Zadanie 1a
Find g(x).
Matematyka AA IB SL — listopad 2022, Paper 1 (zadania wzorowane)
Zadania wzorowane na pytaniach z tego arkusza: te same umiejętności, format i punktacja, inne dane i kontekst. Treści arkusza IB nie publikujemy.
W bazie 20 zadań z tego arkusza, razem 80 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Find h(x).
Zadanie 1c
Find (g∘h)(0).
Zadanie 2
Let g(x) = e2x²−1 for real x. Find g′(−2).
Zadanie 3a
Find (i) the centre of the lower base; (ii) its radius.
Zadanie 3b
Find the exact volume of material remaining after the cone is removed.
Zadanie 4a
Show that (4t)2 + (t2 − 4)2 = (t2 + 4)2.
Zadanie 4b
Find its area in terms of t.
Zadanie 5
A differentiable function satisfies f′(x) = 10x/(x2 + 4) for real x and f(2)=1. Find f(x).
Zadanie 6a
Find P(A∩B) if the events are independent.
Zadanie 6b
Find the smallest possible value of P(A∩B).
Zadanie 6c
Find the largest possible value of P(A∩B), with a justification.
Zadanie 7a
A separate quadratic f has vertex (−2,−3) and a zero at x=1. Find f in the form f(x) = a(x − h)2 + k.
Zadanie 7b
Suppose g(−4)=g(2)=12. (i) Find p,t. (ii) Find the range of g.
Zadanie 7c
Without the restrictions g(−4)=g(2)=12, consider all real t and all real p≠0. Prove that the graphs of g and j(x) = −2x + 5p have two distinct intersections.
Zadanie 8a
Evaluate (i)log3(1/27); (ii)log16 2; (iii).
Zadanie 8b
Positive u,v satisfy u/v≠1 and logu/v u = 4. (i) Show that logu/v v = 3. (ii) Hence find .
Zadanie 9a
Solve f(x)=0.
Zadanie 9b
(i) Find f′(x). (ii) Hence find all points on the graph where f′(x) = 0, including the endpoints when appropriate.
Zadanie 9c
Sketch y=f(x), labelling the intercepts and all points where f′(x) = 0.