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
2025 abitur_nrw__typ_analysis_lk

4 maths questions

Qa 9 marks Areas by integration View
(1) Sketch the graph of $f _ { 0 }$ in the interval [-2;2] in the figure. (2) The graph of $f _ { 0 }$ encloses an area with the $x$-axis in the second quadrant.
Determine the area $A _ { 0 }$ of this region by calculation. [For verification: $A _ { 0 } = 2,25 \mathrm { FE }$.] (3) Given is the line $g : y = - 2 x , x \in \mathbb { R }$.
Determine the ratio in which the line $g$ divides the area from (2).
Qb 12 marks Stationary points and optimisation Find absolute extrema on a closed interval or domain View
(1) Determine the coordinates of the inflection point of the graph of $f _ { k }$ as a function of $k$ by calculation. [For verification: For the $x$-coordinate $x _ { w }$ of the inflection point, we have: $x _ { w } = k$.] The inflection points of the graphs of $f _ { k }$ with $k \geq - 0,5$ lie on the graph of the function $w$ with $w ( x ) = \mathrm { e } ^ { - x } \cdot x ^ { 2 } , x \geq - 0,5$. The graph of $w$ is called the locus curve of the inflection points of the function family. (2) (i) Show: $w ^ { \prime } ( x ) = \mathrm { e } ^ { - x } \cdot \left( - x ^ { 2 } + 2 x \right)$. (ii) Determine the global maximum of $w$ by calculation. (3) Given is the function $w _ { \text {new } }$ with the equation
$$w _ { \text {new } } ( x ) = 3 \cdot \mathrm { e } ^ { - ( x - 2 ) } \cdot ( x - 2 ) ^ { 2 } , x \in \mathbb { R } .$$
The graph of $w _ { \text {new } }$ is the locus curve of the inflection points of another family of functions $v _ { k }$ with $k \geq - 0,5$.
Give a possible function equation for $v _ { k }$.
Qc 5 marks Exponential Equations & Modelling Exponential Growth/Decay Modelling with Contextual Interpretation View
The cooling process in the first 10 minutes is to be modeled for $0 \leq t \leq 10$ by a function $u _ { 1 }$ defined on $\mathbb { R }$ with $u _ { 1 } ( t ) = a + b \cdot \mathrm { e } ^ { - c \cdot t } , a , b , c \in \mathbb { R } , c > 0$.
Here, $t$ denotes the time in minutes since the beginning of the investigation and $u _ { 1 } ( t )$ denotes the temperature of the coffee in ${ } ^ { \circ } \mathrm { C }$.
Explain why $a = 18$ is chosen in the modeling, and then calculate the values of $b$ and $c$ based on the information in the table. [Check solution with rounded values: $u _ { 1 } ( t ) = 18 + 55,86 \cdot \mathrm { e } ^ { - 0,053 \cdot t }$.]
Qd 4 marks Applied differentiation Applied modeling with differentiation View
For $11 \leq t \leq 25$, the cooling process of the mixture can be approximated by the function $u _ { 2 }$ defined on $\mathbb { R }$ with $u _ { 2 } ( t ) = 18 + 36 \cdot \mathrm { e } ^ { - 0,033 \cdot t }$.
Here, $t$ again denotes the time in minutes since the beginning of the investigation and $u _ { 2 } ( t )$ denotes the temperature of the mixture in ${ } ^ { \circ } \mathrm { C }$. The graph of $u _ { 2 }$ is monotonically decreasing.
Determine the average rate of change of the temperature of the coffee in the first 10 minutes of the investigation and the average rate of change of the temperature of the mixture from the 11th to the 21st minute of the investigation.
Compare the results.