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QA 5 marks Travel graphs View

A robot cat starts from the origin on a number line and moves in the positive direction. Its movement method is as follows: With an 8-second cycle, it moves at a constant speed of 4 units per second for 6 seconds, then rests for 2 seconds. Continuing this way, the robot cat will reach the position with coordinate 116 on the number line after $\underbrace{(14)(15)}$ seconds from the start of movement.

QB 5 marks Vectors: Lines & Planes Find Cartesian Equation of a Plane View

In coordinate space, there are two lines $L _ { 1 } , L _ { 2 }$ and a plane $E$. The line $L _ { 1 } : \frac { x } { 2 } = \frac { y } { - 3 } = \frac { z } { - 5 }$, and the parametric equation of $L _ { 2 }$ is $\left\{ \begin{array} { l } x = 1 \\ y = 1 + 2 t \\ z = 1 + 3 t \end{array} \right.$ ($t$ is a real number). If $L _ { 1 }$ lies on $E$ and $L _ { 2 }$ does not intersect $E$, then the equation of $E$ is $x -$ (16) $y +$ (17) $z =$ (18).

QC 5 marks Combinations & Selection Selection with Arithmetic or Divisibility Conditions View

From the nine numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, any three distinct numbers are randomly selected, with each number having equal probability of being selected. The probability that the product of the three numbers is a perfect square is (20). (Express as a fraction in lowest terms)

QD 5 marks Exponential Functions Intersection and Distance between Curves View

On the coordinate plane, $\Gamma$ is a square with side length 4, centered at the point $(1,1)$, with sides parallel to the coordinate axes. The graph of the function $y = a \times 2 ^ { x }$ intersects $\Gamma$, where $a$ is a real number. The maximum possible range of $a$ is (22)(23) $\leq a \leq$ (24).

QE 5 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View

Write $( \sqrt [ 3 ] { 49 } ) ^ { 100 }$ in scientific notation as $( \sqrt [ 3 ] { 49 } ) ^ { 100 } = a \times 10 ^ { n }$, where $1 \leq a < 10$ and $n$ is a positive integer. If the integer part of $a$ is $m$, then the ordered pair $( m , n ) = ($ (25) )(26).

QF 5 marks Radians, Arc Length and Sector Area View

As shown in the figure, a robot starts from a point $P$ on the ground and moves according to the following rules: First, move forward 1 meter in a certain direction, then rotate counterclockwise $45 ^ { \circ }$ in the direction of movement; move forward 1 meter in the new direction, then rotate clockwise $90 ^ { \circ }$ in the direction of movement; move forward 1 meter in the new direction, then rotate counterclockwise $45 ^ { \circ }$ in the direction of movement; move forward 1 meter in the new direction, then rotate clockwise $90 ^ { \circ }$ in the direction of movement, and so on. The path traced by the robot forms a closed region. The area of this closed region is (28) + (29) $\sqrt { (30) }$ square meters. (Express as a fraction in simplest radical form)

QG 5 marks Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View

In tetrahedron $A B C D$, $\overline { A B } = \overline { A C } = \overline { A D } = 4 \sqrt { 6 }$, $\overline { B D } = \overline { C D } = 8$, and $\cos \angle B A C = \frac { 1 } { 3 }$. The distance from point $D$ to plane $A B C$ is (31) $\sqrt { (32) }$. (Express as a fraction in simplest radical form)

Q1 5 marks Matrices Matrix Power Computation and Application View

Let $A = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$. If $A ^ { 4 } = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, what is the value of $a + b + c + d$?
(1) 158
(2) 162
(3) 166
(4) 170
(5) 174

Q2 5 marks Laws of Logarithms Solve a Logarithmic Equation View

A sequence of five real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 } , a _ { 5 }$ where each term is greater than 1, and between any two adjacent terms, one is twice the other. If $a _ { 1 } = \log _ { 10 } 36$, how many possible values can $a _ { 5 }$ have?
(1) 3
(2) 4
(3) 5
(4) 7
(5) 8

