Zadanie 1a
On the same axes, sketch (i) C₁ and (ii) C₂ in the window −2≤x≤5,−5≤y≤5. Mark their intersections with the coordinate axes.
Matematyka AA IB HL — maj 2022, TZ2, 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
(i) State the coordinates of C₂’s two inflection points. (ii) Identify two features that distinguish C₁ from C₂.
Zadanie 1c
Vary the positive integer b in y2 = (x − 1)3 + b. Suggest two features shared by every curve in this family.
Zadanie 1d
Now restrict y2 = (x − 1)3 − 4(x − 1) to its right component x≥3. (i) Show that for x>3, . (ii) Hence explain why neither open branch x>3 has a local maximum or minimum.
Zadanie 1e
Now restrict y2 = (x − 1)3 − 4(x − 1) to its right component x≥3. The two inflection points on this right component share an x-coordinate. Find that coordinate in exact radical form.
Zadanie 1f
A rational point has two rational coordinates. On C : y2 = x3 − 3x + 4, let P=(0,2). The tangent at P meets C again at Q; let S=(0,−2). (i) Find the tangent equation at P. (ii) Hence find Q, expressing its coordinates as fractions.
Zadanie 1g
A rational point has two rational coordinates. On C : y2 = x3 − 3x + 4, let P=(0,2). The tangent at P meets C again at Q; let S=(0,−2). Use the tangent construction to determine Q, then find the third intersection of the full straight line through Q and S with C.
Zadanie 2a
Expand (z−u)(z−v)(z−w) and show that u+v+w=−q/r, uv+vw+wu=p/r, and uvw=−1/r.
Zadanie 2b
You may use u+v+w=−q/r and uv+vw+wu=p/r. (i) Show that u2 + v2 + w2 = (q2 − 2pr)/r2. (ii) Hence show that (u − v)2 + (v − w)2 + (w − u)2 = 2(q2 − 3pr)/r2.
Zadanie 2c
Using (u − v)2 + (v − w)2 + (w − u)2 = 2(q2 − 3pr)/r2, prove that q²<3pr forces at least one original root α,β,γ to be nonreal.
Zadanie 2d
Consider Fc(x) = x3 + cx2 − 6x + 1, where c is real. You may use the sufficient condition: a real cubic x³+px²+qx+r, r≠0, has a nonreal root if q²<3pr. Show that F₁₃(x)=0 has a nonreal root.
Zadanie 2e
Consider Fc(x) = x3 + cx2 − 6x + 1, where c is real. You may use the sufficient condition: a real cubic x³+px²+qx+r, r≠0, has a nonreal root if q²<3pr. A student claims that q²≥3pr guarantees all roots are real. (i) By varying the positive integer c, find the…
Zadanie 2f
A real quartic x4 + px3 + qx2 + rx + s = 0, s≠0, has nonzero roots α,β,γ,δ. Their reciprocals u,v,w,z satisfy u+v+w+z=−r/s and uv+uw+uz+vw+vz+wz=q/s. (i) Express u²+v²+w²+z² in terms of q,r,s. (ii) Hence state a sufficient inequality in q,r,s for the original…
Zadanie 2g
A real quartic with coefficients q,r,s as in x⁴+px³+qx²+rx+s has a nonreal root whenever r²<2qs. Use this to show that x4 + 2x3 + 4x2 − 3x + 2 = 0 has a nonreal root.
Zadanie 2h
Consider x4 + 8x3 − 15x2 − 9x − 2 = 0, which has exactly one integer root. You may use two sufficient criteria: a quartic x⁴+px³+qx²+rx+s has a nonreal root if r²<2qs; a cubic x³+px²+qx+r, r≠0, has one if q²<3pr. (i) State what the quartic criterion tells you here.…