrect application of chain rule for Jacobians

    Earns more

    • Explicitly state the Jacobian chain rule formula
    • Show intermediate determinant calculations clearly

    Extra mark

    • Verify result using direct substitution of x,y in u,v
  2. (a(ii)) Find the value of f(1) from the given integral equation. 5 marks

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

    Must cover

    • Differentiate both sides with respect to x
    • Apply Leibniz rule for variable limits of integration
    • Substitute x=1 into the resulting differential equation
    • Solve for f(1) using the boundary condition

    Loses marks

    • Incorrect application of Leibniz rule for variable limits
    • Failing to substitute x=1 correctly

    Earns more

    • State Leibniz rule explicitly before applying
    • Show the step where f(x) terms cancel or simplify

    Extra mark

    • Verify the solution by substituting back into original equation
  3. (a(iii)) Express the integral in terms of the Beta function. 8 marks

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

    Must cover

    • Perform substitution t = (x-a)/(b-a) to transform limits
    • Express dx in terms of dt and simplify the integrand
    • Identify the resulting form as Beta(m+1, n+1)
    • State the final answer as (b-a)^(m+n+1) B(m+1, n+1)

    Loses marks

    • Incorrect substitution leading to wrong Beta function arguments
    • Failing to account for the (b-a) factor in the final answer

    Earns more

    • Show the substitution steps clearly with limits transformation
    • State the definition of Beta function used

    Extra mark

    • Mention the relationship B(m,n) = Γ(m)Γ(n)/Γ(m+n)
  4. (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

    • Use sphere condition a² + b² + c² = 4r² (diameter property)
    • Find foot of perpendicular from O to plane using formula

    Loses marks

    • Incorrect sphere condition or missing the 4r² factor
    • Failing to eliminate parameters to get the locus equation

    Earns more

    • Justify why a² + b² + c² = 4r² using sphere geometry
    • Show the foot of perpendicular formula derivation

    Extra mark

    • Provide a neat 3D diagram showing sphere, axes, and plane ABC
  5. (c(i)) Prove eigenvectors for distinct eigenvalues of real symmetric matrix are orthogonal. 8 marks

    justify— claim → 3-4 reasons → evidence → conclusion

    Must cover

    • Let Ax = λx and Ay = μy with λ ≠ μ
    • Use symmetry: xᵀAx = (Ax)ᵀx = λxᵀx
    • Show xᵀAy = μxᵀy and (Ax)ᵀy = λxᵀy
    • Conclude (λ - μ)xᵀy = 0, so xᵀy = 0

    Loses marks

    • Failing to use the symmetry property A = Aᵀ
    • Incorrect algebraic manipulation in the proof

    Earns more

    • State the spectral theorem for real symmetric matrices
    • Clearly define the inner product notation used

    Extra mark

    • Mention that this generalizes to Hermitian matrices in complex case
  6. (c(ii)) Show trace(AB) = trace(BA) and AB - BA ≠ I₂. 7 marks

    justify— claim → 3-4 reasons → evidence → conclusion

    Must cover

    • Write A and B as 2x2 matrices with explicit entries
    • Compute AB and BA element-wise
    • Show trace(AB) = a₁₁b₁₁ + a₁₂b₂₁ + a₂₁b₁₂ + a₂₂b₂₂

    Loses marks

    • Failing to show the explicit matrix multiplication
    • Incorrect trace calculation or missing the contradiction step

    Earns more

    • Use general n×n case for trace equality before specializing to 2×2
    • Clearly state the contradiction argument for AB - BA ≠ I₂

    Extra mark

    • Mention that trace is cyclic: trace(ABC) = trace(BCA) = trace(CAB)

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