Zadanie 9b, Paper 2, maj 2025, TZ1, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — maj 2025, TZ1, Paper 2 (zadania wzorowane), pytanie 9(b). 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
(i) Find k.
(ii) Hence give the coordinates of P.
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Odpowiedź z klucza
Minimize D(x) = (x − 1)2 + [(5 − 3x)/(x + 3) + 1]2, the squared distance to A, on the two intervals separated by -3. A numerical minimum search gives candidates x = −6.4013070724… with D ≈ 92.1856, and x = 1.6526096531… with D ≈ 1.44411. Distance tends to infinity near -3 and at both infinite ends, so the latter is the global minimum. Thus k = 1.65 to three significant figures. Using its unrounded value, P = (1.65, 0.00906) to three significant figures.
Schemat punktowania
(i) A1: correct distance or squared-distance function. M1: attempt numerical minimization, or an equivalent valid method; comparing both branches is not required for this method mark. A1: k=1.65 as the global minimum with supporting method; no credit for a bare guessed coordinate without working. (ii) A1: coordinates, following through f(k) from the chosen k; use of a rounded k is accepted consistently.
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.