Zadanie 1a
On common axes, sketch y = log2 x, y = log10 x and y = x2 + e2 for x>0. Label all three graphs and state the nonzero x-intercept.
Matematyka AA IB HL — maj 2023, 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 13 zadań z tego arkusza, razem 55 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Use calculus to find the minimum of x2 + e2 − ln x for x>0. Justify that it is a minimum.
Zadanie 1c
Hence deduce that x2 + e2 > ln x for all x>0. You may use the minimum value e2 + (1 + ln 2)/2 for their difference.
Zadanie 1d
Investigate representative graphs. State the number of intersections in each base range: (i) 0<a<1; (ii) 1<a<1.05; (iii) 1.1<a<2.
Zadanie 1e
The unique contact base lies between 1.05 and 1.1. Find the exact coordinates of the contact point P and the exact base a. You may use the fact that the resulting contact equation has exactly one solution with x>1; an exact candidate must still be verified in…
Zadanie 1f
Give the exact sets of bases for which the graphs have (i) two intersections; (ii) no intersections. You may use the unique contact base a∗ = exp(1/(2e2)).
Zadanie 2a
Show that L(x) = 2x + 8 is admissible.
Zadanie 2b
(i) Show that r=−c/m. (ii) For admissible L, show that L(x) = mx + 2m2/(m − 1). (iii) State the further excluded value of m.
Zadanie 2c
Exactly three admissible L have integer r,m,c. One is L(x) = 3x + 9. Use the general expression to find the other two. For an admissible linear function, c = 2m2/(m − 1) and r = −c/m, with m ≠ 0, 1.
Zadanie 2d
For the cubic family, state (i)r1 + r2 and (ii)r1r2 in terms of a,b,c.
Zadanie 2e
For an admissible cubic F: (i) express r2 − r1 in terms of a,b; (ii) use the sum and difference of the stationary abscissas to show that r1 = (3a2 − 3ab − 2b)/(6a); (iii) use this result to show that b=0 or a=−1/3.
Zadanie 2f
Determine the two admissible cubic functions with b=0.
Zadanie 2g
Now set a=−1/3. (i) Express r₁ in terms of b. (ii) Hence or otherwise find the exact pairs (b,c) making F admissible. Give each value as a rational expression involving a square root.