An opaque bag contains blue and green balls, each marked with a number 1 or 2. The quantities are shown in the table below. For example, there are 2 blue balls marked with number 1.
Blue
Green
Number 1
2
4
Number 2
3
$k$
A ball is randomly drawn from the bag (each ball has an equal probability of being drawn). Given that the event of drawing a blue ball and the event of drawing a ball marked with 1 are independent, what is the value of $k$? (1) 2 (2) 3 (3) 4 (4) 5 (5) 6
On the coordinate plane, $P ( a , 0 )$ is a point on the $x$-axis, where $a > 0$. Let $L _ { 1 }$ and $L _ { 2 }$ be lines passing through point $P$ with slopes $- \frac { 4 } { 3 }$ and $- \frac { 3 } { 2 }$ respectively. Given that the difference in areas of the two right triangles formed by $L _ { 1 }$ and $L _ { 2 }$ with the two coordinate axes is 3, what is the value of $a$? (1) $3 \sqrt { 2 }$ (2) 6 (3) $6 \sqrt { 2 }$ (4) 9 (5) $8 \sqrt { 2 }$
A school is holding a concert with 5 piano performances, 4 violin performances, and 3 vocal performances, totaling 12 different pieces. The school wants to arrange performances of the same type together, and vocal performances must come after either piano or violin performances. How many possible arrangements of pieces are there for this concert? (1) $5 ! \times 4 ! \times 3 !$ (2) $2 \times 5 ! \times 4 ! \times 3 !$ (3) $3 \times 5 ! \times 4 ! \times 3 !$ (4) $4 \times 5 ! \times 4 ! \times 3 !$ (5) $6 \times 5 ! \times 4 ! \times 3 !$
On the coordinate plane, a point whose $x$-coordinate and $y$-coordinate are both integers is called a lattice point. How many lattice points are in the interior (not including the boundary) of the region bounded by the function graph $y = \log _ { 2 } x$, the $x$-axis, and the line $x = \frac { 61 } { 2 }$? (1) 88 (2) 89 (3) 90 (4) 91 (5) 92
In coordinate space, there are three mutually perpendicular vectors $\vec { u } , \vec { v } , \vec { w }$. Given that $\vec { u } - \vec { v } = ( 2 , - 1,0 )$ and $\vec { v } - \vec { w } = ( - 1,2,3 )$. What is the volume of the parallelepiped spanned by $\vec { u } , \vec { v } , \vec { w }$? (1) $2 \sqrt { 5 }$ (2) $5 \sqrt { 2 }$ (3) $2 \sqrt { 10 }$ (4) $4 \sqrt { 5 }$ (5) $4 \sqrt { 10 }$
A sequence $< a _ { n } >$ satisfies $3 a _ { n + 1 } = a _ { n } + n$ (for all positive integers $n$) and $a _ { 1 } = 2$. Let the sequence $< b _ { n } >$ satisfy $b _ { n } = a _ { n } - \frac { n } { 2 } + \frac { 3 } { 4 }$. Select the correct options. (1) $a _ { 2 } = 2$ (2) $b _ { 2 } = \frac { 3 } { 4 }$ (3) The sequence $< b _ { n } >$ is a geometric sequence with common ratio $\frac { 2 } { 3 }$ (4) For any positive integer $n$, $3 ^ { n } a _ { n }$ is always a positive integer (5) $b _ { 10 } < 10 ^ { - 4 }$
Consider points $P ( x , y )$ on the coordinate plane satisfying the equation $\frac { 2 ^ { x ^ { 2 } } } { 8 } = \frac { 4 ^ { x } } { 2 ^ { y ^ { 2 } } }$. Select the correct options. (1) When $x = 3$, there are 2 distinct solutions satisfying this equation (2) If point $( a , b )$ satisfies this equation, then point $( - a , - b )$ also satisfies this equation (3) All possible points $P ( x , y )$ form a circle (4) Point $P ( x , y )$ may lie on the line $x + y = 4$ (5) For all possible points $P ( x , y )$, the maximum value of $x - y$ is $1 + 2 \sqrt { 2 }$
Let $b$ and $c$ be real numbers. The quadratic equation $x ^ { 2 } + b x + c = 0$ has real roots, but the quadratic equation $x ^ { 2 } + ( b + 2 ) x + c = 0$ has no real roots. Select the correct options. (1) $c < 0$ (2) $b < 0$ (3) $x ^ { 2 } + ( b + 1 ) x + c = 0$ has real roots (4) $x ^ { 2 } + ( b + 2 ) x - c = 0$ has real roots (5) $x ^ { 2 } + ( b - 2 ) x + c = 0$ has real roots
Let $\Gamma$ be the function graph of $y = \sin \pi x$ for $0 \leq x \leq 3$. A horizontal line $L : y = k$ intersects $\Gamma$ at three points $P \left( x _ { 1 } , k \right) , Q \left( x _ { 2 } , k \right) , R \left( x _ { 3 } , k \right)$ satisfying $x _ { 1 } < x _ { 2 } < 1 < x _ { 3 }$. Select the correct options. (1) $k > 0$ (2) $L$ and $\Gamma$ have exactly 3 intersection points (3) $x _ { 1 } + x _ { 2 } < 1$ (4) If $2 \overline { P Q } = \overline { Q R }$, then $k = \frac { 1 } { 2 }$ (5) The sum of $x$-coordinates of all intersection points of $L$ and $\Gamma$ is greater than 5
