Given the system $\left\{ \begin{array} { c } ( a + 1 ) x + 4 y = 0 \\ ( a - 1 ) y + z = 3 \\ 4 x + 2 a y + z = 3 \end{array} \right.$, find:\ a) ( 1.25 points) Discuss it as a function of the parameter $a$.\ b) ( 0.5 points) Solve it for $a = 3$.\ c) (0.75 points) Solve it for $a = 5$.

Given the system $\left\{ \begin{array} { l } - 2 x + y + k z = 1 \\ k x - y - z = 0 \\ - y + ( k - 1 ) z = 3 \end{array} \right.$, find:\ a) ( 1.25 points) Discuss it as a function of the parameter $k$.\ b) ( 0.5 points) Solve it for $k = 3$.\ c) ( 0.75 points) Solve it for $k = 3 / 2$ and specify, if possible, a particular solution with $x = 2$.

There are containers of three different sizes to fill a cistern. With six small containers and 2 L we exactly fill one medium container and one large one. With two large containers we fill two medium ones, one small one, and 1 L remains. The cistern is completely filled either with fourteen small containers plus six medium ones, or with five medium ones together with five large ones. It is requested to calculate the capacity of each type of container and, once known, that of the cistern.

10. Let $a, b, c$ be real numbers. Which of the following statements about the linear system $\left\{\begin{array}{c} x + 2y + az = 1 \\ 3x + 4y + bz = -1 \\ 2x + 10y + 7z = c \end{array}\right.$ are correct?
(1) If the linear system has a solution, then it must have exactly one solution
(2) If the linear system has a solution, then $11a - 3b \neq 7$
(3) If the linear system has a solution, then $c = 14$
(4) If the linear system has no solution, then $11a - 3b = 7$
(5) If the linear system has no solution, then $c \neq 14$

In coordinate space, let $O$ be the origin, and let point $P$ be the intersection of three planes $x - 3y - 5z = 0$, $x - 3y + 2z = 0$, $x + y = t$, where $t > 0$. If $\overline{OP} = 10$, then $t =$ （9）（10）（11）. (Express as a simplified radical)

Let $c$ be a real number such that the system of linear equations $\left\{ \begin{array}{c} x - y + z = 0 \\ 2x + cy + 3z = 1 \\ 3x - 3y + cz = 0 \end{array} \right.$ has no solution. Select the value of $c$.
(1) $-3$
(2) $-2$
(3) $0$
(4) $2$
(5) $3$

Let $a , b$ be real numbers, and the system of equations $\left\{ \begin{array} { l } a x + 5 y + 12 z = 4 \\ x + a y + \frac { 8 } { 3 } z = 7 \\ 3 x + 8 y + a z = 1 \end{array} \right.$ has exactly one solution. After a series of Gaussian elimination operations, the original augmented matrix can be transformed to $\left[ \begin{array} { c c c | c } 1 & 2 & b & 7 \\ 0 & b & 5 & - 5 \\ 0 & 0 & b & 0 \end{array} \right]$ . Then $a = ( 14 - 1 ) , b = \frac { ( 14 - 2 ) } { ( 14 - 3 ) }$ . (Express as a fraction in lowest terms)

A newly opened juice shop offers three types of beverages: juice, milk tea, and coffee. The sales volume (in cups) and total revenue (in yuan) for each type of beverage over the first 3 days are shown in the table below. For example, on the first day, the sales of juice, milk tea, and coffee were 60 cups, 80 cups, and 50 cups respectively, with total revenue of 12,900 yuan.
It is known that the price of each type of beverage is the same every day. Then the price per cup of coffee is

	Juice (cups)	Milk Tea (cups)	Coffee (cups)	Total Revenue (yuan)
Day 1	60	80	50	12900
Day 2	30	40	30	6850
Day 3	50	70	40	10800

Let $a, b, c, d$ be real numbers. It is known that the augmented matrices of two systems of linear equations $\left\{\begin{array}{l} ax + by = 2 \\ cx + dy = 1 \end{array}\right.$ and $\left\{\begin{array}{l} ax + by = -1 \\ cx + dy = -1 \end{array}\right.$, after the same row operations, become $\left[\begin{array}{cc|c} 1 & -1 & 3 \\ 0 & 1 & 2 \end{array}\right]$ and $\left[\begin{array}{cc|c} 1 & -1 & 2 \\ 0 & 1 & -1 \end{array}\right]$ respectively. Then the solution to the system of equations $\left\{\begin{array}{l} ax + by = 0 \\ cx + dy = 1 \end{array}\right.$ is $x = $ 9-1, 9-2, $y = $ 9-3.

Consider the system of linear equations in two variables $\left\{\begin{array}{c} ax + 6y = 6 \\ x + by = 1 \end{array}\right.$, where the coefficients $a, b$ are determined by rolling a fair die and flipping a fair coin respectively. Let $a$ be the number of points shown on the die; if the coin shows heads, $b = 1$; if the coin shows tails, $b = 2$. Select the correct options.
(1) The probability of rolling $a = b$ is $\frac{1}{3}$
(2) The probability that the system has no solution is $\frac{1}{12}$
(3) The probability that the system has a unique solution is $\frac{5}{6}$
(4) The probability that the coin shows tails and the system has a solution is $\frac{1}{2}$
(5) Given that the coin shows tails and the system has a solution, the probability that $x$ is positive is $\frac{2}{5}$

$$\frac { 1 } { 2 } - 3 a = \frac { 1 } { 8 } + 3 b$$
Given this, what is the sum $\mathbf { a } + \mathbf { b }$?
A) $\frac { 3 } { 4 }$
B) $\frac { 5 } { 6 }$
C) $\frac { 1 } { 8 }$
D) $\frac { 5 } { 8 }$
E) $\frac { 4 } { 9 }$

$$\begin{aligned} & x ^ { 3 } - 2 y = 7 \\ & x ^ { 4 } - 2 x y = 21 \end{aligned}$$
Given this, what is $\mathbf { x }$?
A) 3
B) 5
C) 7
D) 9
E) 11

