2. To understand the rural economic situation in a certain area, a sample survey was conducted on the annual household income of farmers in that area. The survey data on farmers' annual household income was organized into the following frequency distribution histogram: [Figure] Based on this frequency distribution histogram, which of the following conclusions is incorrect? A. The estimated proportion of farmers with annual household income below 4.5 ten thousand yuan is $6 \%$ B. The estimated proportion of farmers with annual household income not less than 10.5 ten thousand yuan is $10 \%$ C. The estimated average annual household income of farmers in this area does not exceed 6.5 ten thousand yuan D. It is estimated that more than half of the farmers in this area have annual household income between 4.5 and 8.5 ten thousand yuan
Q3
Complex Numbers ArithmeticSolving Equations for Unknown Complex NumbersView
3. Given $( 1 - i ) ^ { 2 } z = 3 + 2 i$, then $z =$ A. $- 1 - \frac { 3 } { 2 } i$ B. $- 1 + \frac { 3 } { 2 } i$ C. $- \frac { 3 } { 2 } + i$ D. $- \frac { 3 } { 2 } - i$
Q4
Curve SketchingVariation Table and Monotonicity from Sign of DerivativeView
4. Which of the following functions is an increasing function? A. $f ( x ) = - x$ B. $f ( x ) = \left( \frac { 2 } { 3 } \right) ^ { x }$ C. $f ( x ) = x ^ { 2 }$ D. $f ( x ) = \sqrt [ 3 ] { x }$
5. The distance from the point $( 3,0 )$ to an asymptote of the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ is A. $\frac { 9 } { 5 }$ B. $\frac { 8 } { 5 }$ C. $\frac { 6 } { 5 }$ D. $\frac { 4 } { 5 }$
Q6
Laws of LogarithmsLogarithmic Formula Application (Modeling)View
6. Vision of adolescents is a matter of widespread social concern. Vision can be measured using a vision chart. Vision data is usually recorded using the five-point recording method and the decimal recording method. The data $L$ in the five-point recording method and the data $V$ in the decimal recording method satisfy $L = 5 + \lg V$. It is known that a student's vision data in the five-point recording method is 4.9. Then the student's vision data in the decimal recording method is approximately ( $\sqrt [ 10 ] { 10 } \approx 1.259$ ) A. 1.5 B. 1.2 C. 0.8 D. 0.6
Q8
Sine and Cosine RulesFind a side length using the cosine ruleView
8. In $\triangle A B C$, it is known that $B = 120 ^ { \circ } , A C = \sqrt { 19 } , A B = 2$, then $B C =$ ( ) A. 1 B. $\sqrt { 2 }$ C. $\sqrt { 5 }$ D. 3
Q9
Geometric Sequences and SeriesFinite Geometric Sum and Term RelationshipsView
9. Let $S _ { n }$ denote the sum of the first $n$ terms of the geometric sequence $\left\{ a _ { n } \right\}$. If $S _ { 2 } = 4 , S _ { 4 } = 6$, then $S _ { 6 } =$ A. 7 B. 8 C. 9 D. 10
Q10
Permutations & ArrangementsProbability via Permutation CountingView
10. Three 1's and two 0's are randomly arranged in a row. The probability that the two 0's are not adjacent is A. 0.3 B. 0.5 C. 0.6 D. 0.8
Q11
Addition & Double Angle FormulaeTrigonometric Equation Solving via IdentitiesView
15. The function $f ( x ) = 2 \cos ( \omega x + \varphi )$ has a partial graph shown in the figure. Then $f \left( \frac { \pi } { 2 } \right) =$ $\_\_\_\_$ . [Figure]
Q16
CirclesArea and Geometric Measurement Involving CirclesView
16. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$. Let $P , Q$ be two points on $C$ that are symmetric with respect to the origin, and $| P Q | = \left| F _ { 1 } F _ { 2 } \right|$. Then the area of quadrilateral $P F _ { 1 } Q F _ { 2 }$ is $\_\_\_\_$ .
III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17--21 are required questions that all students must answer. Questions 22 and 23 are optional questions; students should answer according to the requirements.
17. Two machine tools, Machine A and Machine B, produce the same type of product. Products are classified by quality into first-grade and second-grade products. To compare the quality of products from the two machines, 200 products were produced by each machine. The quality statistics are shown in the table below:
First-grade
Second-grade
Total
Machine A
150
50
200
Machine B
120
80
200
Total
270
130
400
(1) What are the frequencies of first-grade products produced by Machine A and Machine B, respectively? (2) Can we conclude with 99\% confidence that there is a difference in product quality between Machine A and Machine B? Attachment: $\mathrm { K } ^ { 2 } = \frac { n ( a d - b c ) ^ { 2 } } { ( a + b ) ( c + d ) ( a + c ) ( b + d ) }$,
$\mathrm { P } \left( \mathrm { K } ^ { 2 } \geqslant k \right)$
0.050
0.010
0.001
$k$
3.841
6.635
10.828
Q18
Arithmetic Sequences and SeriesProve a Sequence is ArithmeticView
18. Let $S _ { n }$ denote the sum of the first $n$ terms of $\left\{ a _ { n } \right\}$. Given that $a _ { n } > 0 , a _ { 2 } = 3 a _ { 1 }$, and the sequence $\left\{ \sqrt { S _ { n } } \right\}$ is an arithmetic sequence. Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence.
19. In a right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, the lateral face $A A _ { 1 } B _ { 1 } B$ is a square. $A B = B C = 2$. Let $E , F$ be the midpoints of $A C$ and $C C _ { 1 }$ respectively, and $B F \perp A _ { 1 } B _ { 1 }$. (1) Find the volume of the triangular pyramid $F - E B C$; (2) Let $D$ be a point on edge $A _ { 1 } B _ { 1 }$. Prove that $B F \perp D E$. [Figure]
Q20
Stationary points and optimisationDetermine intervals of increase/decrease or monotonicity conditionsView
20. Let $f ( x ) = a ^ { 2 } x ^ { 2 } + a x - 3 \ln x + 1$, where $a > 0$. (1) Discuss the monotonicity of $f ( x )$; (2) If the graph of $y = f ( x )$ has no common points with the $x$-axis, find the range of values for $a$.
21. The parabola $C$ has its vertex at the origin $O$ and its focus on the $x$-axis. The line $l : x = 1$ intersects $C$ at points $P , Q$, and $O P \perp O Q$. Given the point $M ( 2,0 )$, and circle $\odot M$ is tangent to $l$. (1) Find the equations of $C$ and $\odot M$; (2) Let $A _ { 1 } , A _ { 2 } , A _ { 3 }$ be three points on $C$. Lines $A _ { 1 } A _ { 2 }$ and $A _ { 1 } A _ { 3 }$ are both tangent to $\odot M$. Determine the positional relationship between line $A _ {