Zadanie 2e, Paper 3, maj 2023, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2023, TZ2, Paper 3 (zadania wzorowane), pytanie 2(e). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
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Two arithmetic constraints on polynomials are considered. First, for L(x) = mx + c with m,c nonzero, let r be its root. Call L admissible when r,m,c in that order form an arithmetic progression. Second, let F(x) = ax3 + bx2 + cx with real a≠0 and real b,c. Its stationary abscissas r1, r2 are the real roots of F′(x) = 3ax2 + 2bx + c. Call F admissible when a, r1, b, r2, c in that order form an arithmetic progression. The labels r₁,r₂ refer to this order and need not be increasing.
Polecenie
For an admissible cubic F: (i) express r2 − r1 in terms of a,b; (ii) use the sum and difference of the stationary abscissas to show that r1 = (3a2 − 3ab − 2b)/(6a); (iii) use this result to show that b=0 or a=−1/3.
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Odpowiedź z klucza
(i) If the common difference is d, b=a+2d and r₂−r₁=2d=b−a. (ii) Subtract difference from sum:2r₁=−2b/(3a)−(b−a). Combining terms gives the supplied expression. (iii) The arithmetic progression also gives r₁=(a+b)/2. Equating the two expressions yields 3a(a+b)=3a²−3ab−2b, hence 6ab+2b=0, or 2b(3a+1)=0. Therefore b=0 or a=−1/3.
Schemat punktowania
(i) A1:b−a. (ii) M1:eliminate r₂ using the sum and difference. A1:correct algebra yielding the stated r₁. (iii) A1:r₁=(a+b)/2 or an equivalent AP relation. M1:combine with the previous result. A1:factor the relation to establish the alternatives, rather than only restating them. Consistent use of r₁−r₂=a−b earns equivalent marks. In (ii), accept a correct equivalent expression for ±r₁ or ±2r₁ in a forward derivation. In (iii), a common-difference derivation with b=a+2d and r₁=a+d is equivalent. Numerical verification or starting with the target does not prove it.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Exact results require analytical working; graphs and numerical checks support but do not replace proofs.