Consider in $\mathbb{R}$ the equation: $$\ln(6x - 2) + \ln(2x - 1) = \ln(x)$$ Statement 3: the equation has two solutions in the interval $]\frac{1}{2}; +\infty[$. Indicate whether Statement 3 is true or false, justifying your answer.

Consider the two equations numbered [1] and [2]:
$$\begin{aligned} \log _ { 2021 } a & = 2022 - a \\ 2021 ^ { b } & = 2022 - b \end{aligned}$$
(a) Equation [1] has a unique solution.
(b) Equation [2] has a unique solution.
(c) There exists a solution $a$ for [1] and a solution $b$ for [2] such that $a = b$.
(d) There exists a solution $a$ for [1] and a solution $b$ for [2] such that $a + b$ is an integer.

Statements
(17) Let $a = \frac{1}{\ln 3}$. Then $3^a = e$. (18) $\sin(0.02) < 2\sin(0.01)$. (19) $\arctan(0.01) > 0.01$. (20) $4\int_0^1 \arctan(x)\, dx = \pi - \ln 4$.

From the following , select all correct statements. [3 points]
ㄱ. $2 ^ { \log _ { 2 } 1 + \log _ { 2 } 2 + \log _ { 2 } 3 + \cdots + \log _ { 2 } 10 } = 10$ ㄴ. $\log _ { 2 } \left( 2 ^ { 1 } \times 2 ^ { 2 } \times 2 ^ { 3 } \times \cdots \times 2 ^ { 10 } \right) ^ { 2 } = 55 ^ { 2 }$ ㄷ. $\left( \log _ { 2 } 2 ^ { 1 } \right) \left( \log _ { 2 } 2 ^ { 2 } \right) \left( \log _ { 2 } 2 ^ { 3 } \right) \cdots \left( \log _ { 2 } 2 ^ { 10 } \right) = 55$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Which of the following in $\langle$Remarks$\rangle$ are correct? [3 points]
$\langle$Remarks$\rangle$ ㄱ. $2 ^ { \log _ { 2 } 1 + \log _ { 2 } 2 + \log _ { 2 } 3 + \cdots + \log _ { 2 } 10 } = 10 !$ ㄴ. $\log _ { 2 } \left( 2 ^ { 1 } \times 2 ^ { 2 } \times 2 ^ { 3 } \times \cdots \times 2 ^ { 10 } \right) ^ { 2 } = 55 ^ { 2 }$ ㄷ. $\left( \log _ { 2 } 2 ^ { 1 } \right) \left( \log _ { 2 } 2 ^ { 2 } \right) \left( \log _ { 2 } 2 ^ { 3 } \right) \cdots \left( \log _ { 2 } 2 ^ { 10 } \right) = 55$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For an integer $n$, two sets $A ( n ) , B ( n )$ are defined as $$\begin{aligned} & A ( n ) = \left\{ x \mid \log _ { 2 } x \leqq n \right\} \\ & B ( n ) = \left\{ x \mid \log _ { 4 } x \leqq n \right\} \end{aligned}$$ Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $A ( 1 ) = \{ x \mid 0 < x \leqq 1 \}$ ㄴ. $A ( 4 ) = B ( 2 )$ ㄷ. When $A ( n ) \subset B ( n )$, then $B ( - n ) \subset A ( - n )$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ

For an integer $n$, two sets $A ( n ) , B ( n )$ are defined as $$\begin{aligned} & A ( n ) = \left\{ x \mid \log _ { 2 } x \leqq n \right\} \\ & B ( n ) = \left\{ x \mid \log _ { 4 } x \leqq n \right\} \end{aligned}$$ Which of the following statements in are correct? [4 points]
Remarks ㄱ. $A ( 1 ) = \{ x \mid 0 < x \leqq 1 \}$ ㄴ. $A ( 4 ) = B ( 2 )$ ㄷ. When $A ( n ) \subset B ( n )$, we have $B ( - n ) \subset A ( - n )$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ

