1-1. If $A$ and $B$ are two non-empty sets with the condition $A \subset B$, then which of the following relations is incorrect? (1) $B - A' = A$ (2) $A - B' = A$ (3) $A \cap B' = \phi$ (4) $B \cap A' = \phi$
1-2. The set $\Big(A - B\Big) \cup \Big((B \cap C)' \cap \big((B' \cup A) - B\big)\Big)$ is equal to which set? (1) $A \cup B'$ (2) $A \cap B'$ (3) $A$ (4) $B'$
1-3. In the sets of four elements $\Lambda = \{x+2, 1, 4, y\}$ and $B = \{5, 7, z, t-1\}$, suppose $A \times B = B \times A$. The number of sets in the form $\{(x,y), (z,t)\}$ is how many? (1) $2$ (2) $3$ (3) $4$ (4) $6$
1-4. When we divide the polynomial $P(x)$ by $x-1$ and $2x+1$ respectively, the remainders are $8$ and $5$. When we divide $P(x)$ by $2x^2 - x - 1$, the remainder is: (1) $-x+4$ (2) $x+2$ (3) $7x+6$ (4) $7x-3$
1-5. Which statement about the equation $f(x) = x^2 - x^{\frac{1}{2}} = 0$ is correct? (1) The equation has two roots in the interval $[1, \infty)$. (2) The equation has no roots in the interval $[1, \infty)$. (3) The equation has one root in the interval $[1, \infty)$. (4) The equation has at most one root in the interval $[1, \infty)$.
1-6. The area of the region bounded by the graphs of the two functions $y = \sqrt{x^2 - 4x + 4} + 2$ and $y = \dfrac{1}{2}x + 2$ is: (1) $8$ (2) $9$ (3) $10$ (4) $12$
1-7. If $f(x) = x + \sqrt{x}$ and $g(x) = \dfrac{9x+6}{1-x}$, and $(g^{-1} \circ f^{-1})(20)$ is defined, what is its value? (1) $\dfrac{2}{5}$ (2) $\dfrac{3}{5}$ (3) $\dfrac{2}{3}$ (4) $\dfrac{3}{4}$
1-8. The graph of the function $f(x) = \sqrt{x}$ is reflected about the $y$-axis, then the resulting curve is shifted 4 units to the right. The asymptote and the main curve are symmetric with respect to which line? (1) $x = 1$ (2) $x = 1.5$ (3) $x = 2$ (4) $x = 2.5$ %% Page 4
\textbf{1-1.} If $A$ and $B$ are two non-empty sets with the condition $A \subset B$, then which of the following relations is \underline{incorrect}?
\medskip
(1) $B - A' = A$ \hfill (2) $A - B' = A$ \hfill (3) $A \cap B' = \phi$ \hfill (4) $B \cap A' = \phi$
\bigskip
\textbf{1-2.} The set $\Big(A - B\Big) \cup \Big((B \cap C)' \cap \big((B' \cup A) - B\big)\Big)$ is equal to which set?
\medskip
(1) $A \cup B'$ \hfill (2) $A \cap B'$ \hfill (3) $A$ \hfill (4) $B'$
\bigskip
\textbf{1-3.} In the sets of four elements $\Lambda = \{x+2, 1, 4, y\}$ and $B = \{5, 7, z, t-1\}$, suppose $A \times B = B \times A$. The number of sets in the form $\{(x,y), (z,t)\}$ is how many?
\medskip
(1) $2$ \hfill (2) $3$ \hfill (3) $4$ \hfill (4) $6$
\bigskip
\textbf{1-4.} When we divide the polynomial $P(x)$ by $x-1$ and $2x+1$ respectively, the remainders are $8$ and $5$. When we divide $P(x)$ by $2x^2 - x - 1$, the remainder is:
\medskip
(1) $-x+4$ \hfill (2) $x+2$ \hfill (3) $7x+6$ \hfill (4) $7x-3$
\bigskip
\textbf{1-5.} Which statement about the equation $f(x) = x^2 - x^{\frac{1}{2}} = 0$ is correct?
\medskip
(1) The equation has two roots in the interval $[1, \infty)$. \hfill (2) The equation has no roots in the interval $[1, \infty)$.
(3) The equation has one root in the interval $[1, \infty)$. \hfill (4) The equation has at most one root in the interval $[1, \infty)$.
\bigskip
\textbf{1-6.} The area of the region bounded by the graphs of the two functions $y = \sqrt{x^2 - 4x + 4} + 2$ and $y = \dfrac{1}{2}x + 2$ is:
\medskip
(1) $8$ \hfill (2) $9$ \hfill (3) $10$ \hfill (4) $12$
\bigskip
\textbf{1-7.} If $f(x) = x + \sqrt{x}$ and $g(x) = \dfrac{9x+6}{1-x}$, and $(g^{-1} \circ f^{-1})(20)$ is defined, what is its value?
\medskip
(1) $\dfrac{2}{5}$ \hfill (2) $\dfrac{3}{5}$ \hfill (3) $\dfrac{2}{3}$ \hfill (4) $\dfrac{3}{4}$
\bigskip
\textbf{1-8.} The graph of the function $f(x) = \sqrt{x}$ is reflected about the $y$-axis, then the resulting curve is shifted 4 units to the right. The asymptote and the main curve are symmetric with respect to which line?
\medskip
(1) $x = 1$ \hfill (2) $x = 1.5$ \hfill (3) $x = 2$ \hfill (4) $x = 2.5$
%% Page 4