The values of $\alpha$, for which $$2 \alpha + 3 \quad 3 \alpha + 1 \quad 0$$ (1) ( - 2, 1)
(2) ( - 3, 0)
(3) $- \frac { 3 } { 2 } , \frac { 3 } { 2 }$
(4) $( 0,3 )$

Let the points $\left( \frac{11}{2}, \alpha \right)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to:
(1) 44
(2) 22
(3) 33
(4) 55

$x | x + 4 | + 3 | x + 2 | + 10 = 0$ No. of real soln.

Consider the following equation in $x$
$$|ax - 11| = 4x - 10, \tag{1}$$
where $a$ is a constant.
(1) Equation (1) can be rewritten without using the absolute value symbol as
$$\begin{aligned} & \text{when } ax \geqq 11, \quad \text{then } (a - \mathbf{N})x = \mathbf{O}; \\ & \text{when } ax < 11, \quad \text{then } (a + \mathbf{P})x = \mathbf{QR}. \end{aligned}$$
(2) When $a = \sqrt{7}$, the solution of equation (1) is
$$x = \frac{\mathrm{S}}{\mathrm{T}} - \sqrt{\mathrm{U}}.$$
(3) Let $a$ be a positive integer. When equation (1) has a positive integral solution, we have $a = \mathbf{W}$, and that solution $x = \mathbf{X}$.

Consider the following equation in $x$
$$|ax - 11| = 4x - 10, \tag{1}$$
where $a$ is a constant.
(1) Equation (1) can be rewritten without using the absolute value symbol as
$$\begin{aligned} & \text{when } ax \geqq 11, \quad \text{then } (a - \mathbf{N})x = \mathbf{O}; \\ & \text{when } ax < 11, \quad \text{then } (a + \mathbf{P})x = \mathbf{QR}. \end{aligned}$$
(2) When $a = \sqrt{7}$, the solution of equation (1) is
$$x = \frac{\mathrm{S}\left(\frac{\mathrm{T}}{\mathrm{V}} - \sqrt{\mathrm{U}}\right)}{\mathrm{V}}.$$
(3) Let $a$ be a positive integer. When equation (1) has a positive integral solution, we have $a = \mathbf{W}$, and that solution $x = \mathbf{X}$.

Let $a , b , c$ and $d$ be real numbers satisfying $a < b < c < d$. Suppose that the two subsets of real numbers
$$A = \{ x \mid a \leqq x \leqq c \} , \quad B = \{ x \mid b \leqq x \leqq d \}$$
satisfy
$$A \cap B = \left\{ x \mid x ^ { 2 } - 4 x + 3 \leqq 0 \right\} .$$
Then, answer the questions for cases (1) and (2).
(1) Let the union of $A$ and $B$ be
$$A \cup B = \left\{ x \mid x ^ { 2 } - 5 x - 24 \leqq 0 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { \text { NO } } , \quad b = \mathbf { P } , \quad c = \mathbf { Q } , \quad d = \mathbf { Q } .$$
(2) Let the intersection of $A$ and the complement $\bar { B }$ of $B$ be
$$A \cap \bar { B } = \left\{ x \mid x ^ { 2 } + 5 x - 6 \leqq 0 \text { and } x \neq 1 \right\} ,$$
and let the intersection of the complement $\bar { A }$ of $A$ and $B$ be
$$\bar { A } \cap B = \left\{ x \mid x ^ { 2 } - 9 x + 18 \leqq 0 \text { and } x \neq 3 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { S T } , \quad b = \mathbf { U } , \quad c = \mathbf { V } , \quad d = \mathbf { W } .$$

Let $a , b , c$ and $d$ be real numbers satisfying $a < b < c < d$. Suppose that the two subsets of real numbers
$$A = \{ x \mid a \leqq x \leqq c \} , \quad B = \{ x \mid b \leqq x \leqq d \}$$
satisfy
$$A \cap B = \left\{ x \mid x ^ { 2 } - 4 x + 3 \leqq 0 \right\} .$$
Then, answer the questions for cases (1) and (2).
(1) Let the union of $A$ and $B$ be
$$A \cup B = \left\{ x \mid x ^ { 2 } - 5 x - 24 \leqq 0 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { \text { NO } } , \quad b = \mathbf { P } , \quad c = \mathbf { Q } , \quad d = \mathbf { Q } .$$
(2) Let the intersection of $A$ and the complement $\bar { B }$ of $B$ be
$$A \cap \bar { B } = \left\{ x \mid x ^ { 2 } + 5 x - 6 \leqq 0 \text { and } x \neq 1 \right\} ,$$
and let the intersection of the complement $\bar { A }$ of $A$ and $B$ be
$$\bar { A } \cap B = \left\{ x \mid x ^ { 2 } - 9 x + 18 \leqq 0 \text { and } x \neq 3 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { S T } , \quad b = \mathbf { U } , \quad c = \mathbf { V } , \quad d = \mathbf { W } .$$

