Zadanie 1a
Find, in terms of r, (i) the arc length UV; (ii) hence the perimeter of the insert.
- Materiał graficzny
Matematyka AA IB HL — maj 2026, 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 30 zadań z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
The insert has perimeter 20 cm. Find r.
Zadanie 1c
The insert has perimeter 20 cm. Hence find the insert’s area.
Zadanie 2a
Show that b = 1/3.
- Materiał graficzny
Zadanie 2b
Find the area bounded by y=f(x), the axes and x=3π.
Zadanie 3
Let f(x) = x2 + 2x and g(x) = ex − 6, both with real domain. Find all real x satisfying (f∘g)(x) = 24.
Zadanie 4a
Find the other two roots.
- Materiał graficzny
Zadanie 4b
Hence state all three roots of f(−x/3) = 0.
Zadanie 5a
Show that L6 = ln a and determine the positive integer a.
Zadanie 5b
Hence state L98.
Zadanie 6
A field photographer packs 2 distinct lenses from 6 available and 3 distinct filters from 5 available. Two particular filters must not be packed together. How many different kits can be assembled?
Zadanie 7
Prove by mathematical induction that the product n(n+1)(n+2) is divisible by 6 for every positive integer n.
Zadanie 8a
Find dy/dx in terms of x and y.
Zadanie 8b
At P the y-coordinate increases at 4 units per second. Find the instantaneous rate of change of the x-coordinate.
Zadanie 9
Solve on the branch containing x=3, y=√5. Give the result in the form y2 = f(x).
Zadanie 10a
Find the opening’s area when θ=π/6.
- Materiał graficzny
Zadanie 10b
Find its area A(θ) in the form , with positive integers a,b.
Zadanie 10c
(i) Find A′(θ). (ii) Hence find the maximum opening area.
Zadanie 10d
As θ approaches π/2, (i) describe the limiting shape; (ii) find the limiting area.
Zadanie 11a
Use partial fractions to show f(x) = 3/(x + 4) − 5/(x + 2).
Zadanie 11b
Show that the area bounded by y=f(x), the x-axis and x=0, x=1 is ln(486/125).
Zadanie 11c
Show that the binomial expansion through x² is −7/4 + (17/16)x − (37/64)x2.
Zadanie 11d
(i) Use 11c to approximate . (ii) Hence state the area between the polynomial y = −7/4 + (17/16)x − (37/64)x2, the x-axis and x=0, x=1.
Zadanie 11e
The comparison area from 11d is bounded by y = −7/4 + (17/16)x − (37/64)x2, the x-axis and the lines x = 0, x = 1. Given ln(486/125) ≈ 1.36 to three significant figures, decide whether the area from 11d overestimates or underestimates the exact area.…
Zadanie 12a
Solve Q(z)=0, giving the roots (i) in Cartesian form; (ii) in modulus–argument form.
Zadanie 12b
Solve P(z)=0 in modulus–argument form.
Zadanie 12c
Plot these six roots on an Argand diagram. Label them z₁,…,z₆ in counterclockwise order, starting with z₁ on the positive real axis.
Zadanie 12d
Verify that z²−9 is a factor of P(z).
Zadanie 12e
The already-specified quadratic factor is z2 − 9. P is the product of three quadratic polynomials with real coefficients. Find the other two quadratic factors.
Zadanie 12f
Label the six roots counterclockwise, with z1 = 3 on the positive real axis. Find the total length of the five line segments joining z₁ to the other roots.