Given that $\triangle A B C$ is an isosceles right triangle with $AB$ as the hypotenuse, $\triangle A B D$ is an equilateral triangle, and the dihedral angle $C - A B - D$ is $150 ^ { \circ }$, then the tangent of the angle between line $C D$ and plane $A B C$ is
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 2 } } { 5 }$
C. $\frac { \sqrt { 3 } } { 5 }$
D. $\frac { 2 } { 5 }$

A cone has vertex $P$ and base center $O$. $AB$ is a diameter of the base, $\angle APB=120°$, $AP=2$. Point $C$ is on the base circle, and the dihedral angle $P$-$AC$-$O=45°$. Then
A. the volume of the cone is $\pi$
B. the lateral surface area of the cone is $4\sqrt{3}\pi$
C. $AC=2\sqrt{2}$
D. the area of $\triangle PAC$ is $\sqrt{3}$

In the cube $ABCD - A_{1}B_{1}C_{1}D_{1}$ , let $E , F$ be the midpoints of $CD , A_{1}B_{1}$ respectively. The total number of intersection points of the sphere with diameter $EF$ and each edge of the cube is $\_\_\_\_$ .

A factory compares the treatment effects of two processes (Process A and Process B) on the elasticity of rubber products through 10 paired experiments. In each paired experiment, two rubber products of the same material are selected, one is randomly chosen to be treated with Process A and the other with Process B. The elasticity rates of the rubber products treated by Process A and Process B are recorded as $x _ { i } , y _ { i } ( i = 1,2 , \cdots 10 )$ respectively. The experimental results are as follows:

Experiment Number $i$	1	2	3	4	5	6	7	8	9	10
Elasticity Rate $x _ { i }$	545	533	551	522	575	544	541	568	596	548
Elasticity Rate $y _ { i }$	536	527	543	530	560	533	522	550	576	536

Let $z _ { i } = x _ { i } - y _ { i } ( i = 1,2 , \cdots , 10 )$. Let $\bar { z }$ denote the sample mean of $z _ { 1 } , z _ { 2 } , \cdots , z _ { 10 }$ and $s ^ { 2 }$ denote the sample variance.
(1) Find $\bar { z }$ and $s ^ { 2 }$.
(2) Determine whether the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B. (If $\bar { z } \geqslant 2 \sqrt { \frac { s ^ { 2 } } { 10 } }$, then it is considered that the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B; otherwise, it is not considered to be significantly higher.)

As shown in the figure, in the triangular pyramid $P - A B C$, $A B \perp B C$, $A B = 2$, $B C = 2 \sqrt { 2 }$, $P B = P C = \sqrt { 6 }$. The midpoints of $BP$, $AP$, and $BC$ are $D$, $E$, and $O$ respectively. $A D = \sqrt { 5 } D O$. Point $F$ is on $AC$ such that $B F \perp A O$.
(1) Prove that $EF \parallel$ plane $BEF$.
(2) Prove that plane $A D O \perp$ plane $B E F$.
(3) Find the sine of the dihedral angle $D - A O - C$.

[Elective 4-4] (10 points) In the rectangular coordinate system $x O y$, with the origin $O$ as the pole and the positive $x$-axis as the polar axis, establish a polar coordinate system. The polar equation of curve $C _ { 1 }$ is $\rho = 2 \sin \theta \left( \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 2 } \right)$. Curve $C _ { 2 }$ is given (see original paper for full problem statement).

A cylinder and a cone have equal base radii and equal lateral surface areas, and both have height $\sqrt { 3 }$ . Then the volume of the cone is
A. $2 \sqrt { 3 } \pi$
B. $3 \sqrt { 3 } \pi$
C. $6 \sqrt { 3 } \pi$
D. $9 \sqrt { 3 } \pi$

Given a regular triangular frustum $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ with volume $\frac { 52 } { 3 }$, $A B = 6$, $A _ { 1 } B _ { 1 } = 2$, then the tangent of the angle between $A _ { 1 } A$ and plane $A B C$ is ( )
A. $\frac { 1 } { 2 }$
B. 1
C. 2
D. 3

Given a quadrangular pyramid with a square base of side length 4, and lateral edges of lengths $4, 4, 2\sqrt{2}, 2\sqrt{2}$ respectively, find the height of the quadrangular pyramid.

