Zadanie 1a, Paper 3, maj 2022, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2022, TZ2, Paper 3 (zadania wzorowane), pytanie 1(a). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
Consider the plane curves C1 : y2 = (x − 1)3 − 4(x − 1) and C2 : y2 = (x − 1)3 + 4. A curve may have two y-values at one x.
Polecenie
On the same axes, sketch (i) C₁ and (ii) C₂ in the window −2≤x≤5,−5≤y≤5. Mark their intersections with the coordinate axes.
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Odpowiedź z klucza
(i) Put u=x−1. C₁ exists when u(u−2)(u+2)≥0, so x∈[−1,1]∪[3,∞). It has a closed oval between x=−1 and 1, and a separate unbounded right component from x=3; both are symmetric about the x-axis. Its x-intercepts are −1,1,3, and y-intercepts ±√3. (ii) C₂ is symmetric about the x-axis with x≥1−∛4. Its only x-intercept is 1−∛4≈−.587, and its y-intercepts are also ±√3. It is smooth with a vertical tangent at the x-intercept and continues to the right on two branches. The two curves really cross at (0,±√3).
Schemat punktowania
(i) A1:correct symmetric oval plus separate right component. A1:axis intercepts −1,1,3 and ±√3. (ii) A1:correct symmetric smooth curve. A1:axis intercept 1−∛4 and ±√3. Separate sets of axes earn at most 2/4 for the better response. Both branches must be shown. Approximate positions of intercepts relative to labelled axes are sufficient; numerical coordinate labels are not independently required. If only one correct branch of each curve is shown, award at most 1/4. At a genuine x-axis join, an otherwise correct but discontinuous sketch loses the relevant accuracy mark; the separate components of C1 are intentional. Crossing at the actual shared y-intercepts is correct and is not penalized.
Komentarz. Paper 3: graphic display calculator and AA HL formula booklet allowed. Show proof and algebra where requested. Give exact values when requested, otherwise three significant figures. Credit valid alternative methods and appropriate follow-through.