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Q1 5 marks Sequences and Series Evaluation of a Finite or Infinite Sum View

Select the value of $1 . \overline { 5 } \times 5$.
(1) $7 . \overline { 5 }$
(2) $7 . \overline { 6 }$
(3) $7 . \overline { 7 }$
(4) $7 . \overline { 8 }$
(5) $7 . \overline { 9 }$

Q2 5 marks Circles Circle Identification and Classification View

On the coordinate plane, which of the following equations represents a circle passing through the point $(1,1)$?
(1) $( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$
(2) $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
(3) $3 ( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$
(4) $x ^ { 2 } + y ^ { 2 } = 1$
(5) $x ^ { 2 } + 3 y = 4$

Q3 5 marks Conditional Probability Combinatorial Conditional Probability (Counting-Based) View

There are two fair six-sided dice A and B: The numbers on A are $1, 2, 5, 6, 7, 9$, The numbers on B are $1, 3, 4, 5, 6, 9$. The relationship between the numbers on A and B is recorded in the table below. For example: if the numbers on A and B are 5 and 3 respectively, it is recorded as ``A wins''; if both A and B show 5, it is recorded as ``tie''.

\multirow{2}{*}{}		\multicolumn{6}{\|c\|}{A}
	Number	1	2	5	6	7	9
\multirow{6}{*}{B}	1	Tie	A wins	A wins	A wins	A wins	A wins
	3	B wins	B wins	A wins	A wins	A wins	A wins
	4	B wins	B wins	A wins	A wins	A wins	A wins
	5	B wins	B wins	Tie	A wins	A wins	A wins
	6	B wins	B wins	B wins	Tie	A wins	A wins
	9	B wins	B wins	B wins	B wins	B wins	Tie

If a person rolls both dice A and B simultaneously, what is the probability that B shows 6 given that A's number is greater than B's number?
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 9 }$
(3) $\frac { 1 } { 16 }$
(4) $\frac { 1 } { 18 }$
(5) $\frac { 1 } { 32 }$

Q4 5 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View

In space, there is a unit cube with edge length 1. Point $O$ is one vertex, and the remaining 7 vertices are $A, B, C, D, E, F, G$. Given that $\overline { O A } = \overline { A B } = \overline { B C } = \overline { C D } = \overline { D E } = \overline { E F } = \overline { F G } = 1$ and $\overline { O G } > 1$, select the vertex farthest from point $O$.
(1) $C$
(2) $D$
(3) $E$
(4) $F$
(5) $G$

Q5 5 marks Linear regression View

A company collected data on the number of customers $x$ (in units of 100 people) and sales revenue $y$ (in units of 10,000 yuan) from 8 branches last week, obtaining 8 data points $(x, y)$ as follows: $(3,3), (3,5), (3,2), (4,4), (5,8), (6,7), (8,12), (8,7)$. These 8 points are plotted on the coordinate plane, and the regression line equation for $y$ with respect to $x$ is determined to be $y = \frac { 5 } { 4 } x - \frac { 1 } { 4 }$.
The company wants to analyze from another perspective. The 8 data points are sorted separately from smallest to largest for the number of customers and sales revenue, resulting in new 8 data points $(x, y)$ as follows: $(3,2), (3,3), (3,4), (4,5), (5,7), (6,7), (8,8), (8,12)$. Let the regression line equation for $y$ with respect to $x$ for the new 8 data points be $y = m x + b$, where $m, b$ are real numbers. Based on the above, select the correct option.
(1) $m = \frac { 5 } { 4 }$ and $b = - \frac { 1 } { 4 }$
(2) $m > \frac { 5 } { 4 }$ and $b > - \frac { 1 } { 4 }$
(3) $m > \frac { 5 } { 4 }$ and $b < - \frac { 1 } { 4 }$
(4) $m < \frac { 5 } { 4 }$ and $b > - \frac { 1 } { 4 }$
(5) $m < \frac { 5 } { 4 }$ and $b < - \frac { 1 } { 4 }$

Q6 5 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

Select the value of $\sum _ { k = 1 } ^ { 5 } \log _ { 7 } \left( \frac { 2 k - 1 } { 2 k + 1 } \right)$.
(1) $- \log 11$
(2) $\log 11$
(3) $\log \frac { 11 } { 7 }$
(4) $- \frac { \log 11 } { \log 7 }$
(5) $\frac { \log 11 } { \log 7 }$

Q7 8 marks Matrices True/False or Multiple-Select Conceptual Reasoning View

Let the second-order matrices $A = \left[ \begin{array} { l l } 1 & 0 \\ 1 & 0 \end{array} \right], B = \left[ \begin{array} { l l } 0 & 1 \\ 0 & 1 \end{array} \right]$. Select the correct options.
(1) $A ^ { 2 } = A$
(2) $A + B = B + A$
(3) $A B = B A$
(4) $( A - B ) ^ { 2 } = A ^ { 2 } - 2 A B + B ^ { 2 }$
(5) $( A + B ) ^ { 2 } = 2 ( A + B )$

Q8 8 marks Sine and Cosine Rules Prove an inequality or ordering relationship in a triangle View

On a plane, there is a triangle $A B C$ where $\angle A = 91 ^ { \circ }, \angle C = 29 ^ { \circ }$. Let $\overline { B C } = a, \overline { C A } = b, \overline { A B } = c$. Select the correct options.
(1) $a ^ { 2 } > b ^ { 2 } + c ^ { 2 }$
(2) $\frac { c } { a } > \sin 29 ^ { \circ }$
(3) $\frac { b } { a } > \cos 29 ^ { \circ }$
(4) $\frac { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } } { a b } < \sqrt { 3 }$
(5) The circumradius of triangle $A B C$ is less than $c$