For a constant $a$ with $0 < a < \frac { 1 } { 2 }$, let the point where the line $y = x$ meets the curve $y = \log _ { a } x$ be $( p , p )$, and let the point where the line $y = x$ meets the curve $y = \log _ { 2 a } x$ be $( q , q )$. Which of the following statements in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. If $p = \frac { 1 } { 2 }$, then $a = \frac { 1 } { 4 }$. ㄴ. $p < q$ ㄷ. $a ^ { p + q } = \frac { p q } { 2 ^ { q } }$
(1) ㄱ
(2) ㄱ, 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

8. Let real numbers $a , b , t$ satisfy $| a + 1 | = | \sin b | = t$
[Figure]
(Figure for Question 7)
A. If $t$ is determined, then $b ^ { 2 }$ is uniquely determined
B. If $t$ is determined, then $a ^ { 2 } + 2 a$ is uniquely determined
C. If $t$ is determined, then $\sin \frac { b } { 2 }$ is uniquely determined
D. If $t$ is determined, then $a ^ { 2 } + a$ is uniquely determined

II. Fill-in-the-Blank Questions (This section contains 7 questions. Multi-blank questions are worth 6 points each, single-blank questions are worth 4 points each, 36 points total.)

9. In a container, there are two types of bacteria, A and B. At any time, the product of the number of bacteria A and B is a constant $10 ^ { 10 }$. For simplicity, scientists use $P _ { A } = \log \left( n _ { A } \right)$ to record data about the number of bacteria A, where $n _ { A }$ is the number of bacteria A. Which of the following statements are correct?
(1) $1 \leq P _ { A } \leq 10$
(2) When $P _ { A } = 5$, the number of bacteria B equals the number of bacteria A
(3) If $P _ { A }$ measured last Monday was 4 and $P _ { A }$ measured last Friday was 8, then the number of bacteria A on Friday is twice the number on Monday
(4) If today's $P _ { A }$ value increases by 1 compared to yesterday, then today's bacteria A count is 10 more than yesterday's
(5) If the scientist controls the number of bacteria B to be 50,000, then $5 < P _ { A } < 5.5$

Let $\Gamma$ be the graph formed by points $(x, y)$ satisfying $y = \log x$ on the coordinate plane. Which of the following relationships produce graphs that are completely identical to $\Gamma$?
(1) $y + \frac{1}{2} = \log(5x)$
(2) $2y = \log\left(x^{2}\right)$
(3) $3y = \log\left(x^{3}\right)$
(4) $x = 10^{y}$
(5) $x^{3} = 10^{\left(y^{3}\right)}$

In a Mathematics lesson, Canan performed operations by following the steps below in order. I. $\operatorname { step } \quad : \quad 6 = 1 \cdot 2 \cdot 3 = \mathrm { e } ^ { \ln 1 } \cdot \mathrm { e } ^ { \ln 2 } \cdot \mathrm { e } ^ { \ln 3 }$ II. $\operatorname { step } : \quad e ^ { \ln 1 } \cdot e ^ { \ln 2 } \cdot e ^ { \ln 3 } = e ^ { \ln 1 + \ln 2 + \ln 3 }$ III. step: $\quad e ^ { \ln 1 + \ln 2 + \ln 3 } = e ^ { \ln 6 }$ IV. $\operatorname { step } : : \mathrm { e } ^ { \ln 6 } = \mathrm { e } ^ { \ln ( 2 + 4 ) }$ V. step: $\mathrm { e } ^ { \ln ( 2 + 4 ) } = \mathrm { e } ^ { \ln 2 + \ln 4 }$ VI. $\operatorname { step } : \quad e ^ { \ln 2 + \ln 4 } = e ^ { \ln 2 } \cdot e ^ { \ln 4 }$ VII. step: $e ^ { \ln 2 } \cdot e ^ { \ln 4 } = 2 \cdot 4 = 8$
At the end of these steps, Canan obtained the result $6 = 8$. Accordingly, in which of the numbered steps did Canan make an error?
A) II
B) III
C) IV
D) V
E) VI

Where $a$ and $b$ are positive real numbers different from 1,
$$\log_a 2 < 0 < \log_2 b$$
the inequality is satisfied. Accordingly, I. $a + b > 1$ II. $b - a > a$ III. $a \cdot b > 1$ Which of the following statements are always true?
A) Only I
B) Only II
C) I and II
D) I and III
E) II and III