Zadanie 1h, Paper 3, listopad 2023, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2023, Paper 3 (zadania wzorowane), pytanie 1(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
Investigate the real curves y = Fa,b(x) = x4 + ax2 + b, where a,b are real and a≠0. An x-axis intercept is counted once even if its root is repeated. A repeated real root is a real value of x for which both F(x)=0 and F′(x)=0.
Polecenie
Prove that a<0 and 0<b<a²/4 imply exactly four distinct x-axis intercepts.
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Set u=x². The intercept equation becomes u²+au+b=0. Its discriminant a²−4b is positive, so it has two distinct real roots u₁,u₂. Their product is b>0, hence they have the same nonzero sign; their sum is −a>0, so both are positive. Each therefore gives two distinct nonzero real x-values, ±√u₁ and ±√u₂. Because u₁≠u₂, all four x-values are distinct, and a quartic has no further roots. Alternatively the positive central maximum, two negative minima and upward tails give one crossing in each of the four monotonic intervals.
Schemat punktowania
M1:reduce to the quadratic in x², or analyze stationary heights. A1:establish two distinct real u-roots using a²−4b>0 (or the strict extremal signs). A1:use product and sum to establish both u-roots positive (or show all four monotonic intervals). R1:each positive root yields a pair of real x-values (or an actual crossing in each interval). R1:justify exactly four distinct intercepts and conclude the forward implication. Proving only the converse earns at most 3/5.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Exact claims and proofs require analytical working. Counts of intercepts mean distinct real points, unless repeated roots are explicitly requested.