Paper I — Q1
(a) If A= 1 & -1 & 1 2 & -1 & 0 1 & 0 & 0 , then show that A² = A⁻¹ (without finding A⁻¹). (10 marks) (b) Find the matrix…
If A= 1 & -1 & 1 2 & -1 & 0 1 & 0 & 0 , then show that A² = A⁻¹ (without finding A⁻¹). 10 marks
Find the matrix associated with the linear operator on V₃(R) defined by T(a, b, c) = (a+b, a-b, 2c) with respect to the ordered basis B = (0, 1, 1), (1, 0, 1), (1, 1, 0). 10 marks
Given: Δ(x)= f(x+α) & f(x+2α) & f(x+3α) f(α) & f(2α) & f(3α) f'(α) & f'(2α) & f'(3α) where f is a real valued differentiable function and α is a constant. Find displaystylelimₓ → 0 (Δ(x))/x. 10 marks
Show that between any two roots of e^x cos x = 1, there exists at least one root of e^x sin x - 1 = 0. 10 marks
Find the equation of the cylinder whose generators are parallel to the line x = -y/2 = z/3 and whose guiding curve is x² + 2y² = 1, z = 0. 10 marks
हिंदी में प्रश्न पढ़ें
यदि A= 1 & -1 & 1 2 & -1 & 0 1 & 0 & 0 है, तो A⁻¹ को ज्ञात किए बिना दर्शाइए कि A² = A⁻¹। (10 अंक)
क्रमित आधारक B = (0, 1, 1), (1, 0, 1), (1, 1, 0) के सापेक्ष V₃(R) पर परिभाषित रैखिक संकारक : T(a, b, c) = (a+b, a-b, 2c) से संबंधित आव्यूह ज्ञात कीजिए। (10 अंक)
दिया गया है : Δ(x)= f(x+α) & f(x+2α) & f(x+3α) f(α) & f(2α) & f(3α) f'(α) & f'(2α) & f'(3α) जहाँ f एक वास्तविक-मान अवकलनीय फलन है तथा α एक अचर है। displaystylelimₓ → 0 (Δ(x))/x को ज्ञात कीजिए। (10 अंक)
दर्शाइए कि e^x cos x = 1 के किन्हीं दो मूलों के बीच में e^x sin x - 1 = 0 का कम से कम एक मूल विद्यमान है। (10 अंक)
उस बेलन का समीकरण ज्ञात कीजिए जिसके जनक, रेखा : x = -y/2 = z/3 के समानांतर हैं तथा जिसका निर्देशक-वक्र x² + 2y² = 1, z = 0 है। (10 अंक)
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) A² = A⁻¹ without finding A⁻¹
It suffices to show A³ = I, since then A·A² = I and so A² is the inverse of A.
A² = A·A = [0 0 1; 0 −1 2; 1 −1 1] (rows separated by semicolons).
A³ = A²·A: row 1 gives (1, 0, 0); row 2 gives (0, 1, 0); row 3 gives (0, 0, 1). So A³ = I.
Hence A·A² = A²·A = I, and therefore A² = A⁻¹.
(b) Matrix of T relative to the basis B
Let v₁ = (0,1,1), v₂ = (1,0,1), v₃ = (1,1,0). Then T(v₁) = (1,−1,2), T(v₂) = (1,1,2), T(v₃) = (2,0,0).
Since αv₁ + βv₂ + γv₃ = (β + γ, α + γ, α + β), a vector (x, y, z) has coordinates α = (y + z − x)/2, β = (x + z − y)/2, γ = (x + y − z)/2.
T(v₁): (α, β, γ) = (0, 2, −1). T(v₂): (1, 1, 0). T(v₃): (−1, 1, 1).
These are the columns, so [T]_B = [0 1 −1; 2 1 1; −1 0 1].
(c) Limit of Δ(x)/x
At x = 0 the first row of Δ equals the second row, so Δ(0) = 0 and the limit is Δ′(0).
Only the first row depends on x; its derivative at 0 is (f′(α), f′(2α), f′(3α)), which equals the third row. Therefore Δ′(0) is a determinant with two equal rows, and
lim Δ(x)/x = 0.
