(a) If $A=\begin{bmatrix} 1 & -1 & 1 \ 2 & -1 & 0 \ 1 & 0 & 0 \end{bmatrix}$, then show that $A^2 = A^{-1}$ (without finding $A^{-1}$). (10 marks)

(b) Find the matrix associated with the linear operator on $V_3(R)$ defined by $T(a, b, c) = (a+b, a

Question

(a) If $A=\begin{bmatrix} 1 & -1 & 1 \ 2 & -1 & 0 \ 1 & 0 & 0 \end{bmatrix}$, then show that $A^2 = A^{-1}$ (without finding $A^{-1}$). (10 marks)

(b) Find the matrix associated with the linear operator on $V_3(R)$ 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)

(c) Given: $$\Delta(x)=\begin{vmatrix} f(x+\alpha) & f(x+2\alpha) & f(x+3\alpha) \ f(\alpha) & f(2\alpha) & f(3\alpha) \ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix}$$ where $f$ is a real valued differentiable function and $\alpha$ is a constant. Find $\displaystyle\lim_{x 	o 0} \frac{\Delta(x)}{x}$. (10 marks)

(d) 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)

(e) Find the equation of the cylinder whose generators are parallel to the line $x = -\frac{y}{2} = \frac{z}{3}$ and whose guiding curve is $x^2 + 2y^2 = 1$, $z = 0$. (10 marks)

UPSC Answer Check · Accepted Answer

Prove the required results across all five sub-parts with rigorous mathematical reasoning. For (a), compute A² and show A³=I implying A²=A⁻¹ without explicit inversion; for (b), construct the change-of-basis matrix and apply T to basis vectors; for (c), use Taylor expansion and determinant properties; for (d), apply Rolle's theorem to the appropriate auxiliary function; for (e), use the standard cylinder equation with given generator direction. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e), reflecting the analytical depth required for the calculus and analysis components.
- For (a): Compute A² by matrix multiplication, then verify A³ = I₃ (identity), hence A² = A⁻¹ without computing inverse explicitly
- For (b): Apply T to each basis vector of B, express results as linear combinations of B, and assemble coefficients as columns of the matrix representation
- For (c): Expand f(x+kα) = f(kα) + xf'(kα) + O(x²), substitute into determinant, identify leading term as x times a 3×3 determinant involving f and f' values
- For (d): Define g(x) = e⁻ˣ - cos x, note roots of g correspond to roots of eˣcos x = 1, apply Rolle's theorem to g between consecutive roots
- For (e): Use direction ratios (1, -2, 3) from generator line, write cylinder as locus of points at fixed distance from axis with given guiding curve x² + 2y² = 1, z = 0
- For (c): The limit equals the determinant with rows [f'(α), f'(2α), f'(3α)], [f(α), f(2α), f(3α)], [f'(α), f'(2α), f'(3α)] which simplifies appropriately
- For (d): Show g'(x) = -e⁻ˣ + sin x = 0 implies eˣsin x = 1, completing the proof via intermediate value property
- For (e): Final equation eliminates parameter to give 13(x² + 2y²) - (x - 2y + 3z)² = 13 or equivalent standard form

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies A³ verification for (a), constructs proper change-of-basis framework for (b), sets up Taylor expansion for (c), defines appropriate auxiliary function g(x) for (d), and extracts correct direction ratios (1,-2,3) for (e)	Minor errors in basis identification for (b) or direction ratio extraction for (e); incomplete setup for Taylor expansion in (c)	Wrong approach entirely for any sub-part, such as attempting direct matrix inversion in (a) or missing the Rolle's theorem connection in (d)
Method choice	20%	10	Selects optimal methods: A³=I for (a), matrix of T in standard basis then change-of-basis or direct computation for (b), determinant linearity properties for (c), Rolle's theorem with correct auxiliary function for (d), and distance-from-axis or envelope method for (e)	Correct but inefficient methods, such as computing A⁻¹ explicitly in (a) or using Lagrange interpolation unnecessarily in (c)	Inappropriate methods like direct limit evaluation without expansion in (c) or ignoring the geometric constraint in (e)
Computation accuracy	20%	10	Flawless matrix multiplications in (a) and (b), correct determinant evaluation in (c) yielding \|f'(α) f'(2α) f'(3α); f(α) f(2α) f(3α); f'(α) f'(2α) f'(3α)\|, precise derivative calculations in (d), and accurate algebraic manipulation for cylinder equation in (e)	Minor arithmetic slips in matrix entries or determinant expansion, sign errors in derivative calculations, or algebraic slips in final cylinder simplification	Major computational errors such as wrong matrix products, incorrect determinant evaluation, or fundamentally wrong final equations
Step justification	20%	10	Explicitly states why A³=I implies A²=A⁻¹, justifies basis change formula T_B = P⁻¹[T]_stdP or direct computation, explains determinant multilinearity for (c), clearly invokes Rolle's theorem hypotheses for (d), and verifies all points on cylinder satisfy the generator condition for (e)	States key theorems but with gaps in verifying hypotheses, particularly for Rolle's theorem application or missing verification that constructed cylinder contains all generators	Unjustified leaps, missing theorem statements, or logical gaps such as asserting existence of root without proper intermediate value argument
Final answer & units	20%	10	Presents clean final forms: explicit verification A²=A⁻¹ for (a), correct 3×3 matrix for (b), simplified determinant expression for limit in (c), clear statement of Rolle's theorem conclusion for (d), and standard form 13x² + 22y² + 9z² + 4xy - 6xz + 12yz = 13 or equivalent for (e)	Correct answers but in unsimplified form, or missing final boxed/conclusive statements for one or two sub-parts	Missing final answers, wrong conclusions, or failure to explicitly state what was to be proved in each sub-part

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