Matematyka AA IB HL — maj 2025, TZ2, Paper 3 (zadania wzorowane)

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane4 pktobliczeniowetrudne (szac.)

Zadanie 1a

For A = 9: (i) Sketch y = F₉(x) on −7 ≤ x ≤ 7. (ii) State all its intersections with the horizontal axis. (iii) State the local maximum and local minimum coordinates.

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane2 pktobliczeniowetrudne (szac.)

Zadanie 1b

Is F₉ odd, even or neither? Justify your classification.

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane4 pktobliczeniowetrudne (szac.)

Zadanie 1c

For general A > 0, prove that $F_A^{\prime}(\sqrt A)$ has the same value for every A.

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane4 pktobliczeniowetrudne (szac.)

Zadanie 1d

(i) Prove the identity $F_A(x)=\frac32x-\frac{2Ax}{x^2+A}$. (ii) Hence give the equation of its oblique asymptote. (iii) Give one point on the curve at which the tangent is parallel to that asymptote.

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane5 pktobliczeniowetrudne (szac.)

Zadanie 1e

Consider $4x^2\frac{dy}{dx}=3x^2+8xy-4y^2$ with x ≠ 0, y ≠ 3x/2 and y ≠ −x/2. Under the substitution y = vx, this becomes $x\frac{dv}{dx}=(3/2-v)(v+1/2)$. Using partial fractions, show that…

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane5 pktobliczeniowetrudne (szac.)

Zadanie 1f

Consider $4x^2\frac{dy}{dx}=3x^2+8xy-4y^2$ with x ≠ 0, y ≠ 3x/2 and y ≠ −x/2. Under the substitution y = vx, this becomes $x\frac{dv}{dx}=(3/2-v)(v+1/2)$. Hence show that a solution relation is $x^2=\left|\frac{A(y+x/2)}{3x/2-y}\right|$ for some A > 0.

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane4 pktobliczeniowetrudne (szac.)

Zadanie 1g

The solution relation is $x^2=\left|\frac{A(y+x/2)}{3x/2-y}\right|$, with A > 0, x ≠ 0, y ≠ 3x/2 and y ≠ −x/2. Restrict the relation in (f) to $\frac{A(y+x/2)}{3x/2-y}>0$. Show algebraically that it produces y = F_A(x).

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane5 pktobliczeniowetrudne (szac.)

Zadanie 2a

On the flat map, W is 9000 km due west of N, while M is 3500 km due north of N. (i) Find the straight-line distance from W to M. (ii) Find the bearing of M from W, in degrees.

• Materiał graficzny

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane4 pktobliczeniowetrudne (szac.)

Zadanie 2b

In the spherical model the world has radius 5000 km. Coordinates are in thousands of kilometres, with origin at its centre. The north pole P and station N have vectors $\mathbf p=(0,0,5)$ and $\mathbf n=(0,5,0)$. The positive x direction points west…

• Materiał graficzny

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane5 pktobliczeniowetrudne (szac.)

Zadanie 2c

In the spherical model the world has radius 5000 km. Coordinates are in thousands of kilometres, with origin at its centre. The north pole P and station N have vectors $\mathbf p=(0,0,5)$ and $\mathbf n=(0,5,0)$. The positive x direction points west…

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane2 pktobliczeniowetrudne (szac.)

Zadanie 2d

In the spherical model the world has radius 5000 km. Coordinates are in thousands of kilometres, with origin at its centre. The north pole P and station N have vectors $\mathbf p=(0,0,5)$ and $\mathbf n=(0,5,0)$. The positive x direction points west…

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane5 pktobliczeniowetrudne (szac.)

Zadanie 2e

In the spherical model the world has radius 5000 km. Coordinates are in thousands of kilometres, with origin at its centre. The north pole P and station N have vectors $\mathbf p=(0,0,5)$ and $\mathbf n=(0,5,0)$. The positive x direction points west…

Matematyka AA IBPaper 3, maj 2025, TZ2wzorowane6 pktobliczeniowetrudne (szac.)

Zadanie 2f

The method defines the spherical angle at W as the angle between the normals $\mathbf w\times\mathbf p$ and $\mathbf w\times\mathbf m$. In the spherical model the world has radius 5000 km. Coordinates are in thousands of kilometres, with origin at its…