Q2
(a) If the matrix of a linear transformation T : IR³→IR³ relative to the basis {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is $$\begin{bmatrix} 1 & 1 & 2 \\ -1 & 2 & 1 \\ 0 & 1 & 3 \end{bmatrix},$$ then find the matrix of T relative to the basis {(1, 1, 1), (0, 1, 1), (0, 0, 1)}. (15 marks) (b) Evaluate the triple integral which gives the volume of the solid enclosed between the two paraboloids Z = 5(x² + y²) and Z = 6 – 7x² – y². (15 marks) (c)(i) Show that the equation 2x² + 3y² – 8x + 6y – 12z + 11 = 0 represents an elliptic paraboloid. Also find its principal axis and principal planes. (10 marks) (c)(ii) The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ meets the coordinate axes in A, B, C respectively. Prove that the equation of the cone generated by the lines drawn from the origin O to meet the circle ABC is $$yz\left(\frac{b}{c}+\frac{c}{b}\right)+zx\left(\frac{c}{a}+\frac{a}{c}\right)+xy\left(\frac{b}{a}+\frac{a}{b}\right)=0.$$ (10 marks)
हिंदी में प्रश्न पढ़ें
(a) यदि आधार {(1, 0, 0), (0, 1, 0), (0, 0, 1)} के सापेक्ष रैखिक रूपांतरण T : IR³→IR³ का आव्यूह $$\begin{bmatrix} 1 & 1 & 2 \\ -1 & 2 & 1 \\ 0 & 1 & 3 \end{bmatrix}$$ है, तब आधार {(1, 1, 1), (0, 1, 1), (0, 0, 1)} के सापेक्ष T का आव्यूह ज्ञात कीजिए। (15 अंक) (b) दो परवलयजों Z = 5(x² + y²) और Z = 6 – 7x² – y² के बीच घिरे ठोस के आयतन को दर्शाने वाले त्रिशः समाकल का मान निकालिए। (15 अंक) (c)(i) दर्शाइए कि समीकरण 2x² + 3y² – 8x + 6y – 12z + 11 = 0 एक दीर्घवृत्तीय परवलयज प्रदर्शित करता है। साथ ही मुख्य अक्ष और मुख्य समतलों को भी ज्ञात कीजिए। (10 अंक) (c)(ii) समतल $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$, निर्देशांक अक्षों को क्रमशः A, B, C में मिलता है। सिद्ध कीजिए कि मूल बिंदु O से वृत्त ABC को मिलाने वाली रेखाओं द्वारा जनित शंकु का समीकरण $$yz\left(\frac{b}{c}+\frac{c}{b}\right)+zx\left(\frac{c}{a}+\frac{a}{c}\right)+xy\left(\frac{b}{a}+\frac{a}{b}\right)=0$$ है। (10 अंक)
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How this answer will be evaluated
Approach
Solve all three parts systematically, allocating approximately 30% time to part (a) on change of basis matrices, 30% to part (b) on triple integration using cylindrical coordinates, and 40% to part (c) covering both the elliptic paraboloid identification and the cone equation derivation. Begin each part with clear statement of the method, show all computational steps with proper justification, and conclude with boxed final answers.
Key points expected
- Part (a): Correct construction of change of basis matrix P using new basis vectors, computation of P⁻¹, and application of similarity transformation P⁻¹AP to find the new matrix representation
- Part (b): Identification of intersection curve x² + y² = 0.5, correct setup of cylindrical coordinate bounds (r: 0 to 1/√2, θ: 0 to 2π, z: 5r² to 6-7r²), and accurate evaluation of the triple integral
- Part (c)(i): Completion of squares to standard form, verification of elliptic paraboloid by identifying one linear term and two squared terms with same sign, identification of principal axis (z-axis direction) and principal planes
- Part (c)(ii): Correct determination of coordinates A(a,0,0), B(0,b,0), C(0,0,c), equation of sphere through OABC, intersection with plane to get circle ABC, and homogenization to derive the cone equation
- Proper matrix notation and determinant calculations for part (a) with verification of non-singularity
- Clear geometric interpretation in part (b) showing the paraboloid intersection forms a bounded solid
- Systematic algebraic manipulation in (c)(ii) showing the homogenization process with λ = 0 substitution
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies and writes all initial setups: for (a) constructs proper change of basis matrix P with columns as new basis vectors; for (b) accurately finds intersection curve and sets correct integration limits; for (c)(i) completes squares correctly and for (c)(ii) correctly identifies coordinates of A, B, C and sphere equation | Sets up most problems correctly but has minor errors in basis matrix construction, integration limits, or coordinate identification that don't fundamentally derail the solution | Major setup errors such as wrong basis vectors in P, incorrect intersection curve in (b), or fundamental misunderstanding of how to identify A, B, C coordinates in (c)(ii) |
| Method choice | 20% | 10 | Selects optimal methods throughout: similarity transformation P⁻¹AP for (a), cylindrical coordinates for (b) recognizing rotational symmetry, completing squares for standard form in (c)(i), and homogenization technique for cone generation in (c)(ii) | Uses acceptable methods that lead to solutions but may be inefficient (e.g., Cartesian coordinates in (b)) or misses optimal approaches without causing failure | Chooses inappropriate methods such as attempting row reduction for change of basis in (a), or fails to recognize the need for homogenization in (c)(ii) |
| Computation accuracy | 20% | 10 | Executes all calculations with precision: correct matrix inversion and multiplication in (a), accurate integration including proper antiderivatives and evaluation in (b), error-free algebraic manipulation in completing squares and homogenization in (c) | Generally correct computations with occasional arithmetic slips in matrix entries, integration constants, or algebraic terms that are partially recoverable | Frequent computational errors leading to wrong determinants, incorrect integrals, or fundamentally flawed algebraic results that cannot be salvaged |
| Step justification | 20% | 10 | Provides clear justification for each key step: explains why P⁻¹AP gives the new matrix, shows derivation of intersection curve and why cylindrical coordinates are appropriate, proves the surface is elliptic paraboloid by analyzing the standard form, and demonstrates the homogenization process with explicit substitution | Shows most key steps with minimal justification, or has gaps in reasoning that leave the examiner to infer the logic, particularly in the geometric reasoning parts | Minimal or missing justification with unjustified leaps between steps, particularly failing to explain why transformations are valid or how geometric conclusions follow |
| Final answer & units | 20% | 10 | Presents all final answers clearly: 3×3 matrix for (a), exact simplified volume expression (π/2) for (b), explicit principal axis (x=2, y=-1, z arbitrary) and principal planes for (c)(i), and fully derived cone equation matching the required form for (c)(ii) | Final answers present but with minor formatting issues, unsimplified expressions, or missing components such as not specifying principal planes | Missing final answers, answers in wrong format, or significant errors in the final expressions that suggest fundamental misunderstanding of what was asked |
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