In September 1987, Goiânia was the site of the largest radioactive accident that occurred in Brazil, when a sample of caesium-137, removed from an abandoned radiotherapy device, was inadvertently handled by part of the population. The half-life of a radioactive material is the time required for the mass of that material to be reduced to half. The half-life of caesium-137 is 30 years and the amount of remaining mass of a radioactive material, after $t$ years, is calculated by the expression $M(t) = A \cdot (2.7)^{kt}$, where $A$ is the initial mass and $k$ is a negative constant.
Consider 0.3 as an approximation for $\log_{10} 2$.
What is the time required, in years, for an amount of caesium-137 mass to be reduced to 10\% of the initial amount?
(A) 27 (B) 36 (C) 50 (D) 54 (E) 100

To take the trip of her dreams, a person needed to take out a loan in the amount of $\mathrm{R}\$ 5000.00$. To pay the installments, she has at most $\mathrm{R}\$ 400.00$ monthly. For this loan amount, the installment value ($P$) is calculated as a function of the number of installments ($n$) according to the formula
$$P = \frac { 5000 \times 1.013 ^ { n } \times 0.013 } { \left( 1.013 ^ { n } - 1 \right) }$$
If necessary, use 0.005 as an approximation for $\log 1.013$; 2.602 as an approximation for $\log 400$; 2.525 as an approximation for $\log 335$.
According to the given formula, the smallest number of installments whose values do not compromise the limit defined by the person is
(A) 12.
(B) 14.
(C) 15.
(D) 16.
(E) 17.

An entrepreneur uses machines whose internal pressure $P$, in atmosphere, depends on the continuous time of use $t$, in hour, and on a positive parameter $K$, which defines the model of the machine, according to the expression: $$P = 4 \cdot \log[-K \cdot (t + 1) \cdot (t - 19)]$$
The manufacturer of these machines recommends to the user that the internal pressure of this type of machine does not exceed 10 atmospheres during its operation.
The entrepreneur intends to buy new machines of this type that should operate, daily, for a continuous period of 10 hours. For this, he needs to define the model of machine to be acquired by choosing the largest possible value of the parameter $K$, in accordance with the manufacturer's recommendation. The largest value to be chosen for $K$ is
(A) $10^{0.5}$
(B) $10^{8}$
(C) $\dfrac{10^{2.5}}{84}$
(D) $\dfrac{10^{2.5}}{99}$
(E) $25 \times 10^{-2}$

When sound passes through a building wall, a certain proportion is transmitted into the interior while the rest is reflected or absorbed. The ratio of sound transmitted into the interior is called the transmission rate. When the acoustic output of a speaker is $W$ (watts), the intensity $P$ (decibels) of sound transmitted into the interior at a distance of $r$ (m) from the speaker in a building with transmission rate $\alpha$ is as follows. $$\begin{aligned} & P = 10 \log \frac { \alpha W } { I _ { 0 } } - 20 \log r - 11 \\ & \text{(where } I _ { 0 } = 10 ^ { - 12 } \text{ (watts/m}^2\text{) and } r > 1 \text{.)} \end{aligned}$$ A speaker is emitting sound with an acoustic output of 100 (watts). When the intensity of sound transmitted into the interior of a building with transmission rate $\frac { 1 } { 100 }$ is 59 (decibels) or less, what is the minimum distance between the speaker and the building? (Assume that sound spreads uniformly in space and that factors other than transmission rate are not considered.) [4 points]
(1) $10 ^ { 2 } \mathrm{~m}$
(2) $10 ^ { \frac { 17 } { 8 } } \mathrm{~m}$
(3) $10 ^ { \frac { 13 } { 6 } } \mathrm{~m}$
(4) $10 ^ { \frac { 9 } { 4 } } \mathrm{~m}$
(5) $10 ^ { \frac { 5 } { 2 } } \mathrm{~m}$