Let the set $M = \{ ( i , j , s , t ) \mid i \in \{ 1,2 \} , j \in \{ 3,4 \} , s \in \{ 5,6 \} , t \in \{ 7,8 \} \}$. For a given finite sequence $A$ and sequence $\Omega : \omega _ { 1 } , \omega _ { 2 } , \cdots , \omega _ { k } , \omega _ { k } = \left( i _ { k } , j _ { k } , s _ { k } , t _ { k } \right) \in M$, define transformation $T$: add 1 to columns $i _ { 1 } , j _ { 1 } , s _ { 1 } , t _ { 1 }$ of sequence $A$ to obtain sequence $T _ { 1 } ( A )$; add 1 to columns $i _ { 2 } , j _ { 2 } , s _ { 2 } , t _ { 2 }$ of sequence $T _ { 1 } ( A )$ to obtain sequence $T _ { 2 } T _ { 1 } ( A )$; repeat the above operations to obtain sequence $T _ { k } \cdots T _ { 2 } T _ { 1 } ( A )$, denoted as $\Omega ( A )$.
(3) If $a _ { 1 } + a _ { 3 } + a _ { 5 } + a _ { 7 }$ is even, prove that ``$\Omega ( A )$ is a constant sequence'' is a necessary and sufficient condition for ``$a _ { 1 } + a _ { 2 } = a _ { 3 } + a _ { 4 } = a _ { 5 } + a _ { 6 } = a _ { 7 } + a _ { 8 }$''.

In the right triangular prism $ABC - A_1B_1C_1$, let $D$ be the midpoint of $BC$. Then
A. $AD \perp A_1C$
B. $BC \perp$ plane $AA_1D$
C. $CC_1 \parallel$ plane $AA_1D$
D. $AD \parallel A_1B_1$

In the right triangular prism $ABC - A_1B_1C_1$, let $D$ be the midpoint of $BC$. Then
A. $AD \perp A_1C$
B. $BC \perp$ plane $AA_1D$
C. $AD \parallel A_1B_1$
D. $CC_1 \parallel$ plane $AA_1D$

A closed cylindrical container with base radius 4 cm and height 9 cm (container wall thickness is negligible) contains two iron spheres of equal radius. The maximum radius of the iron spheres is \_\_\_\_ cm.

For energy reasons, the distance between the two points where the two boreholes meet the water-bearing rock layer should be at least 1500 m. Decide on the basis of the model whether this condition is satisfied for every possible second borehole.

Figure 2 shows the floor plan of the hall model in the $x _ { 1 } x _ { 2 }$-plane. Using the results obtained so far, represent the shadow region of the floodlight in the figure exactly.

Fill in the missing values in the table.

1. Examination Part

For the first examination part, schools receive a set of tasks to be completed without aids for download for both basic and advanced courses. These task sets consist of a mandatory part and an optional part.
The following regulations apply:

• The subject teacher determines which task set corresponds to their course (basic course or advanced course).
• For the basic course, the task set in the mandatory part contains three tasks that must be completed: one task for each subject area (Analysis, Vector Geometry and Statistics).

The optional part contains six tasks: two tasks for each subject area (Analysis, Vector Geometry and Statistics). From these six tasks in the optional part, students select two tasks to work on. It is possible to focus on one of the three subject areas (Analysis, Vector Geometry and Statistics).
Overall, students in the basic course work on five tasks in the 1st examination part.

• For the advanced course, the task set in the mandatory part contains four tasks that must be completed: two tasks in the subject area Analysis and one task each in the subject areas Vector Geometry and Statistics.

The optional part contains six tasks: two tasks for each subject area (Analysis, Vector Geometry and Statistics). From these six tasks in the optional part, students select two tasks to work on. It is possible to focus on one of the three subject areas (Analysis, Vector Geometry and Statistics).
Overall, students in the advanced course work on six tasks in the 1st examination part.

• Task selection by teachers is not provided for in the first examination part.

2. Examination Part

Furthermore, four additional task sets are offered for download for the second examination part: one GDC task set and one CAS task set each for the basic and advanced courses. The following regulations apply with regard to these task sets:

• The subject teacher determines which task set corresponds to their course (basic course or advanced course) and the aid used in instruction (GDC or CAS).
• Each task set (for the basic course and the advanced course, for GDC and for CAS) contains five tasks each: two analysis tasks, one task on vector geometry and one task on statistics, which may relate to the focal points ``characteristics of probability distributions'' and ``binomial distribution'' (in the basic course) or the focal points ``characteristics of probability distributions'', ``binomial distribution and normal distribution'' and ``hypothesis testing'' (in the advanced course). Furthermore, an additional analysis task is provided.
• Basic course and advanced course: The second examination part consists of three tasks from the corresponding task set: From the two analysis tasks mentioned first above, the teacher selects exactly one task. Furthermore, the teacher selects two tasks from the remaining tasks (task on vector geometry, task on statistics, additional analysis task).
• Task selection by students is not provided for. c) Aids
• Dictionary for German spelling
• GDC (Graphics Calculator) or CAS (Computer Algebra System)
• Mathematical formula collection

d) Duration of Written Examination

The working time including selection time is 255 minutes in the basic course and 300 minutes in the advanced course. ${ } ^ { 1 }$

