Mathematics 2021 Paper I 50 marks Prove

Q2

(a) Show that the planes, which cut the cone $ax^2 + by^2 + cz^2 = 0$ in perpendicular generators, touch the cone $\frac{x^2}{b+c} + \frac{y^2}{c+a} + \frac{z^2}{a+b} = 0$. (20 marks) (b) Given that $f(x,y) = |x^2 - y^2|$. Find $f_{xy}(0,0)$ and $f_{yx}(0,0)$. Hence show that $f_{xy}(0,0) = f_{yx}(0,0)$. (15 marks) (c) Show that $S = \{(x, 2y, 3x) : x, y$ are real numbers$\}$ is a subspace of $R^3(R)$. Find two bases of $S$. Also find the dimension of $S$. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) दर्शाइए कि वे समतल, जो कि शंकु $ax^2 + by^2 + cz^2 = 0$ को लंब जनकों में काटते हैं, शंकु $\frac{x^2}{b+c} + \frac{y^2}{c+a} + \frac{z^2}{a+b} = 0$ को स्पर्श करते हैं। (20 अंक) (b) दिया गया है : $f(x,y) = |x^2 - y^2|$, तब $f_{xy}(0,0)$ तथा $f_{yx}(0,0)$ ज्ञात कीजिए। अतः दर्शाइए कि $f_{xy}(0,0) = f_{yx}(0,0)$। (15 अंक) (c) दर्शाइए कि $S = \{(x, 2y, 3x) : x, y$ वास्तविक संख्याएँ हैं$\}$ $R^3(R)$ का एक उपसमष्टि है। $S$ के दो आधार ज्ञात कीजिए। $S$ की विमा भी ज्ञात कीजिए। (15 अंक)

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Directive word: Prove

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How this answer will be evaluated

Approach

Prove the three mathematical statements systematically, allocating approximately 40% of effort to part (a) given its 20 marks, and 30% each to parts (b) and (c). Begin each part with clear statement of what is to be proved, develop the proof through logical steps with proper mathematical notation, and conclude with explicit verification of the required result. For (a), establish the condition for perpendicular generators first; for (b), carefully handle the absolute value through case analysis; for (c), verify all three subspace axioms before finding bases.

Key points expected

  • Part (a): Derive condition for perpendicular generators of cone ax² + by² + cz² = 0 using direction cosines and orthogonality condition l₁l₂ + m₁m₂ + n₁n₂ = 0
  • Part (a): Show that tangent plane condition leads to the reciprocal cone x²/(b+c) + y²/(c+a) + z²/(a+b) = 0 using the determinant condition for tangency
  • Part (b): Analyze f(x,y) = |x² - y²| in four quadrants/regions to compute partial derivatives fx and fy near origin
  • Part (b): Calculate mixed partial derivatives f_xy(0,0) and f_yx(0,0) using limit definition, showing both equal zero despite |x²-y²| not being C²
  • Part (c): Verify S is subspace of R³ by checking: (i) non-empty/contains zero, (ii) closed under addition, (iii) closed under scalar multiplication
  • Part (c): Express S as span{(1,0,3), (0,2,0)} = span{(1,0,3), (0,1,0)}, verify linear independence, conclude dim(S) = 2 with two distinct bases

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies cone geometry for (a) with proper generator parametrization; for (b) correctly partitions plane into regions |x|>|y| and |x|<|y|; for (c) correctly writes S = {(x,2y,3x)} = span with proper vector identificationBasic setup for each part is present but missing key geometric insights for (a), incomplete case analysis for (b), or imprecise span description for (c)Misidentifies the cone type in (a), fails to handle absolute value in (b), or confuses S with all of R³ or wrong subspace in (c)
Method choice20%10Uses optimal methods: homogeneous coordinates and reciprocal cone theory for (a); limit definition of partial derivatives for (b); standard subspace test and basis extraction via row reduction or direct inspection for (c)Uses workable but suboptimal methods, such as brute force coordinate approach for (a), informal differentiation for (b), or verification without systematic basis construction for (c)Chooses inappropriate methods like trying to use single variable calculus for (b), or fails to use any recognizable subspace test for (c)
Computation accuracy20%10Flawless calculations: correct determinant expansion for tangency condition in (a), accurate limit evaluations yielding f_xy(0,0) = f_yx(0,0) = 0 in (b), correct dimension count and verified linear independence in (c)Minor computational slips such as sign errors in determinants, incorrect limit evaluation at one boundary, or arithmetic errors in linear combination verificationMajor computational errors including wrong tangency condition, incorrect mixed partial values, or claiming wrong dimension for (c)
Step justification20%10Every critical step justified: why perpendicular generators imply specific plane equation, rigorous ε-δ argument or clear limit reasoning for partial derivatives, explicit verification of all three subspace axioms with general elementsSome steps justified but gaps remain: asserts tangency without showing discriminant zero, sketches limit without proper justification, or verifies subspace properties with specific examples onlyBare assertions without proof: states results without showing work, or simply declares S is a subspace without verification
Final answer & units20%10Clear concluding statements: explicit proof completion for (a), verified equality f_xy = f_yx = 0 with explicit values for (b), two distinct explicitly written bases and dim(S)=2 for (c); proper mathematical notation throughoutAnswers present but poorly organized or missing explicit final statements; correct values but not clearly labeled as final answersMissing or incorrect final answers, or failure to address all parts of the question (especially 'two bases' requirement in (c))

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