(d) Roots of eˣ sin x − 1 = 0 between roots of eˣ cos x = 1
Let φ(x) = e⁻ˣ − cos x. The roots of eˣ cos x = 1 are exactly the zeros of φ. Let a < b be two such roots, so φ(a) = φ(b) = 0.
φ is continuous on [a, b] and differentiable on (a, b), so by Rolle's theorem there is c in (a, b) with φ′(c) = 0, that is, −e⁻ᶜ + sin c = 0.
Multiplying by eᶜ gives eᶜ sin c − 1 = 0, so c is a root of eˣ sin x − 1 = 0.
(e) Equation of the cylinder
The generators have direction ratios (1, −2, 3), since x = −y/2 = z/3.
A point (u, v, 0) of the guiding curve satisfies u² + 2v² = 1. The generator through it is (x, y, z) = (u + λ, v − 2λ, 3λ), so λ = z/3, u = x − z/3, v = y + 2z/3.
Substituting: (x − z/3)² + 2(y + 2z/3)² = 1.
Expanding and multiplying by 3 gives 3x² + 6y² + 3z² − 2xz + 8yz = 3.
How this answer will be evaluated
What earns full marks: All parts fully solved with correct methods and no errors.
Evaluation rubric
Each sub-part is marked on its own, against the marks and word limit printed on the paper.
- (1(a)) Show A² = A⁻¹ without computing the inverse. 10 marks
justify— claim → 3-4 reasons → evidence → conclusion
Must cover
- Compute A² correctly
- Show A³ = I
- Conclude A² = A⁻¹
Loses marks
- Finding A⁻¹ directly
- Arithmetic error in A²
Earns more
- Use of associativity (A·A² = A²·A)
- Clear matrix multiplication steps
Extra mark
- Remark that A satisfies a cubic equation
- (1(b)) Find the matrix of T relative to the ordered basis B. 10 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Compute T of each basis vector
- Express each image in basis B
- Write the columns as [T]_B = [0 1 −1; 2 1 1; −1 0 1]
Loses marks
- Writing images as rows instead of columns
- Using the standard basis
Earns more
- General coordinate formula for (x,y,z)
- Check of one column
Extra mark
- Verification via P⁻¹MP
- (1(c)) Find the limit of Δ(x)/x as x tends to 0. 10 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Observe Δ(0) = 0 since two rows coincide
- Identify the limit as Δ′(0)
- Show Δ′(0) has two equal rows so the limit is 0
Loses marks
- Expanding the determinant unnecessarily
- Stating 0 without justification
Earns more
- Row-wise differentiation of the determinant
- (1(d)) Show a root of eˣ sin x − 1 = 0 lies between two roots of eˣ cos x = 1. 10 marks
justify— claim → 3-4 reasons → evidence → conclusion
Must cover
- Define φ(x) = e⁻ˣ − cos x with φ(a) = φ(b) = 0
- Apply Rolle's theorem
- Derive eᶜ sin c − 1 = 0
Loses marks
- Using intermediate value theorem wrongly
- Incorrect φ′
Earns more
- Check continuity and differentiability
- (1(e)) Find the equation of the cylinder with given generator direction and guiding curve. 10 marks
derive— given → assumptions → stepwise derivation → result → check
Must cover
- Direction ratios (1, −2, 3)
- Eliminate parameters to get (x − z/3)² + 2(y + 2z/3)² = 1
- Final form 3x² + 6y² + 3z² − 2xz + 8yz = 3
Loses marks
- Wrong direction ratios
- Expansion errors
Earns more
- Parametric form of the generator
What "Prove" is asking you to do
Establish that the statement holds for every case it claims, not for one representative case. The argument must be closed: each line follows from a definition, a hypothesis, or a named theorem you are entitled to use.
Structure that answers it
Given and to prove, restated → theorem or construction to be used, named → the argument line by line → conclusion stated as proved
Where marks are lost
Testing one example, which illustrates but proves nothing. On an if and only if claim, proving one direction and stopping forfeits that half outright, and degenerate cases — zero, the empty set, the equality case — have to be disposed of rather than assumed away.
Practice this exact question
Write your answer and it is marked point by point against the model answer above — what you covered, what you missed, what you got wrong.
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