For $\mathbf { H } , \mathbf { I }$ in question (1), and for $\mathbf { J }$, $\mathbf { K }$ in question (2), choose the appropriate answer from among (0) $\sim$ (3) at the bottom of this page.
For $\mathbf { L } \sim \mathbf { R }$ in question (3), enter the appropriate number.
Consider the following three possible conditions on two real numbers $x$ and $y$:
$p : x$ and $y$ satisfy the equation $( x + y ) ^ { 2 } = a \left( x ^ { 2 } + y ^ { 2 } \right) + b x y$, where $a$ and $b$ are real constants.
$$\begin{aligned} & q : x = 0 \text { and } y = 0 . \\ & r : x = 0 \text { or } y = 0 . \end{aligned}$$
(1) Suppose that in condition $p$, $a = b = 1$. Then $p$ is $\mathbf { H }$ for $q$, and $p$ is $\mathbf { I }$ for $r$.
(2) Suppose that in condition $p$, $a = b = 2$. Then $p$ is $\mathbf { J }$ for $q$, and $p$ is $\mathbf { K }$ for $r$.
(3) If in condition $p$ we set $a = 2$, we can transform the equation in $p$ into
$$\left( x + \frac { b - \mathbf { L } } { \mathbf { L } } y \right) ^ { 2 } + \left( \mathbf { L } - \frac { ( b - \mathbf { Q} ) ^ { 2 } } { \mathbf { L } } \right) y ^ { 2 } = 0 .$$
Hence $p$ is a necessary and sufficient condition for $q$ if and only if $b$ satisfies
$$\mathbf { Q } < b < \mathbf { R } .$$
(0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition

Let $a$ be a constant. Consider the quadratic inequality
$$x ^ { 2 } - 2 ( a + 2 ) x + 25 > 0 . \tag{1}$$
The left-hand side of inequality (1) can be transformed into
$$( x - a - \mathbf { A } ) ^ { 2 } - a ^ { 2 } - \mathbf { B } a + \mathbf { C D } .$$
Hence, we have the following results.
(1) The condition under which inequality (1) holds for all real numbers $x$ is
$$\mathbf { E F } < a < \mathbf { G } .$$
(2) The condition under which inequality (1) holds for all real numbers $x$ satisfying $x \geqq - 1$ is
$$\mathbf { H I J } < a < \mathbf { K } .$$

For each of $\mathbf{A} \sim \mathbf{D}$ in the following questions, choose the correct answer from among (0) $\sim$ (5) below each question.
Consider the three quadratic inequalities
$$\begin{aligned} x ^ { 2 } + 3 x - 18 & < 0 \tag{1}\\ x ^ { 2 } - 2 x - 8 & > 0 \tag{2}\\ x ^ { 2 } + a x + b & < 0 . \tag{3} \end{aligned}$$
(1) The range of $x$ which satisfies both of the inequalities (1) and (2) is $\mathbf { A }$. Also, the range of $x$ which satisfies neither inequality (1) nor (2) is $\mathbf{B}$. (0) $3 \leqq x \leqq 4$
(1) $- 6 \leqq x \leqq - 2$
(2) $3 < x < 4$
(3) $2 < x < 6$
(4) $- 6 < x < - 2$
(5) $- 4 \leqq x \leqq - 3$
(2) The range of $x$ that satisfies at least one of the inequalities (1) and (3) will be $- 6 < x < 7$, if and only if $a$ and $b$ satisfy the equation $\square \mathbf{C}$, and $a$ satisfies the inequality $\square \mathbf{D}$. (0) $b = 6 a - 36$
(1) $b = 7 a - 49$
(2) $b = - 7 a - 49$
(3) $- 10 < a \leqq - 3$
(4) $- 10 < a \leqq - 1$
(5) $- 1 \leqq a < 3$

