Zadanie 1a
Write down f(0).
- Materiał graficzny
Matematyka AA IB HL — maj 2025, TZ2, 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 31 zadań z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Write down the equation of the horizontal asymptote of the graph of f.
Zadanie 1c
Find the range of g.
Zadanie 2a
Give a vector equation for L1.
Zadanie 2b
Find the intersection of the two lines.
Zadanie 3a
Use the sine rule to show that .
- Materiał graficzny
Zadanie 3b
Hence find .
Zadanie 3c
Find PD.
Zadanie 4a
Calculate P(B).
Zadanie 4b
Now assume that A and B are independent. Find P(A′ ∣ B).
Zadanie 5a
Find the allowable values of k.
Zadanie 5b
Find the allowable values of k if the two solutions also have the same sign.
Zadanie 6a
Show that .
Zadanie 6b
Hence determine the x-coordinate of the point on the curve nearest to A.
Zadanie 7
The polynomial F(x) = x4 + px3 − 2x2 + qx − 8 is divisible by (x − 2)2. Determine the real coefficients p and q.
Zadanie 8a
Sketch for −1 ≤ x ≤ 1, labelling the endpoints and the intersection with the vertical axis.
Zadanie 8b
Solve on −1/2 ≤ x ≤ 1/2.
Zadanie 9
Prove by contradiction that for every real t > 0.
Zadanie 10a
Find f−1(9) as a fraction of integers.
Zadanie 10b
(i) Find g−1(x). (ii) Describe a sequence of transformations taking the graph of g−1 to the graph of f.
Zadanie 10c
Show that (f∘g)(x) = 729x3.
Zadanie 10d
A separate function is h(x) = 9x2/(3x + 2), x ≠ −2/3. Region R is bounded by its curve, the line y = 1, and x = 1, x = 2. (i) Establish . (ii) Hence or otherwise, calculate the area of R in the form p + qln r, with positive…
- Materiał graficzny
Zadanie 11a
Find the first four terms of the binomial expansion of F(x) in ascending powers of x.
Zadanie 11b
Expand R(x) in ascending powers of x through x2.
Zadanie 11c
The first three terms of these expansions agree. Prove that q = 2.
Zadanie 11d
Using p = 4, q = 2, the rational expression approximates F(x) near zero, with ∣x∣ < 1/6. (i) Choose a suitable x and use this approximation to estimate as a fraction of integers. (ii) Explain why the same approximation is less accurate for…
Zadanie 12a
Solve , expressing both answers as .
Zadanie 12b
In Cartesian form, (i) find z1, z2; (ii) deduce z3, z4.
Zadanie 12c
(i) Plot the four roots on an Argand diagram. (ii) Find the area of their convex quadrilateral.
Zadanie 12d
These four roots obey . Their reciprocals obey , where p, q, r are integers with no common factor and p > 0. Find p, q, r.
Zadanie 12e
(i) Find 1/z1 in Cartesian form. (ii) Hence deduce the area of the convex quadrilateral whose vertices represent 1/z1, 1/z2, 1/z3, 1/z4.