Zadanie 2h, Paper 3, maj 2024, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2024, TZ1, Paper 3 (zadania wzorowane), pytanie 2(h). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
This investigation uses the sextic curve G(x) = x6 − (5/2)x4 + 2x and its generalization F(x) = x6 − mx4 + nx, where m>0 and n is real. For either curve, B and C are its two points of inflection in increasing x order. The line BC meets the curve at two further real points A,D, ordered xA < xB < xC < xD. A zero of the second derivative must be checked for a change of concavity.
Polecenie
For F, show that .
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Odpowiedź z klucza
EquatingF to its secant gives x⁶−mx⁴+12m³/125=0. Let z=x²≥0. Since the known inflection points havez=2m/5, this cubic has factorz−2m/5. Division gives (z−2m/5)[z²−(3m/5)z−6m²/25]=0. The quadratic roots are z=m(3±√33)/10. The minus choice is negative and cannot bex². The positive one exceeds 2m/5 because √33>1, giving the outer intersectionsx=±√[m(3+√33)/10]. The leftmost is the negative value stated. Alternatively use Vieta on the cubic after identifying 2m/5 as a root.
Schemat punktowania
M1:form the curve-minus-line polynomial equal tozero. M1:usez=x² and identify the known root/factor. A1:factorz−2m/5. M1:divide or use root sums/products to obtain the remaining quadratic. A1:correct quadratic. M1:solve it. A1:reject negativez, take the negative outerx and establish the target. Directly inserting the claimed answer without deriving it earns only the initial method mark. Equivalent Vieta/root-sum and product derivations earn the corresponding marks. The sign selection/rejection may be implicit in a complete correct derivation ending at the given negative outer root; intermediate ± notation is condoned if the stated target is correctly selected.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Retain unrounded intermediate values. Give exact answers or three significant figures unless a different precision is requested.