Q2
(a) Let T : ℝ³ → ℝ² be a linear transformation such that T(1, 1, -1) = (1, 0), T(4, 1, 1) = (0, 1) and T(1, -1, 2) = (1, 1). Find T. 15 marks (b) Using Mean Value Theorem, prove that π/6 + √3/15 < sin⁻¹(3/5) < π/6 + 1/8 15 marks (c) (i) Find the equation of the cylinder whose generators are parallel to the line x/1 = y/2 = z/3 and that passes through the curve x² + y² = 16, z = 0. 10 marks (ii) Find the shortest distance between the straight lines (x-3)/3 = (y-8)/(-1) = (z-3)/1 and (x+3)/(-3) = (y+7)/2 = (z-6)/4. 10 marks
हिंदी में प्रश्न पढ़ें
(a) माना T : ℝ³ → ℝ² एक ऐसा रैखिक रूपांतरण है कि T(1, 1, -1) = (1, 0), T(4, 1, 1) = (0, 1) तथा T(1, -1, 2) = (1, 1) है। T ज्ञात कीजिए। 15 अंक (b) माध्यमान प्रमेय का प्रयोग करते हुए सिद्ध कीजिए कि π/6 + √3/15 < sin⁻¹(3/5) < π/6 + 1/8 15 अंक (c) (i) उस बेलन का समीकरण ज्ञात कीजिए जिसके जनक, रेखा x/1 = y/2 = z/3 के समांतर हैं और जो वक्र x² + y² = 16, z = 0 से होकर गुजरता है। 10 अंक (ii) सरल रेखाओं (x-3)/3 = (y-8)/(-1) = (z-3)/1 और (x+3)/(-3) = (y+7)/2 = (z-6)/4 के बीच की न्यूनतम दूरी ज्ञात कीजिए। 10 अंक
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How this answer will be evaluated
Approach
Solve all four sub-parts systematically, allocating approximately 30% time to part (a) on linear transformation, 30% to part (b) on Mean Value Theorem proof, 20% to part (c)(i) on cylinder equation, and 20% to part (c)(ii) on shortest distance. Begin each part with clear statement of the approach, show complete working with proper mathematical notation, and conclude with boxed final answers.
Key points expected
- For (a): Verify that {(1,1,-1), (4,1,1), (1,-1,2)} forms a basis for ℝ³, then express T as a 2×3 matrix by solving for images of standard basis vectors or using linearity directly
- For (b): Apply MVT to f(x) = sin⁻¹x on [1/2, 3/5], showing f'(c) = 1/√(1-c²) lies between 4/5 and 5/√39, then manipulate to obtain the required bounds
- For (c)(i): Use the condition that for any point (x,y,z) on cylinder, its distance from the axis line x/1=y/2=z/3 equals the radius 4, giving (2x-y)² + (3y-2z)² + (3x-z)² = 16×14 or equivalent simplified form
- For (c)(ii): Verify the lines are skew, then apply SD formula |(a₂-a₁)·(b₁×b₂)|/|b₁×b₂| with correct identification of points and direction vectors, obtaining exact numerical value
- Clear demonstration of linearity properties in (a), proper interval selection in (b), correct generator direction handling in (c)(i), and accurate cross product computation in (c)(ii)
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies basis verification in (a), proper interval [1/2, 3/5] and function f(x)=sin⁻¹x in (b), accurate axis line and radius 4 in (c)(i), correct identification of skew lines with valid points (3,8,3) and (-3,-7,6) in (c)(ii) | Partially correct setups with minor errors such as wrong interval endpoints in (b) or incorrect point identification in (c)(ii), but shows understanding of core concepts | Fundamental setup errors like assuming given vectors are standard basis in (a), wrong function or interval in (b), treating cylinder as circular cylinder about z-axis in (c)(i), or assuming lines are parallel in (c)(ii) |
| Method choice | 20% | 10 | Uses matrix representation T(x)=Ax with systematic solving in (a), applies MVT with clever algebraic manipulation for bounds in (b), employs distance-from-axis method for cylinder in (c)(i), and uses vector triple product formula for skew lines in (c)(ii) | Correct general methods chosen but with inefficient approaches like solving three systems separately in (a) or using general point method instead of formula in (c)(ii) | Incorrect methods such as assuming T is determined by values on non-basis in (a), using Taylor expansion instead of MVT in (b), or using wrong distance formula in (c)(ii) |
| Computation accuracy | 20% | 10 | Flawless arithmetic: correct 2×3 matrix for T in (a), precise inequality manipulation yielding exact bounds in (b), accurate expansion to standard cylinder equation in (c)(i), and exact shortest distance value with proper simplification in (c)(ii) | Minor computational slips like sign errors in matrix entries, arithmetic errors in bound calculation, or numerical evaluation errors in final answers, but salvageable working shown | Major computational failures such as incorrect matrix inversion, algebraic errors destroying inequality direction in (b), or completely wrong numerical answers without valid intermediate steps |
| Step justification | 20% | 10 | Explicitly justifies basis property in (a), clearly states MVT conditions and shows f'(c) bounds derivation in (b), proves distance preservation under translation in (c)(i), and verifies non-parallelism before applying skew line formula in (c)(ii) | Some logical gaps like assuming without proof that vectors form basis, or stating MVT applies without checking continuity/differentiability, but overall structure is sound | Missing critical justifications: no verification of linear independence, application of MVT without conditions, or use of formulas without checking applicability (e.g., parallel lines assumption) |
| Final answer & units | 20% | 10 | All four answers clearly presented: explicit T(x,y,z) formula in (a), proven double inequality in (b), simplified Cartesian equation of cylinder in (c)(i), and exact shortest distance with proper units in (c)(ii); all boxed or highlighted | Correct answers present but poorly formatted, or one part missing final answer while others complete, or answers embedded in working without clear extraction | Missing multiple final answers, or answers without supporting working, or fundamentally wrong answers presented as final without indication of uncertainty |
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