To remove bacteria living in a water tank, a chemical is to be administered. Let $C _ { 0 }$ be the initial number of bacteria per 1 mL of water in the tank, and let $C$ be the number of bacteria per 1 mL at time $t$ hours after the chemical is administered. The following relationship holds: $$\log \frac { C } { C _ { 0 } } = - k t \quad ( k \text { is a positive constant } )$$ The initial number of bacteria per 1 mL of water is $8 \times 10 ^ { 5 }$, and at time 3 hours after the chemical is administered, the number of bacteria per 1 mL becomes $2 \times 10 ^ { 5 }$. After $a$ hours from administering the chemical, the number of bacteria per 1 mL first becomes $8 \times 10 ^ { 3 }$ or less. Find the value of $a$. (Here, calculate using $\log 2 = 0.3$.) [4 points]

The average number of earthquakes $N$ with magnitude $M$ or greater occurring in a region over one year satisfies the following equation.
$$\log N = a - 0.9 M ( \text{ where } a \text{ is a positive constant } )$$
In this region, earthquakes with magnitude 4 or greater occur on average 64 times per year. Earthquakes with magnitude $x$ or greater occur on average once per year. Find the value of $9 x$. (Use $\log 2 = 0.3$ for the calculation.) [4 points]

Shellfish filter suspended matter. When the water temperature is $t \left( { } ^ { \circ } \mathrm { C } \right)$ and the individual weight is $w ( \mathrm {~g} )$, the amounts (in L) filtered in 1 hour by shellfish A and B are denoted as $Q _ { \mathrm { A } }$ and $Q _ { \mathrm { B } }$ respectively, and the following relationships hold. $$\begin{aligned} & Q _ { \mathrm { A } } = 0.01 t ^ { 1.25 } w ^ { 0.25 } \\ & Q _ { \mathrm { B } } = 0.05 t ^ { 0.75 } w ^ { 0.30 } \end{aligned}$$ When the water temperature is $20 ^ { \circ } \mathrm { C }$ and the individual weights of shellfish A and B are each 8 g, the value of $\frac { Q _ { \mathrm { A } } } { Q _ { \mathrm { B } } }$ is $2 ^ { a } \times 5 ^ { b }$. What is the value of $a + b$? (Here, $a$ and $b$ are rational numbers.) [3 points]
(1) 0.15
(2) 0.35
(3) 0.55
(4) 0.75
(5) 0.95

Shellfish filter suspended matter. When the water temperature is $t \left( { } ^ { \circ } \mathrm { C } \right)$ and the individual weight is $w ( \mathrm {~g} )$, the amounts filtered in 1 hour by shellfish A and B (in L) are $Q _ { \mathrm { A } }$ and $Q _ { \mathrm { B } }$ respectively, and the following relational equations hold. $$\begin{aligned} Q _ { \mathrm { A } } & = 0.01 t ^ { 1.25 } w ^ { 0.25 } \\ Q _ { \mathrm { B } } & = 0.05 t ^ { 0.75 } w ^ { 0.30 } \end{aligned}$$ When the water temperature is $20 { } ^ { \circ } \mathrm { C }$ and the individual weights of shellfish A and B are each 8 g, the value of $\frac { Q _ { \mathrm { A } } } { Q _ { \mathrm { B } } }$ is $2 ^ { a } \times 5 ^ { b }$. What is the value of $a + b$? (Here, $a$ and $b$ are rational numbers.) [3 points]
(1) 0.15
(2) 0.35
(3) 0.55
(4) 0.75
(5) 0.95

To determine the relative density of soil, a test device is inserted into the soil for investigation. When the effective vertical stress of the soil is $S$ and the resistance force received by the test device as it enters the soil is $R$, the relative density $D ( \% )$ of the soil can be calculated as follows. $$D = - 98 + 66 \log \frac { R } { \sqrt { S } }$$ (Here, the units of $S$ and $R$ are metric ton/$\mathrm{m}^{2}$.) The effective vertical stress of soil A is 1.44 times the effective vertical stress of soil B, and the resistance force received by the test device as it enters soil A is 1.5 times the resistance force received as it enters soil B. When the relative density of soil B is $65 ( \% )$, what is the relative density of soil A (in $\%$)? (Use $\log 2 = 0.3$ for calculation.) [3 points]
(1) 81.5
(2) 78.2
(3) 74.9
(4) 71.6
(5) 68.3

