Zadanie 9e, Paper 2, maj 2025, TZ1, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — maj 2025, TZ1, Paper 2 (zadania wzorowane), pytanie 9(e). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
Define for x ≠ −3. Let P = (k, f(k)) be the point on its graph nearest to A = (1, −1). A second family is , with nonzero real c, d, x ≠ −d/c, and 4d + 7c ≠ 0. The last condition rules out cancellation to a constant. Where stated, impose g−1 = g.
Polecenie
Impose g−1 = g to determine d.
Take c = 2 and the value of d from (d). Sketch y = 1/g(x), showing intercepts, any excluded points, and all asymptotes with their equations.
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Odpowiedź z klucza
1/g(x) = (2x + 4)/(7 − 4x), with x ≠ −2 (g is undefined there) and x ≠ 7/4 (g is zero there). The vertical asymptote is x = 7/4 and the horizontal asymptote is y = −1/2. The y-intercept is (0, 4/7). There is a hole at (−2, 0), not an actual x-intercept. The left branch approaches the horizontal asymptote from above as x tends to negative infinity, and tends to positive infinity at the vertical asymptote; the right branch runs from negative infinity and approaches the horizontal asymptote from below.
Schemat punktowania
Only with a broadly correct two-branch rational-curve shape, award independently: A1 for labelled vertical asymptote, A1 for labelled horizontal asymptote, A1 for labelled hole (-2,0), A1 for labelled y-intercept (0,4/7). As in the source allocation, a labelled x-intercept at the hole location is accepted for that single mark. Follow through the value of d for the excluded point x=-d/2 and y-intercept d/7.
Komentarz. Paper 2: calculators permitted. Show the equations or numerical procedure used, and retain intermediate precision. Give answers to three significant figures unless stated otherwise.