Q3 5 marks Sine and Cosine Rules Circumradius or incircle radius computation View

As shown in the figure, $\triangle A B C$ is an acute triangle, $P$ is a point outside the circumcircle $\Gamma$ of $\triangle A B C$, and both $\overline { P B }$ and $\overline { P C }$ are tangent to circle $\Gamma$. Let $\angle B P C = \theta$. What is the value of $\cos A$?
(1) $\sin 2 \theta$
(2) $\frac { \sin \theta } { 2 }$
(3) $\sin \frac { \theta } { 2 }$
(4) $\frac { \cos \theta } { 2 }$
(5) $\cos \frac { \theta } { 2 }$

Q4 5 marks Vectors Introduction & 2D Area Computation Using Vectors View

Let $\vec { a }$ and $\vec { b }$ be non-zero vectors in the plane. If the area of the triangle formed by $2 \vec { a } + \vec { b }$ and $\vec { a } + 2 \vec { b }$ is 6, what is the area of the triangle formed by $3 \vec { a } + \vec { b }$ and $\vec { a } + 3 \vec { b }$?
(1) 8
(2) 9
(3) 12
(4) 13.5
(5) 16

Q5 5 marks Factor & Remainder Theorem Remainder by Quadratic or Higher Divisor View

Let $f ( x )$ be a real polynomial function of degree 3 satisfying the condition that the remainder when $( x + 1 ) f ( x )$ is divided by $x ^ { 3 } + 2$ is $x + 2$. If $f ( 0 ) = 4$, what is the value of $f ( 2 )$?
(1) 8
(2) 10
(3) 15
(4) 18
(5) 20

Q6 5 marks Circles Intersection of Circles or Circle with Conic View

On the coordinate plane, there is a regular hexagon $A B C D E F$ with side length 3, where $A ( 3,0 ) , D ( - 3,0 )$. How many intersection points does the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$ have with the regular hexagon $A B C D E F$?
(1) 0
(2) 2
(3) 4
(4) 6
(5) 8

Q7 5 marks Conditional Probability Confusion Matrix / Misidentification Probability Table View

A psychologist conducted an experiment with 1000 subjects in a dark room, where each subject had to observe and identify three digit cards: 6, 8, and 9. The probability of mistaking the actual digit for another digit is shown in the following table:

\backslashbox{Actual Digit}{Seen as}	6	8	9	Other
6	0.4	0.3	0.2	0.1
8	0.3	0.4	0.1	0.2
9	0.2	0.2	0.5	0.1

For example: The actual digit 6 is seen as 6, 8, 9 with probabilities 0.4, 0.3, 0.2 respectively, and is seen as another digit with probability 0.1. Based on the above experimental results, select the correct options.
(1) If the actual digit is 8, then there is at least a 50\% chance it will be seen as 8
(2) If the actual digit is 6, then there is a 60\% chance it will be seen as not 6
(3) Among the three digits 6, 8, 9, the digit 9 has the lowest probability of being misidentified
(4) If the digit seen is 6, then the probability that it is actually 6 is less than 50\%
(5) If the digit seen is 9, then the probability that it is actually 9 is greater than $\frac { 2 } { 3 }$

Q8 5 marks Circles Chord Length and Chord Properties View

As shown in the figure, $L$ is a line passing through the origin $O$ on the coordinate plane, $\Gamma$ is a circle centered at $O$, and $L$ and $\Gamma$ have one intersection point $A ( 3,4 )$. It is known that $B , C$ are two distinct points on $\Gamma$ satisfying $\overrightarrow { B C } = \overrightarrow { O A }$. Select the correct options.
(1) The other intersection point of $L$ and $\Gamma$ is $( - 4 , - 3 )$
(2) The slope of line $B C$ is $\frac { 3 } { 4 }$
(3) $\angle A O C = 60 ^ { \circ }$
(4) The area of $\triangle A B C$ is $\frac { 25 \sqrt { 3 } } { 2 }$
(5) $B$ and $C$ are in the same quadrant