To determine the relative density of soil, a method of inserting a test device into the soil for investigation is used. When the effective vertical stress of the soil is $S$ and the resistance force received by the test device as it enters the soil is $R$, the relative density $D ( \% )$ of the soil can be calculated as follows. $$D = - 98 + 66 \log \frac { R } { \sqrt { S } }$$ (where the units of $S$ and $R$ are metric ton $/ \mathrm { m } ^ { 2 }$.) The effective vertical stress of soil A is 1.44 times the effective vertical stress of soil B, and the resistance force received by the test device as it enters soil A is 1.5 times the resistance force received as it enters soil B. When the relative density of soil B is $65 ( \% )$, what is the relative density of soil A (in $\%$)? (Use $\log 2 = 0.3$ for calculation.) [3 points]
(1) 81.5
(2) 78.2
(3) 74.9
(4) 71.6
(5) 68.3

The female silkworm moth secretes pheromone to attract males. When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation.
$$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$
When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance of 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance of $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

The female silkworm moth secretes pheromone to attract males.
When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation.
$$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$
When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

The temperature of a fire room changes over time. For a certain fire room, let the initial temperature be $T _ { 0 } \left( {}^{\circ}\mathrm{C} \right)$ and the temperature $t$ minutes after the fire starts be $T \left( {}^{\circ}\mathrm{C} \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8t + 1 ) \quad ($ where $k$ is a constant.$)$ In this fire room with an initial temperature of $20^{\circ}\mathrm{C}$, the temperature was $365^{\circ}\mathrm{C}$ after $\frac{9}{8}$ minutes from the start of the fire, and the temperature was $710^{\circ}\mathrm{C}$ after $a$ minutes from the start of the fire. What is the value of $a$? [3 points]
(1) $\frac{99}{8}$
(2) $\frac{109}{8}$
(3) $\frac{119}{8}$
(4) $\frac{129}{8}$
(5) $\frac{139}{8}$

The temperature of a fire room changes over time. Let the initial temperature of a certain fire room be $T _ { 0 } \left( { } ^ { \circ } \mathrm { C } \right)$, and the temperature $t$ minutes after the fire starts be $T \left( { } ^ { \circ } \mathrm { C } \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8 t + 1 ) \quad ($ where $k$ is a constant. $)$ In this fire room with an initial temperature of $20 ^ { \circ } \mathrm { C }$, the temperature was $365 ^ { \circ } \mathrm { C }$ at $\frac { 9 } { 8 }$ minutes after the fire started, and the temperature was $710 ^ { \circ } \mathrm { C }$ at $a$ minutes after the fire started. What is the value of $a$? [3 points]
(1) $\frac { 99 } { 8 }$
(2) $\frac { 109 } { 8 }$
(3) $\frac { 119 } { 8 }$
(4) $\frac { 129 } { 8 }$
(5) $\frac { 139 } { 8 }$

Water flows completely through a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$. Let $v _ { c }$ be the speed of water at the center of the cross-section, and let $v$ be the speed of water at a point $x ( 0 < x \leq R )$ away from the wall toward the center. The following relationship holds:
$$\frac { v _ { c } } { v } = 1 - k \log \frac { x } { R }$$
(Here, $k$ is a positive constant, the unit of length is m, and the unit of speed is m/s.)
When the speed of water at a point $R ^ { \frac { 27 } { 23 } }$ away from the wall toward the center is $\frac { 1 } { 2 }$ of the speed at the center, the speed of water at a point $R ^ { a }$ away from the wall toward the center is $\frac { 1 } { 3 }$ of the speed at the center. What is the value of $a$? [3 points]
(1) $\frac { 39 } { 23 }$
(2) $\frac { 37 } { 23 }$
(3) $\frac { 35 } { 23 }$
(4) $\frac { 33 } { 23 }$
(5) $\frac { 31 } { 23 }$