For each of $\mathbf { A } \sim \mathbf { M }$ in the following statements, choose the correct answer from among (0) ~ (9) at the bottom of this page.
We are to solve the following simultaneous inequalities
$$\left\{ \begin{aligned} x ^ { 2 } - 2 x < 3 & \cdots \cdots \cdots (1)\\ a x ^ { 2 } - a x - x + 1 > 0 , & \cdots \cdots \cdots (2) \end{aligned} \right.$$
where $0 < a < 1$.
When we solve (1), we have
$$\mathbf { A } < x < \mathbf { B } .$$
Next, when we transform (2), we have
$$( a x - \mathbf { C } ) ( x - \mathbf { D } ) > 0 .$$
Hence, noting that $0 < a < 1$, we see that the solution of (2) is
$$x < \mathbf { E } \text { or } \mathbf { F } < x .$$
Thus, when $0 < a \leqq \mathbf { G }$, the solution of the simultaneous inequalities is
$$\mathbf { H } < x < \mathbf { I }$$
and when $\mathbf { G } < a < 1$, the solution is
$$\mathbf { J } < x < \mathbf { K } \text { or } \mathbf { L } < x < \mathbf { M }$$
where $\mathbf { K } < \mathbf { M }$.
(0) 0
(1) 1
(2) 2
(3) 3
(4) $- 1$
(5) $\frac { 1 } { 2 }$ (6) $\frac { 1 } { 3 }$ (7) $\frac { 1 } { a }$ (8) $\frac { 2 } { a }$ (9) $\frac { 3 } { a }$

Consider the two functions
$$\begin{aligned} & f ( x ) = x ^ { 2 } + 2 a x + 4 a - 3 \\ & g ( x ) = 2 x + 1 \end{aligned}$$
We are to find the condition on $a$ for which $f ( x ) \geqq g ( x )$ for all $x$ and also find the range of values of the minimum of $f ( x )$ under this condition.
We must find the condition under which
$$x ^ { 2 } + \mathbf { A } ( a - \mathbf { A } ) x + \mathbf { A C } a - \mathbf { A D } \geq 0$$
for all $x$.
For each of $\mathbf { E } \sim \mathbf{ H }$ in the following questions, choose the correct answer from among (0) $\sim$ (7) below each question.
(1) The required condition is that $a$ satisfy the quadratic inequality $\mathbf{E}$. Hence $a$ is in the range $\mathbf { F }$. (0) $a ^ { 2 } - 5 a + 4 \geqq 0$
(1) $a ^ { 2 } - 6 a + 5 \geqq 0$
(2) $a ^ { 2 } - 5 a + 4 \leqq 0$
(3) $a ^ { 2 } - 6 a + 5 \leqq 0$
(4) $a \leqq 1$ or $5 \leqq a$
(5) $1 \leqq a \leqq 5$ (6) $1 \leqq a \leqq 4$ (7) $a \leqq 1$ or $4 \leqq a$
(2) Let $m$ be the minimum value of $f ( x )$. Then, since $m = \mathbf { G }$, the range of values which $m$ can take under the condition in (1) is $\mathbf { H }$. (0) $a ^ { 2 } + 4 a - 3$
(1) $4 a ^ { 2 } + 4 a - 3$
(2) $- a ^ { 2 } + 4 a - 3$
(3) $2 a ^ { 2 } - 4 a + 3$
(4) $- 5 \leqq m \leqq 1$
(5) $- 8 \leqq m \leqq 1$ (6) $- 8 \leqq m \leqq - 1$ (7) $- 5 \leqq m \leqq - 1$

Q1 Let $a = \sqrt { 5 } + \sqrt { 3 }$ and $b = \sqrt { 5 } - \sqrt { 3 }$. We are to find the integers $x$ satisfying
$$2 \left| x - \frac { a } { b } \right| + x < 10$$
(1) We see that $\frac { a } { b } = \mathbf { A } + \sqrt { \mathbf { BC } }$. Hence the largest integer less than $\frac { a } { b }$ is $\mathbf{D}$.
(2) For $\mathbf { F }$ and $\mathbf { H }$ in the following sentence, choose the correct answer from among choices (0) $\sim$ (7) below, and for $\mathbf { E }$ and $\mathbf { G }$, enter the correct numbers.
When $x$ is an integer, the left side of the inequality can be expressed without using the absolute value symbol as follows:
$$\left\{ \begin{array} { l } \text { if } x \leqq \mathbf { E } , \text { then } 2 \left| x - \frac { a } { b } \right| + x = \mathbf { F } , \\ \text { if } x \geqq \mathbf { G } , \text { then } 2 \left| x - \frac { a } { b } \right| + x = \mathbf { H } . \end{array} \right.$$
(0) $x - 6 - 2 \sqrt { 10 }$
(1) $x + 8 + 2 \sqrt { 15 }$
(2) $- x + 8 + 2 \sqrt { 15 }$
(3) $- x + 6 + 2 \sqrt { 10 }$
(4) $3 x - 6 - 2 \sqrt { 10 }$
(5) $3 x - 8 - 2 \sqrt { 15 }$ (6) $- 3 x + 8 + 2 \sqrt { 15 }$ (7) $- 3 x + 6 + 2 \sqrt { 10 }$
(3) Thus, the integers $x$ satisfying inequality $2 \left| x - \frac { a } { b } \right| + x < 10$ are those greater than or equal to $\mathbf { I }$ and less than or equal to $\mathbf { J }$.