III. Overview - Content Focal Points of the Core Curriculum and Focuses

The focuses indicated below each relate to the content focal points specified in Chapter 2 of the core curriculum, which in their entirety are mandatory for written abitur examinations. In the overview below, they are therefore listed in full. The overarching competence expectations as well as the content focal points with their assigned concrete competence expectations remain binding, regardless of whether focuses have been established.
\footnotetext{${ } ^ { 1 }$ The duration of the written examination is indicated for uniform presentation in all subjects with student selection including selection time. This is done analogously to the KMK agreement on the design of the upper secondary level and the abitur examination (Resolution of the Conference of Ministers of Education from 07.07.1972 as amended on 18.02.2021). }

Basic Course

Functions and Analysis	Analytical Geometry and Linear Algebra	Statistics
Functions as mathematical models	Linear systems of equations	Characteristics of probability distributions
\begin{tabular}{l} Continuation of differential calculus
- Investigation of polynomial functions
- Investigation of functions of the type $f ( x ) = p ( x ) e ^ { a x + b }$, where $p ( x )$ is a polynomial with at most three terms
- Investigation of functions that result as simple sums of the above-mentioned function types
- Interpretation and determination of parameters of the above-mentioned functions
- Necessary differentiation rules (product rule, chain rule)

& Representation and investigation of geometric objects & Binomial distribution \hline Basic understanding of the integral concept & Position relationships & \hline Integral calculus & Scalar product & \hline \end{tabular}

Advanced Course

Functions and Analysis	Analytical Geometry and Linear Algebra	Statistics
Functions as mathematical models	Linear systems of equations	Characteristics of probability distributions
\begin{tabular}{l} Continuation of differential calculus
- Treatment of polynomial functions, natural exponential and logarithmic functions and their combinations or compositions with investigation of properties depending on parameters
- Necessary differentiation rules (product rule, chain rule)

& Representation and investigation of geometric objects & Binomial distribution and normal distribution \hline Basic understanding of the integral concept & Position relationships and distances & Hypothesis testing \hline Integral calculus & Scalar product & \hline \end{tabular}

Establish that $\tau = \tau _ { 0 } \cup \tau _ { 1 }$.

Represent on the same figure $\tau _ { 0 } , \tau _ { 1 } , \tau$.

Throughout the problem, the set $\mathbf { C }$ of complex numbers is considered as the Euclidean affine plane equipped with its canonical orthonormal coordinate system $(0,1,i)$ (where $i^2 = -1$). We denote by $K$ the set of triplets $(\alpha, \beta, \gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha + \beta + \gamma = 1$. If $(a,b,c) \in \mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma) \in K\}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
a) Let $a \in \mathbf{C}$ and $\theta \in \mathbf{R}$. Prove that the image $z'$ of the complex number $z$ by the reflection whose axis is the line passing through $a$ and directed by $\mathrm{e}^{\mathrm{i}\theta}$ satisfies the relation: $$z' - a = \mathrm{e}^{2\mathrm{i}\theta} \overline{(z-a)}$$
b) Establish a relation analogous to that of the previous question between a complex number $z$ and its image $z'$ by the homothety with center $a$ and ratio $\rho > 0$.
c) Prove that $\phi_0$ is the composition of a reflection whose axis we will specify and a homothety with strictly positive ratio to be specified and whose center belongs to the axis of the reflection. Prove an analogous property for $\phi_1$. Are these decompositions unique?

Throughout the problem, the set $\mathbf{C}$ of complex numbers is considered as the Euclidean affine plane equipped with its canonical orthonormal coordinate system $(0,1,i)$ (where $i^2=-1$). We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
What is the image of a filled triangle $\widehat{abc}$ by $\phi_0$ and by $\phi_1$? Determine $\phi_0(\tau)$ and $\phi_1(\tau)$.

We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$.
a) Prove that $K$ is a compact subset of $\mathbf{R}^3$ for its usual topology.
b) Prove that $K$ is convex, that is, for every real $t\in[0,1]$ and every pair $(u,v)$ of elements of $K$, $tu+(1-t)v$ belongs to $K$.
c) Establish that, if $(a,b,c)\in\mathbf{C}^3$, $\widehat{abc}$ is a compact convex subset of $\mathbf{C}$ equipped with its usual topology.
d) With the same notation prove the existence of: $$\delta(\widehat{abc}) = \max\left\{|z'-z| / (z,z')\in\widehat{abc}^2\right\}$$

We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$.
a) Prove that, if we fix $z\in\mathbf{C}$ and $(a,b,c)\in\mathbf{C}^3$: $$\max\left\{|z'-z| / z'\in\widehat{abc}\right\} = \max(|z-a|,|z-b|,|z-c|)$$
b) Deduce from this a simple expression for $\delta(\widehat{abc})$.

We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
Let $(r_n)_{n\geq 1}$ be an element of $\{0,1\}^{\mathbf{N}^*}$. For each non-zero natural number $n$, we denote $\tilde{\tau}_n = \phi_{r_1}\circ\phi_{r_2}\circ\cdots\circ\phi_{r_n}(\tau)$.
Show that $\bigcap_{n\geq 1}\tilde{\tau}_n$ is reduced to a single point belonging to $\tau$.