csat-suneung 2009 Q12

csat-suneung · South-Korea · csat__math-humanities 4 marks Laws of Logarithms Verify Truth of Logarithmic Statements
Let the set $U$ be
$$U = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, a , b , c , d \text { are positive numbers other than } 1 \right\}$$
Let the subset $S$ of $U$ be
$$S = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, \log _ { a } d = \log _ { b } c , \quad a \neq b , \quad b c \neq 1 \right\}$$
Which of the following statements in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. If $A = \left( \begin{array} { l l } 4 & 9 \\ 3 & 2 \end{array} \right)$, then $A \in S$. ㄴ. If $A \in U$ and $A$ has an inverse matrix, then $A \in S$. ㄷ. If $A \in S$, then $A$ has an inverse matrix.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Let the set $U$ be

$$U = \left\{ \left. \left( \begin{array} { l l } 
a & b \\
c & d
\end{array} \right) \right\rvert \, a , b , c , d \text { are positive numbers other than } 1 \right\}$$

Let the subset $S$ of $U$ be

$$S = \left\{ \left. \left( \begin{array} { l l } 
a & b \\
c & d
\end{array} \right) \right\rvert \, \log _ { a } d = \log _ { b } c , \quad a \neq b , \quad b c \neq 1 \right\}$$

Which of the following statements in $\langle$Remarks$\rangle$ are correct? [4 points]

$\langle$Remarks$\rangle$\\
ㄱ. If $A = \left( \begin{array} { l l } 4 & 9 \\ 3 & 2 \end{array} \right)$, then $A \in S$.\\
ㄴ. If $A \in U$ and $A$ has an inverse matrix, then $A \in S$.\\
ㄷ. If $A \in S$, then $A$ has an inverse matrix.\\
(1) ㄱ\\
(2) ㄴ\\
(3) ㄱ, ㄷ\\
(4) ㄴ, ㄷ\\
(5) ㄱ, ㄴ, ㄷ