Let $\mathrm { A } = \{ ( x , y ) \in \mathbf { R } \times \mathbf { R } : | x + y | \geqslant 3 \}$ and $\mathrm { B } = \{ ( x , y ) \in \mathbf { R } \times \mathbf { R } : | x | + | y | \leq 3 \}$. If $\mathrm { C } = \{ ( x , y ) \in \mathrm { A } \cap \mathrm { B } : x = 0$ or $y = 0 \}$, then $\sum _ { ( x , y ) \in \mathrm { C } } | x + y |$ is : (1) 15 (2) 24 (3) 18 (4) 12
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 any $x$ and $y$ satisfying $x > 0$ and $y > 0$, let $m$ be the smallest value among $\frac { y } { x } , x$ and $\frac { 8 } { y }$. Also, let $A$ be the set of points $( x , y )$ where $m = \frac { y } { x }$, and let $B$ be the set of points $( x , y )$ where $m = \frac { 8 } { y }$. (1) For $\mathbf { M } \sim$ S in the following sentence, choose the correct answer from among choices (0) $\sim$ (7) below. $A$ and $B$ can be expressed as follows: $$\begin{aligned}
& A = \{ ( x , y ) \mid \mathbf { M } \leqq \mathbf { N } , \quad \mathbf { O } \leqq 8\mathbf { P } \} \\
& B = \{ ( x , y ) \mid 8 \mathbf { Q } \leqq \mathbf { R } , \quad 8 \leqq \mathbf { S } \} .
\end{aligned}$$ (0) $x$ (1) $y$ (2) $x + y$ (3) $x - y$ (4) $x ^ { 2 }$ (5) $x y$ (6) $y ^ { 2 }$ (7) $x ^ { 2 } + y ^ { 2 }$ (2) For $\mathbf { T }$ and $\mathbf { U }$ in the following sentence, choose the correct answer from among choices (0) $\sim$ (8). When sets $A$ and $B$ are indicated on the $xy$-plane, $A$ is the shaded portion of graph $\mathbf{T}$ and $B$ is the shaded portion of graph $\mathbf{U}$. Note that the $x$ and $y$ axes are not included in the shaded portions. (3) We are to find the maximum value of $m$ when a point $\mathrm { P } ( x , y )$ moves within $A \cup B$. When $\mathrm { P } ( x , y ) \in A$, since $y = m x$, we need to find the point P which maximizes the slope of the straight line passing through the origin O and P. Also, when $\mathrm { P } ( x , y ) \in B$, since $m = \frac { 8 } { y }$, we need to find the point P at which the $y$ coordinate of P is minimized. From the above, at $( x , y ) = ( \mathbf { V } , \mathbf { W } ) , m$ takes the maximum value $\mathbf { X }$.