Let $\mathrm { a } , \mathrm { b } , \mathrm { x }$ and y be positive numbers such that
$$\begin{aligned} & \frac { x } { a } \cdot \frac { b } { y } = 2 \\ & \frac { a ^ { 2 } } { x ^ { 2 } } + \frac { b ^ { 2 } } { y ^ { 2 } } = 20 \end{aligned}$$
Given this, which of the following is the value of x in terms of a?
A) $\frac { a } { 2 }$
B) $\frac { 3 a } { 4 }$
C) $\frac { 3 a } { 5 }$
D) $\frac { 4 a } { 5 }$
E) $\frac { 5 a } { 6 }$

$$\begin{array}{r} 2x + 2y - z = 1 \\ x + y + z = 2 \\ y - z = 1 \end{array}$$
In the solution of the system of equations above, what is $x$?
A) $-3$
B) $-2$
C) $-1$
D) $0$
E) $3$

The vertices of a parallelogram with diagonals $[ AC ]$ and $[ BD ]$ are $\mathrm { A } ( 3,1 ) , \mathrm { B } ( 5,3 ) , \mathrm { C } ( 2,5 )$ and $\mathrm { D } ( \mathrm { a } , \mathrm { b } )$. What is the length of diagonal $[ BD ]$ in units?
A) 1
B) 2
C) 3
D) 4
E) 5

Let x and y be real numbers such that
$$\begin{aligned} & x ^ { 3 } - 3 x ^ { 2 } y = 3 \\ & y ^ { 3 } - 3 x y ^ { 2 } = 11 \end{aligned}$$
Accordingly, what is the difference $x - y$?
A) 3
B) 2
C) 1
D) - 2
E) - 3

$\frac{a - 1}{b} = \frac{c}{a}$
$$\frac{a}{c - 2} = \frac{b + 3}{a - 1}$$
Given that, what is the value of the expression $3c - 2b$?
A) 8 B) 9 C) 6 D) 3 E) 4

For distinct numbers a and b
$$\frac{a^{2}}{b} - \frac{b^{2}}{a} = b - a$$
Given that, what is the value of the expression $\frac{a}{b} + \frac{b}{a}$?
A) $-2$ B) $-1$ C) 0 D) 1 E) 4

$$\frac { a - 1 } { a - 3 } = \frac { a - 5 } { a - 4 }$$
Given that, what is a?
A) $\frac { 8 } { 5 }$
B) $\frac { 13 } { 4 }$
C) $\frac { 9 } { 4 }$

Let x and y be real numbers such that
$$\begin{aligned} & x ^ { 2 } - 4 y = - 7 \\ & y ^ { 2 } - 2 x = 2 \end{aligned}$$
Given this, what is the sum $x + y$?
A) 3
B) 4
C) 5
D) $\frac { 4 } { 3 }$
E) $\frac { 5 } { 3 }$

For real numbers $x , y$ and $z$
$$\begin{aligned} & x \cdot y = 14 \\ & x \cdot z = 20 \\ & 3x + 2y + z = 24 \end{aligned}$$
Given that, what is x?
A) $\frac { 8 } { 3 }$
B) $\frac { 14 } { 5 }$
C) 3

$x , y$ are positive real numbers and
$$\frac { 2 y } { x + \frac { 1 } { y } } - \frac { 3 x } { y + \frac { 1 } { x } } = \frac { 5 x ^ { 2 } } { x \cdot y + 1 }$$
Given this, what is the ratio $\frac { x } { y }$?
A) $\frac { 2 } { 5 }$
B) $\frac { 1 } { 5 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 1 } { 3 }$
E) $\frac { 1 } { 2 }$

For integers $\mathrm { x } , \mathrm { y }$ and z
$$2 x = 3 y = 5 z$$
Given this, which of the possible values of the sum $x + y + z$ is closest to 100?
A) 93
B) 96
C) 98
D) 103
E) 105

In a laboratory, the following are known about a drug experiment conducted on male and female guinea pig mice.

• Male mice were given 1 tablet of medicine every 12 hours, and female mice were given 1 tablet every 8 hours.
• Male mice were given 0.5 gram tablets, and female mice were given 1 gram tablets.
• A total of 85 grams of medicine was given to these mice in one day, in the form of 95 tablets.

Accordingly, how many mice were used in the experiment?
A) 20
B) 25
C) 30
D) 35
E) 40

Stationery materials are to be distributed to students in a class. Enough materials are brought to the class so that each of the 36 students receives one pencil, one pencil sharpener, and one eraser. However, on the distribution day, since some students are absent, each student present in the class is given 3 pencils, 2 pencil sharpeners, and 1 eraser.
If a total of 42 items of these materials remain after distribution, how many pencil sharpeners are left?
A) 10
B) 11
C) 12
D) 13
E) 14