Q9 5 marks Probability Definitions Probability Using Set/Event Algebra View

A village mayor election has two polling stations. The proportion of valid votes received by the two candidates at each polling station is shown in the following table (invalid votes are not counted):

	Candidate A	Candidate B
First Polling Station	$40 \%$	$60 \%$
Second Polling Station	$55 \%$	$45 \%$

Assume the number of valid votes at the first and second polling stations are $x$ and $y$ respectively (where $x > 0 , y > 0$), and the candidate with the higher total votes wins. Based on the above table, select the correct options.
(1) When the total number of valid votes $x + y$ is known, the winner can be determined
(2) When the ratio $x : y$ is less than $\frac { 1 } { 2 }$, the winner can be determined
(3) When $x > y$, the winner can be determined
(4) When Candidate A's valid votes at the first polling station exceed those at the second polling station, the winner can be determined
(5) When Candidate B's valid votes at the second polling station exceed those at the first polling station, the winner can be determined

Q10 5 marks Sine and Cosine Rules Ambiguous case and triangle existence/uniqueness View

In $\triangle A B C$, it is known that $\overline { A B } = 4$ and $\overline { A C } = 6$, which is insufficient to determine the shape and size of $\triangle A B C$. However, knowing certain additional conditions (for example, knowing the length of $\overline { B C }$) would uniquely determine the shape and size of $\triangle A B C$. Select the correct options.
(1) If we additionally know the value of $\cos A$, then $\triangle A B C$ can be uniquely determined
(2) If we additionally know the value of $\cos B$, then $\triangle A B C$ can be uniquely determined
(3) If we additionally know the value of $\cos C$, then $\triangle A B C$ can be uniquely determined
(4) If we additionally know the area of $\triangle A B C$, then $\triangle A B C$ can be uniquely determined
(5) If we additionally know the circumradius of $\triangle A B C$, then $\triangle A B C$ can be uniquely determined

Q11 5 marks Vectors Introduction & 2D True/False or Multiple-Statement Verification View

On a plane, there is a trapezoid $A B C D$ with upper base $\overline { A B } = 10$, lower base $\overline { C D } = 15$, and leg length $\overline { A D } = \overline { B C } + 1$. Select the correct options.
(1) $\angle A > \angle B$
(2) $\angle B + \angle D < 180 ^ { \circ }$
(3) $\overrightarrow { B A } \cdot \overrightarrow { B C } < 0$
(4) The length of $\overline { B C }$ could be 2
(5) $\overrightarrow { C B } \cdot \overrightarrow { C D } < 30$

Q12 5 marks Permutations & Arrangements Probability via Permutation Counting View

Let $P ( X )$ denote the probability of event $X$ occurring, and $P ( X \mid Y )$ denote the probability of event $X$ occurring given that event $Y$ has occurred. There are 7 balls of the same size: 2 black balls, 2 white balls, and 3 red balls arranged in a row. Let event $A$ be the event that the 2 black balls are adjacent, event $B$ be the event that the 2 black balls are not adjacent, and event $C$ be the event that no two red balls are adjacent. Select the correct options.
(1) $P ( A ) > P ( B )$
(2) $P ( C ) = \frac { 2 } { 7 }$
(3) $2 P ( C \mid A ) + 5 P ( C \mid B ) < 2$
(4) $P ( C \mid A ) > 0.2$
(5) $P ( C \mid B ) > 0.3$

Q13 5 marks Roots of polynomials Existence or counting of roots with specified properties View

Let the polynomial function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b , c$ are all rational numbers. Select the correct options.
(1) The graph of $y = f ( x )$ and the parabola $y = x ^ { 2 } + 100$ may have no intersection points
(2) If $f ( 0 ) f ( 1 ) < 0 < f ( 0 ) f ( 2 )$, then the equation $f ( x ) = 0$ must have three distinct real roots
(3) If $1 + 3 i$ is a complex root of the equation $f ( x ) = 0$, then the equation $f ( x ) = 0$ has a rational root
(4) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form an arithmetic sequence in order
(5) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form a geometric sequence in order

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