Zadanie 1a
Verify that the start-of-year-2 estimate is 5000 pages.
Matematyka AA IB HL — listopad 2024, Paper 3 (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 15 zadań z tego arkusza, razem 55 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Find the start-of-year estimates for (i) year 3; (ii) year 4.
Zadanie 1c
The recurrence gives . (i) Show that Tn = 1600(9/8)n−1 + 3200. (ii) Estimate the work at the start of year 6, to the nearest whole page.
Zadanie 1d
For quota 750, Dn = −1200(9/8)n−1 + 6000. (i) Show that Dn+1 − Dn = −150(9/8)n−1. (ii) Explain why the algebraic sequence is strictly decreasing.
Zadanie 1e
Under quota 750, Dn = −1200(9/8)n−1 + 6000. At the end of which processing year does the queue first clear?
Zadanie 1f
With a general quota d, the algebraic estimate is Cn = (4800 − 8d)(9/8)n−1 + 8d. Find the quota that keeps the start-of-year work constant.
Zadanie 1g
Consider the general recurrence un+1 = run − d, with r,d>0 and r≠1. Prove by induction that for every integer n≥1.
Zadanie 2a
(i) Solve x2 − 6x + 1 = 0 exactly. (ii) Verify that the two roots are reciprocals.
Zadanie 2b
Show that the complex roots of x2 − x + 1 = 0 form a reciprocal pair.
Zadanie 2c
For p(x) = ax2 + bx + a, where a≠0, verify p(x) = x2p(1/x) for x≠0.
Zadanie 2d
If α≠0 is a root of a palindromic polynomial p of degree n, prove that 1/α is also a root.
Zadanie 2e
Prove that the product of two palindromic polynomials is palindromic.
Zadanie 2f
Write F(x) = (x4 + ux2 + 1)(x4 + vx2 + 1), where u,v are integers with u<v. Determine u,v by forming and solving an appropriate system of equations.
Zadanie 2g
Hence find all roots of F(x) = 0 exactly, stating multiplicities.
Zadanie 2h
Prove that every monic palindromic polynomial of odd degree has −1 as a root.