(a) Show that $x ^ { 2 } + 4 x + 4 \geq 0$ for all values of $x$.\ (b) For which values of the constant $a$ is there at least one solution, $x$, of the inequality
$$a x ^ { 2 } + 4 x + 3 \leq x ^ { 2 } - 1 ?$$
(c) Suppose that $1 < a \leq 2$. Find all values of $x$ for which the inequality in (b) holds.

On the coordinate plane, there is a polygonal region $\Gamma$ (including boundary) as shown in the figure. If $k > 0$ , the line $7 x + 2 y = k$ and the two coordinate axes form a triangular region such that the polygonal region $\Gamma$ lies within this triangular region (including boundary), then the minimum positive real number $k =$ (7)(8).

On a number line, there is the origin $O$ and three points $A ( - 2 ) , B ( 10 ) , C ( x )$, where $x$ is a real number. Given that the lengths of segments $\overline { B C } , \overline { A C } , \overline { O B }$ satisfy $\overline { B C } < \overline { A C } < \overline { O B }$, then the maximum range of $x$ is (8) $< x <$ (9).

How many integers $x$ satisfy $2|x| + x < 10$?
(1) 13
(2) 14
(3) 15
(4) 16
(5) Infinitely many

Let integer $n$ satisfy $| 5 n - 21 | \geq 7 | n |$ . Select the correct options.
(1) $| 5 n - 7 n | \geq 21$
(2) $- 1 \leq \frac { 7 n } { 5 n - 21 } \leq 1$
(3) $7 n \leq 5 n - 21$
(4) $( 5 n - 21 ) ^ { 2 } \geq 49 n ^ { 2 }$
(5) There are infinitely many integers $n$ satisfying the given inequality

Let $a , b$ be real numbers. It is known that the four numbers $- 3 , - 1, 4, 7$ all satisfy the inequality $| x - a | \leq b$ in $x$. Select the correct options.
(1) $\sqrt { 10 }$ also satisfies the inequality $| x - a | \leq b$ in $x$
(2) $3, 1 , - 4 , - 7$ satisfy the inequality $| x + a | \leq b$ in $x$
(3) $- \frac { 3 } { 2 } , - \frac { 1 } { 2 } , 2 , \frac { 7 } { 2 }$ satisfy the inequality $| x - a | \leq \frac { b } { 2 }$ in $x$
(4) $b$ could equal 4
(5) $a$ and $b$ could be equal

A point $P$ on a number line satisfies the condition that the distance from $P$ to 1 plus the distance from $P$ to 4 equals 4. How many such points $P$ are there?
(1) 0
(2) 1
(3) 2
(4) 3
(5) Infinitely many

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) $(16,8)$

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 $y$.

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.

4. The complete set of values of $x$ for which $\left( x ^ { 2 } - 1 \right) ( x - 2 ) > 0$ is
A $x < - 1,1 < x < 2$
B $x < - 1 , x > 2$
C $- 1 < x < 2$
D $x < 1 , x > 2$
E $\quad - 1 < x < 1 , x > 2$

15. For any real numbers $a , b$, and $c$ where $a \geq b$, consider these three statements:
$$\begin{array} { l l } 1 & - b \geq - a \\ 2 & a ^ { 2 } + b ^ { 2 } \geq 2 a b \\ 3 & a c \geq b c \end{array}$$
Which of the statements 1,2 , and 3 must be true?
A none
B 1 only
C 2 only
D 3 only
E 1 and 2 only F 1 and 3 only G 2 and 3 only H 1,2 and 3