germany-abitur

Papers (25)
2025
abitur__nrw_analysis-gk 3 abitur_nrw__typ_analysis_lk 4 abitur_nrw__typ_stochastik_gk 3 abitur_nrw__typ_stochastik_lk 3
2024
abitur__bayern_geometrie 9 abitur__bayern_infinitesimalrechnung 2 abitur__bayern_stochastik 9
2023
abitur__bayern_geometrie 9 abitur__bayern_infinitesimalrechnung 2 abitur__bayern_stochastik 10
2022
abitur__bayern_geometrie 9 abitur__bayern_infinitesimalrechnung 1 abitur__bayern_stochastik 8
2021
abitur__bayern_geometrie 10 abitur__bayern_infinitesimalrechnung 2 abitur__bayern_stochastik 9
2020
abitur__bayern_geometrie 7 abitur__bayern_infinitesimalrechnung 2 abitur__bayern_stochastik 9
2019
abitur__bayern_geometrie 9 abitur__bayern_infinitesimalrechnung 2 abitur__bayern_stochastik 2
2018
abitur__bayern_geometrie 1 abitur__bayern_infinitesimalrechnung 2 abitur__bayern_stochastik 2
2022 abitur__bayern_geometrie

9 maths questions

Q1 3 marks Circles Sphere and 3D Circle Problems View
Given is the sphere $K$ with center $M ( 3 | - 6 | 5 )$ and radius $2 \sqrt { 6 }$.
Give an equation of $K$ in coordinate form and show that the point $P ( 5 | - 4 | 1 )$ lies on $K$.
Q2 2 marks Circles Sphere and 3D Circle Problems View
Investigate whether $K$ intersects the $x _ { 1 } x _ { 2 }$-plane.
Q3 4 marks Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Given are the points $P ( 4 | 5 | - 19 ) , Q ( 5 | 9 | - 18 )$ and $R ( 3 | 7 | - 17 )$, which lie in the plane $E$, as well as the line $g : \vec { X } = \left( \begin{array} { c } - 12 \\ 11 \\ 0 \end{array} \right) + \lambda \cdot \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \lambda \in \mathbb { R }$.
Determine the length of the line segment $[ \mathrm { PQ } ]$. Show that the triangle PQR is right-angled at $R$, and use this to justify that the line segment $[ \mathrm { PQ } ]$ is the diameter of the circumcircle of triangle PQR.
(for verification: $\overline { \mathrm { PQ } } = 3 \sqrt { 2 }$ )
Q4 5 marks Vectors 3D & Lines Normal Vector and Plane Equation View
Determine an equation of $E$ in coordinate form and show that the line $g$ lies in $E$.
(for verification: $E : 2 x _ { 1 } - x _ { 2 } + 2 x _ { 3 } + 35 = 0$ )
Q5 1 marks Vectors 3D & Lines MCQ: Perpendicularity or Parallelism of Lines and Planes View
Justify without calculation that $g$ lies in the $x _ { 1 } x _ { 2 }$-plane.
Q6 3 marks Vectors 3D & Lines Dihedral Angle Computation View
In a model for a coastal section by the sea, the $x _ { 1 } x _ { 2 }$-plane describes the horizontal water surface and the line $g$ describes the shoreline. The plane $E$ represents the sea floor in the considered section. A buoy floats on the water surface at the location corresponding to the coordinate origin $O$. One unit of length corresponds to one meter in reality.
Determine the angle at which the sea floor slopes down relative to the water surface.
Q7 4 marks Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
A photographer is to take underwater photos for a travel magazine.
The photographer swims along the shortest possible path from the shoreline to the buoy. Determine the length of this path.
Q8 5 marks Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
From the buoy, the photographer dives vertically with respect to the water surface downward to a location whose distance to the sea floor is exactly three meters and is represented in the model by the point $K$.
Determine by calculation what depth below the water surface the photographer reaches during this dive.
Q9 3 marks Vectors 3D & Lines MCQ: Cross-Section or Surface Area of a Solid View
Three small colorful starfish are located on the sea floor and are represented in the model by the points $P , Q$ and $R$. The photographer moves for his shots from the location described in the model by the point $K$, parallel to the sea floor. The camera lens points perpendicular to the sea floor and has a cone-shaped field of view with an opening angle of $90 ^ { \circ }$.
Assess whether the photographer can reach a location in this way where he can see all three starfish simultaneously in the camera's field of view.