Zadanie 1
Solve for 0° ≤ x ≤ 120°.
Matematyka AA IB HL — maj 2024, 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 2
Solve 2 ⋅ 16x + 7 ⋅ 4x − 4 = 0 over the reals.
Zadanie 3a
(i) Find the midpoint of AC. (ii) Find the Cartesian equation of OB.
Zadanie 3b
Find the exact area enclosed by OABC.
Zadanie 4a
State the probability of no alert from (i) check A; (ii) check B.
Zadanie 4b
(i) Show that 16k2 − 40k + 9 = 0. (ii) The roots of this equation are 1/4 and 9/4. State the valid value of k and justify your choice.
Zadanie 5a
State the equation of the horizontal asymptote of f.
Zadanie 5b
(i) For m > 0, state the number of real solutions of f(x) = g(x). (ii) Find m for which there is exactly one real solution. (iii) Find all m for which there are two distinct solutions and both are nonnegative.
Zadanie 6a
Estimate the percentage of type A pellets with mass greater than 16 g.
Zadanie 6b
A randomly selected pellet has mass greater than 16 g. Estimate the probability that it is type A. Give your answer as a fraction.
Zadanie 7a
State α + β + γ.
Zadanie 7b
Show that p=2.
Zadanie 7c
It is further given that h(−1/3) = 0 and α, β are positive integers with α < β. (i) Show that αβ = 3. (ii) Find α and β.
Zadanie 8
Using l’Hôpital’s rule, evaluate . Angles are in radians.
Zadanie 9a
Express the number of assignments in terms of n.
Zadanie 9b
Two named engineers must now work in different crews. The number of permitted assignments becomes 8/15 of the original number. Find n.
Zadanie 10a
Show that 2p − q = a.
Zadanie 10b
Show that s2 = at.
Zadanie 10c
Show that p>2.
Zadanie 10d
Suppose a=16 and s>0. List the first four terms of (i) the arithmetic sequence; (ii) the geometric sequence.
Zadanie 10e
Suppose a=16 and s>0, and define un = An + ln Gn. (i) Show that u_n is arithmetic and find its common difference. (ii) Show that .
Zadanie 11a
Show that A lies on Π₁.
Zadanie 11b
Suppose p=−2, Π₁ and Π₂ are perpendicular, and A lies on Π₂. Find k and q.
Zadanie 11c
Suppose k=2 and Π₂ is parallel to Π₁. Find p.
Zadanie 11d
Suppose k=2, p=−3/2 and q=75/2. The normal line to Π₁ through A meets Π₂ at B. (i) Find B. (ii) Show that the distance between the planes is .
Zadanie 11e
The plane 2x + 4y − z = 25 is at distance from Π₁. Find the other plane Π₃ parallel to Π₁ at the same distance.
Zadanie 12a
Prove by induction that for every positive integer n, .
Zadanie 12b
Use derivatives at zero or another valid method to show that the Maclaurin expansion begins .
Zadanie 12c
Hence, by multiplying the relevant expansions, show that when terms beyond degree 2 are discarded.
Zadanie 12d
Let P(x) = (1 − x)−3/2(1 + x)−3/2. The two Maclaurin series in this product are those of f1(x) and f−1(x). State the range of x for which both Maclaurin series used in the product converge.
Zadanie 12e
Let P(x) = (1 − x)−3/2(1 + x)−3/2. The quadratic approximation to P is P(x) ≈ 1 + 3x2/2. Use x=1/5 in the quadratic approximation 1 + 3x2/2 to obtain a rational approximation to .