Q3
(a) Reduce the following matrix to echelon form: $$A = \begin{bmatrix} 2 & -2 & 2 & 1 \\ -3 & 6 & 0 & -1 \\ 1 & -7 & 10 & 2 \end{bmatrix}$$ (15 marks) (b) Find the equations of the spheres which pass through the circle $x^2 + y^2 + z^2 - 2x + 2y + 4z - 3 = 0$, $2x + y + z = 4$ and touch the plane $3x + 4y = 14$. (15 marks) (c) (i) Evaluate $\displaystyle\iint\limits_R y\, dx\, dy$, where R is the region bounded by $y = x$ and $y = 4x - x^2$. (10 marks) (ii) If $u(x,y) = x\, f\left(\dfrac{y}{x}\right) + g\left(\dfrac{y}{x}\right)$, where f and g are arbitrary functions, then show that I. $\quad x\,\dfrac{\partial u}{\partial x} + y\,\dfrac{\partial u}{\partial y} = x\, f\left(\dfrac{y}{x}\right)$, II. $\quad x^2\,\dfrac{\partial^2 u}{\partial x^2} + 2xy\,\dfrac{\partial^2 u}{\partial x\,\partial y} + y^2\,\dfrac{\partial^2 u}{\partial y^2} = 0$. (10 marks)
हिंदी में प्रश्न पढ़ें
(a) निम्नलिखित आव्यूह को सोपानक (एशेलोन) रूप में समानीत कीजिए : $$A = \begin{bmatrix} 2 & -2 & 2 & 1 \\ -3 & 6 & 0 & -1 \\ 1 & -7 & 10 & 2 \end{bmatrix}$$ (15 अंक) (b) उन गोलों के समीकरण ज्ञात कीजिए जो वृत्त $x^2 + y^2 + z^2 - 2x + 2y + 4z - 3 = 0$, $2x + y + z = 4$ से होकर गुजरते हैं और समतल $3x + 4y = 14$ को स्पर्श करते हैं। (15 अंक) (c) (i) $\displaystyle\iint\limits_R y\, dx\, dy$ का मान ज्ञात कीजिए, जहाँ R, $y = x$ तथा $y = 4x - x^2$ से परिवृत क्षेत्र है। (10 अंक) (ii) यदि $u(x,y) = x\, f\left(\dfrac{y}{x}\right) + g\left(\dfrac{y}{x}\right)$ है, जहाँ f और g स्वेच्छ फलन हैं, तो दर्शाइए कि I. $\quad x\,\dfrac{\partial u}{\partial x} + y\,\dfrac{\partial u}{\partial y} = x\, f\left(\dfrac{y}{x}\right)$ है, II. $\quad x^2\,\dfrac{\partial^2 u}{\partial x^2} + 2xy\,\dfrac{\partial^2 u}{\partial x\,\partial y} + y^2\,\dfrac{\partial^2 u}{\partial y^2} = 0$ है। (10 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve all four sub-parts systematically, allocating approximately 30% time to part (a) matrix reduction, 30% to part (b) sphere equations, 25% to part (c)(i) double integration, and 15% to part (c)(ii) partial derivative proofs. Begin each sub-part with clear identification of the mathematical technique, show complete working with row operations for (a), sphere family parameterization for (b), region sketching and limits for (c)(i), and chain rule applications for (c)(ii). Conclude with boxed final answers for each part.
Key points expected
- For (a): Correct application of elementary row operations to achieve row echelon form with leading entries 1, 2, 0 in successive rows and proper identification of pivot positions
- For (b): Formation of sphere family equation S + λP = 0, correct application of tangency condition using distance from center to plane equals radius, yielding two valid sphere equations
- For (c)(i): Accurate sketch of region R bounded by line y=x and parabola y=4x-x², correct intersection points (0,0) and (3,3), proper order of integration with limits 0 to 3 for x and x to 4x-x² for y
- For (c)(ii): Verification of Euler's homogeneous function theorem for part I using substitution v=y/x and chain rule, and confirmation of part II as second-degree Euler equation for homogeneous functions
- Clear presentation of all row operation steps in (a) with explicit notation (R₂ → R₂ + ³⁄₂R₁ etc.) to enable partial credit tracing
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): Correct initial augmented matrix setup with proper identification of 3×4 dimensions; for (b): Accurate sphere family equation S + λ(2x+y+z-4)=0 with correct center (-1-λ, 1-λ/2, 2-λ/2) and radius formula; for (c)(i): Precise region identification with intersection solving; for (c)(ii): Correct substitution v=y/x and proper partial derivative setup | Minor errors in sphere family constant term or radius extraction; incomplete region sketch for (c)(i); partial substitution in (c)(ii) with some derivative terms correct | Fundamental setup errors such as wrong matrix dimensions, incorrect sphere family form, misidentified region boundaries, or failure to use substitution v=y/x |
| Method choice | 20% | 10 | For (a): Strategic row operations choosing pivot a₂₁=1 for numerical stability; for (b): Elegant use of tangency condition |3α+4β-14|/5 = r rather than discriminant method; for (c)(i): dx dy order chosen over dy dx for simpler limits; for (c)(ii): Direct application of Euler's theorem rather than brute-force differentiation | Correct but suboptimal methods: using a₁₁=2 as pivot creating fractions, solving quadratic for plane-sphere intersection, or full expansion of second derivatives in (c)(ii) | Incorrect method selection such as attempting eigenvalue decomposition for echelon form, using wrong tangency condition, or integrating in wrong order with unsolvable limits |
| Computation accuracy | 20% | 10 | For (a): Flawless arithmetic yielding echelon form with entries 0, 1, -3, -1/2 in final row; for (b): Exact λ values (2 and -4/3) with simplified sphere equations; for (c)(i): Precise evaluation yielding 27/4 or 6.75; for (c)(ii): Complete cancellation verifying both identities | Single arithmetic slip (e.g., sign error in row operation, one incorrect λ value, integration arithmetic error) with remaining work correct | Multiple computational errors: incorrect row sums, wrong quadratic solutions, major integration mistakes, or failure to simplify final expressions |
| Step justification | 20% | 10 | Explicit row operation notation at each step (R₁ ↔ R₃, R₂ → R₂ + 3R₁, etc.); clear statement of tangency condition |d|=r; labeled diagram or verbal description of region R; explicit citation of Euler's homogeneous function theorem with degree verification for (c)(ii) | Some steps shown but with gaps (e.g., row operations implied but not stated, tangency condition used without explicit formula, region described without intersection calculation) | Missing justifications: no row operation indicators, unexplained jumps to answers, unlabeled calculations, or assertion of identities without derivation |
| Final answer & units | 20% | 10 | For (a): Clear echelon matrix with boxed final form; for (b): Two explicit sphere equations in standard form with centers and radii stated; for (c)(i): Exact answer 27/4 or 6.75 square units; for (c)(ii): Both identities clearly verified with Q.E.D. or equivalent; all answers properly labeled and cross-referenced to sub-parts | Correct answers but poorly formatted (unboxed, missing units, or buried in working); one sub-part answer missing or incorrect | Missing final answers, incorrect extraction of results from working, or failure to present answers in required format (standard form for spheres, simplified fractions) |
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