Mathematics 2021 Paper II 50 marks Solve

Q2

(a) Find the maximum and minimum values of f(x) = x³ - 9x² + 26x - 24 for 0 ≤ x ≤ 1. (15 marks) (b) Let F be a field and f(x) ∈ F[x] a polynomial of degree > 0 over F. Show that there is a field F' and an imbedding q : F → F' s.t. the polynomial f^q ∈ F'[x] has a root in F', where f^q is obtained by replacing each coefficient a of f by q(a). (15 marks) (c) Find the Laurent series expansion of f(z) = (z² - z + 1)/[z(z² - 3z + 2)] in the powers of (z+1) in the region |z+1| > 3. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) f(x) = x³ - 9x² + 26x - 24 का, 0 ≤ x ≤ 1 के लिए, अधिकतम तथा न्यूनतम मान निकालिए। (15 अंक) (b) मान लीजिए कि F एक क्षेत्र है तथा f(x) ∈ F[x], क्षेत्र F के ऊपर घात > 0 का एक बहुपद है। दर्शाइए कि एक क्षेत्र F' तथा एक अंतःस्थापन q : F → F' इस प्रकार से अस्तित्व में है कि बहुपद f^q ∈ F'[x] का एक मूल F' में है, जहाँ f^q, f के प्रत्येक गुणांक a को q(a) द्वारा प्रतिस्थापित करने से प्राप्त होता है। (15 अंक) (c) क्षेत्र |z+1| > 3 में f(z) = (z² - z + 1)/[z(z² - 3z + 2)] का लॉरें श्रेणी प्रसार, (z+1) की घातों में ज्ञात कीजिए। (20 अंक)

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Directive word: Solve

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How this answer will be evaluated

Approach

Solve this three-part problem by allocating time proportionally to marks: approximately 15 minutes for part (a) on cubic optimization, 15 minutes for part (b) on field extension theory, and 20 minutes for part (c) on Laurent series expansion. Begin each part with clear statement of the mathematical approach, show all working steps with proper justification, and conclude with boxed final answers. For part (c), explicitly note the substitution w = z+1 and verify convergence in the specified annular region.

Key points expected

  • Part (a): Correctly find f'(x) = 3x² - 18x + 26, determine no critical points in [0,1] since discriminant < 0 and f'(x) > 0 throughout, hence extrema occur at endpoints with f(0) = -24 (minimum) and f(1) = -6 (maximum)
  • Part (b): Construct the field extension F' = F[x]/(p(x)) where p(x) is an irreducible factor of f(x), define the natural embedding q: F → F', and prove that the coset α = x + (p(x)) is a root of f^q in F' using the evaluation homomorphism
  • Part (c): Substitute z = w - 1 where w = z + 1, rewrite f(z) in terms of w, perform partial fraction decomposition, and expand each term as geometric series valid for |w| > 3 (i.e., |z+1| > 3)
  • Part (c): Identify singularities at z = 0, 1, 2 which correspond to w = 1, 2, 3, confirming |w| > 3 excludes all singularities and ensures convergence
  • Part (c): Obtain Laurent series with only negative powers of w (analytic part vanishes), presenting coefficients explicitly as rational numbers

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a): correctly identifies f'(x) and verifies no critical points in [0,1]; for (b): properly defines F' = F[x]/(p) with irreducible p|f; for (c): makes substitution z = w-1 and identifies region |w|>3 correctlySets up most parts correctly but misses verification of critical point absence in (a) or has minor errors in field construction in (b) or substitution in (c)Major setup errors: wrong derivative in (a), incorrect field quotient in (b), or fails to transform to (z+1) powers in (c)
Method choice20%10Uses endpoint analysis for (a) since no interior critical points; employs Kronecker's construction/field extension theorem for (b); applies partial fractions followed by geometric series expansion for (c) with convergence justificationUses appropriate methods but with suboptimal choices (e.g., attempts second derivative test unnecessarily in (a), or doesn't justify irreducibility assumption in (b))Inappropriate methods: tries to find critical points numerically in (a), attempts root formula for general polynomials in (b), or uses Taylor instead of Laurent expansion in (c)
Computation accuracy20%10Accurate arithmetic throughout: f(0)=-24, f(1)=-6 in (a); correct coset arithmetic in (b); precise partial fraction coefficients 1/2, -1, 1/2 and correct geometric series coefficients in (c)Minor computational slips: sign errors in function evaluation, arithmetic mistakes in partial fraction decomposition, or coefficient errors in series expansionMajor computational errors: wrong endpoint values in (a), incorrect quotient ring operations in (b), or fundamentally wrong series expansion in (c)
Step justification20%10Rigorous justification: proves f'(x)>0 on [0,1] via discriminant or monotonicity in (a); proves evaluation homomorphism kernel and isomorphism in (b); verifies convergence condition |w|>3>|singularities| for geometric series in (c)States key results but with gaps in proof: asserts monotonicity without verification in (a), omits homomorphism property proof in (b), or assumes convergence without checkingMissing or circular justifications: no verification of critical point analysis in (a), no proof that α is a root in (b), or no convergence argument in (c)
Final answer & units20%10Clear presentation: states max = -6 at x=1, min = -24 at x=0 for (a); explicitly constructs F', q, and root α for (b); gives complete Laurent series Σ a_n/(z+1)^n with explicit coefficients and valid region for (c)Correct answers but poorly formatted: missing 'maximum/minimum' labels, unclear field description, or series without explicit coefficient formulaMissing or wrong final answers: incorrect extrema values, incomplete field construction, or series not in powers of (z+1) as required

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