Zadanie 2g, Paper 3, maj 2023, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2023, TZ2, Paper 3 (zadania wzorowane), pytanie 2(g). 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
Now set a=−1/3. (i) Express r₁ in terms of b. (ii) Hence or otherwise find the exact pairs (b,c) making F admissible. Give each value as a rational expression involving a square root.
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Odpowiedź z klucza
(i)r1 = (a + b)/2 = (3b − 1)/6. (ii) The progression gives r₂=(9b+1)/6 and c=2b+1/3. Vieta gives [(3b − 1)(9b + 1)]/36 = −c = −2b − 1/3, or 27b²+66b+11=0. Hence . Then , with matching signs. Both pairs yield real derivative roots at the stipulated r₁,r₂ and satisfy all progression differences.
Schemat punktowania
(i) M1:substitute a=−1/3 into an AP formula for r₁. A1:(3b−1)/6. (ii) M1:substitute a stationary abscissa into F′=0 or use their product. A1:c=2b+1/3. M1:solve the resulting quadratic. A1:both exact b values. A1:both exact c values with matching signs. For follow-through from an incorrect part (i), award at most M1A1M1A0A0=3/5 in part (ii), including A1 for a correct AP relation for c; decimal-only pairs lose the exact-answer accuracy marks. Equivalent methods through a common difference, either derivative root, factoring F′, or equating quadratic-formula and AP expressions receive the same marks. In (i), using a=+1/3 does not earn M1.
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.