Q9 8 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View

There is a card-drawing prize activity with the following rules: In an opaque box, there are 2 cards marked with ``1000 yuan'' and 3 cards marked with ``0 yuan''. A participant randomly draws one card from the box. Without knowing the amount marked on the drawn card, the host then places a card marked with ``500 yuan'' into the box. At this point, the participant has two choices: (I) Keep the originally drawn card; the amount marked on that card is the prize won. (II) Discard the originally drawn card without returning it, and randomly draw another card from the box; the amount marked on that card is the prize won.
A participant joins this activity. Assume each card has an equal chance of being drawn. Select the correct options.
(1) If the participant chooses (I), the probability of winning 0 yuan is $\frac { 3 } { 5 }$
(2) If the participant chooses (I), the expected value of the prize is 500 yuan
(3) If the participant chooses (II), the probability of winning 1000 yuan is $\frac { 2 } { 5 }$
(4) If the participant chooses (II), the probability of winning 0 yuan is $\frac { 12 } { 25 }$
(5) If the participant chooses (II), the expected value of the prize is 420 yuan

Q10 6 marks Complex Numbers Arithmetic Complex Division/Multiplication Simplification View

Let $i = \sqrt { - 1 }$. Given that the complex number $\frac { 1 - 7 i } { - 1 + i } = a + b i$, where $a, b$ are real numbers. Then $a =$ (10–1)(10–2), $b =$ (10–3).

Q11 6 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View

A washing machine cycle must select one from 5 different fabric types (1, 2, 3, 4, 5), paired with one of 4 different modes (A, B, C, D), and there are 3 additional functions (A, B, C) that can be freely chosen to enable or disable. However, ``fabric type 1'' cannot be used simultaneously with additional function ``A''. For example, ``fabric type 2'' paired with ``mode A'', with both ``A'' and ``B'' additional functions enabled is a valid cycle; but ``fabric type 1'' paired with ``mode A'', with both ``A'' and ``B'' additional functions enabled is not a valid cycle. Based on the above, this washing machine has how many valid cycles?

Q12 6 marks Vectors Introduction & 2D Dot Product Computation View

On a plane, there are three non-collinear points $A, B, C$. Given that the dot product of vectors $\overrightarrow { A B }$ and $\overrightarrow { A C }$ is 16, the dot product of $\overrightarrow { C B }$ and $\overrightarrow { A C }$ is 3, then $\overline { A C } = \sqrt{\text{(12--1)}}$ (12–2). (Simplify to simplest radical form)

Q13 3 marks Stationary points and optimisation Find concavity, inflection points, or second derivative properties View

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.
Regarding the statements about $f ^ { \prime \prime } ( - 3 )$ and $f ^ { \prime \prime } ( 1 )$, select the correct option. (Single choice)
(1) $f ^ { \prime \prime } ( - 3 ) = f ^ { \prime \prime } ( 1 ) = 0$
(2) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(3) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) < 0$
(4) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(5) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) < 0$

Q14 6 marks Chain Rule Finding Composition Parameters from Derivative Conditions View

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.
Given that the slope of the tangent line at the inflection point of the graph of $y = f ( x )$ is 4, find $f ^ { \prime } ( x )$.

Q15 4 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$. Given that the slope of the tangent line at the inflection point of the graph of $y = f ( x )$ is 4, so that $f ^ { \prime } ( x )$ is as found in question 14.
Find the value of $\int _ { - 3 } ^ { 1 } f ^ { \prime } ( x ) \, d x$.

Q16 3 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View

A person wants to plant two types of fruits, A and B, on farmland, and sets the planting area of fruit A (area A) and the planting area of fruit B (area B) to satisfy the following three conditions: (I) Area A does not exceed 15 hectares. (II) The sum of area A and area B does not exceed 24 hectares. (III) Area A does not exceed 3 times area B, and area B does not exceed 2 times area A. Let area A be $x$ hectares and area B be $y$ hectares.
Which of the following options for the ordered pair $(x, y)$ satisfies the above three conditions? (Single choice)
(1) $(7,15)$
(2) $(12,13)$
(3) $(14,10)$
(4) $(15,4)$
(5) $(16,8)$

Q17 4 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View

A person wants to plant two types of fruits, A and B, on farmland, and sets the planting area of fruit A (area A) and the planting area of fruit B (area B) to satisfy the following three conditions: (I) Area A does not exceed 15 hectares. (II) The sum of area A and area B does not exceed 24 hectares. (III) Area A does not exceed 3 times area B, and area B does not exceed 2 times area A. Let area A be $x$ hectares and area B be $y$ hectares.
Express the three conditions set by the person for area A and area B as a system of linear inequalities in $x$ and $y$.

Q18 8 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View

A person wants to plant two types of fruits, A and B, on farmland, and sets the planting area of fruit A (area A) and the planting area of fruit B (area B) to satisfy the following three conditions: (I) Area A does not exceed 15 hectares. (II) The sum of area A and area B does not exceed 24 hectares. (III) Area A does not exceed 3 times area B, and area B does not exceed 2 times area A. Let area A be $x$ hectares and area B be $y$ hectares.
Given that when the farmland is harvested, fruit A yields a profit of 6 ten-thousand yuan per hectare and fruit B yields a profit of 7 ten-thousand yuan per hectare, find the maximum profit from planting both fruits in ten-thousand yuan. Show the calculation process in the solution area of the answer sheet, and draw the feasible region in the diagram area of the answer sheet, marking all vertices of the region and shading the region with diagonal lines.

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