In $\triangle A B C$, $\overline { A B } = 6 , \overline { A C } = 5 , \overline { B C } = 4$. Let $D$ be the midpoint of $\overline { A B }$, and $P$ be the intersection of the angle bisector of $\angle A B C$ and $\overline { C D }$, as shown in the figure. Select the correct options. (1) $\overline { C P } = \frac { 3 } { 7 } \overline { C D }$ (2) $\overrightarrow { A P } = \frac { 3 } { 7 } \overrightarrow { A B } + \frac { 2 } { 7 } \overrightarrow { A C }$ (3) $\cos \angle B A C = \frac { 3 } { 4 }$ (4) The area of $\triangle A C P$ is $\frac { 15 } { 14 } \sqrt { 7 }$ (5) (Dot product) $\overrightarrow { A P } \cdot \overrightarrow { A C } = \frac { 120 } { 7 }$
A certain alloy is composed of two metals, A and B. A student wants to know the relationship between the metal ratio and the wavelength of the alloy. He conducted an experiment measuring ``the wavelength $y$ (in nanometers) of an alloy with A comprising $x\%$'' and plotted 20 data points $(x_k, y_k)$, $k = 1, \cdots, 20$, on the $xy$ plane. The regression line (best-fit line) is $y = 21.3 x - 40$. To comply with submission standards, the report must be described as ``the wavelength $v$ (in micrometers) of an alloy with B comprising $u\%$''. He converted the data $(x_k, y_k)$ to $(u_k, v_k)$, $k = 1, \cdots, 20$, and obtained the regression line on the $uv$ plane as $v = a u + b$. Given that 1 nanometer $= 10 ^ { - 9 }$ meter and 1 micrometer $= 10 ^ { - 6 }$ meter. Select the correct options. (1) $u _ { k } = 100 - x _ { k } , k = 1 , \cdots , 20$ (2) $v _ { k } = 1000 y _ { k } , k = 1 , \cdots , 20$ (3) The standard deviation of $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots , u _ { 20 }$ equals the standard deviation of $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots , x _ { 20 }$ (4) $b = 2.09$ (5) The student found another data point $(u _ { 21 } , v _ { 21 })$ satisfying $v _ { 21 } = a u _ { 21 } + b$; if these 21 data points $(u_k, v_k)$, $k = 1, \cdots, 21$, are plotted on the $uv$ plane, the regression line is still $v = a u + b$
A real-coefficient cubic polynomial $f ( x )$ divided by $x + 6$ gives quotient $q ( x )$ and remainder 3. If $q ( x )$ has a maximum value of 8 at $x = - 6$, then the coordinates of the center of symmetry of the graph $y = f ( x )$ are ((13-1) (13-2), (13-3)).
In coordinate space, point $A$ has coordinates $(a, b, c)$, where $a, b, c$ are all negative real numbers. Point $A$ is at distance 6 from each of the three planes $E _ { 1 } : 4 y + 3 z = 2$, $E _ { 2 } : 3 y + 4 z = - 5$, and $E _ { 3 } : x + 2 y + 2 z = - 2$. Then $a + b + c =$ (14-1) (14-2) (14-3).
A holiday market stall offers ``test your luck—cute dolls regularly priced at 480 yuan can be purchased for as low as 240 yuan''. The rules are: customers flip a fair coin up to 5 times. If 3 consecutive heads are obtained in the first 3 flips, they can purchase a doll for 240 yuan. If 3 heads are accumulated by the 4th flip, they can purchase for 320 yuan. If 3 heads are accumulated by the 5th flip, they can purchase for 400 yuan. If 3 heads are not accumulated after 5 flips, they can purchase for 480 yuan. The expected value of the amount a customer spends to purchase a doll is (15-1) (15-2) (15-3) yuan.
On the coordinate plane, let $L _ { 1 }$ and $L _ { 2 }$ be two lines passing through point $(3, 1)$ with slopes $m$ and $- m$ respectively, where $m$ is a real number. Let $\Gamma$ be a circle with center at the origin. Given that $\Gamma$ intersects $L _ { 1 }$ at two distinct points $A$ and $B$, and the distance from the center to $L _ { 1 }$ is 1, and $\Gamma$ is tangent to $L _ { 2 }$, then the length of chord $\overline { A B }$ is (express as a fraction in lowest terms).
In $\triangle A B C$, $\overline { A B } = \overline { B C } = 3$ and $\cos \angle A B C = - \frac { 1 } { 8 }$. On the circumcircle of $\triangle A B C$ there is a point $D$ satisfying $\overline { B D } = 4$ and $\overline { A D } \leq \overline { C D }$. Then $\overline { C D } = $ (17-1) $+$ $\sqrt{\text{(17-2)}}$. (Express as a simplified radical form.)
Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers. Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$. What is the value of $c$? (Single choice question, 3 points) (1) 0 (2) $- 1$ (3) 1 (4) $- \frac { 1 } { 2 }$ (5) $\frac { 1 } { 2 }$
Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers. Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$. Find the coordinates of point $Q$ and the angle between $\overrightarrow { O R }$ and the vector $(1, 0)$. (Non-multiple choice question, 6 points)
Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers. Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$. Let $L$ be the line passing through point $P$ and parallel to line $OQ$. Let point $S$ be the intersection of $L$ and line $OR$. Find $\angle O S P$ and the coordinates of point $S$. (Non-multiple choice question, 6 points)