In a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$, water flows completely full. Let $v _ { c }$ be the speed of water at the center of the cross-section, and let $v$ be the speed of water at a point $x ( 0 < x \leq R )$ away from the wall toward the center. The following relationship holds: $$\frac { v _ { c } } { v } = 1 - k \log \frac { x } { R }$$ (Here, $k$ is a positive constant, and the unit of length is m and the unit of speed is m/s.) In this water pipe where $R < 1$, when the speed of water at a point $R ^ { \frac { 27 } { 23 } }$ away from the wall toward the center is $\frac { 1 } { 2 }$ of the speed at the center, the speed of water at a point $R ^ { a }$ away from the wall toward the center is $\frac { 1 } { 3 }$ of the speed at the center.
Find the value of $23 a$. [3 points]

When compressing digital images, let $P$ denote the peak signal-to-noise ratio, which is an indicator of the difference between the original and compressed images, and let $E$ denote the mean squared error between the original and compressed images. The following relationship holds: $$P = 20 \log 255 - 10 \log E \quad ( E > 0 )$$ When two original images $A$ and $B$ are compressed, let $P _ { A }$ and $P _ { B }$ denote their peak signal-to-noise ratios, and let $E _ { A } \left( E _ { A } > 0 \right)$ and $E _ { B } \left( E _ { B } > 0 \right)$ denote their mean squared errors. If $E _ { B } = 100 E _ { A }$, what is the value of $P _ { A } - P _ { B }$? [3 points]
(1) 30
(2) 25
(3) 20
(4) 15
(5) 10

When compressing digital images, let $P$ be the peak signal-to-noise ratio, which is an index indicating the degree of difference between the original and compressed images, and let $E$ be the mean squared error between the original and compressed images. The following relationship holds: $$P = 20 \log 255 - 10 \log E \quad ( E > 0 )$$ When two original images $A$ and $B$ are compressed, let the peak signal-to-noise ratios be $P _ { A }$ and $P _ { B }$ respectively, and the mean squared errors be $E _ { A } \left( E _ { A } > 0 \right)$ and $E _ { B } \left( E _ { B } > 0 \right)$ respectively. When $E _ { B } = 100 E _ { A }$, find the value of $P _ { A } - P _ { B }$. [3 points]

In astronomy, the brightness of a celestial body can be described by magnitude or luminosity. The magnitude and luminosity of two stars satisfy $m _ { 2 } - m _ { 1 } = \frac { 5 } { 2 } \lg \frac { E _ { 1 } } { E _ { 2 } }$, where the luminosity of a star with magnitude $m _ { k }$ is $E _ { k } ( k = 1,2 )$. Given that the magnitude of the Sun is $- 26.7$ and the magnitude of Sirius is $- 1.45$, the ratio of the luminosity of the Sun to that of Sirius is (A) $10 ^ { 10.1 }$ (B) 10.1 (C) $\lg 10.1$ (D) $10 ^ { - 10.1 }$

6. Vision of adolescents is a matter of widespread social concern. Vision can be measured using a vision chart. Vision data is usually recorded using the five-point recording method and the decimal recording method. The data $L$ in the five-point recording method and the data $V$ in the decimal recording method satisfy $L = 5 + \lg V$. It is known that a student's vision data in the five-point recording method is 4.9. Then the student's vision data in the decimal recording method is approximately ( $\sqrt [ 10 ] { 10 } \approx 1.259$ )
A. 1.5
B. 1.2
C. 0.8
D. 0.6

4. Adolescent vision is a matter of widespread social concern. Vision can be measured using a vision chart. Vision data is usually recorded using the five-point recording method and the decimal recording method. The data $l$ in the five-point recording method and the data $v$ in the decimal recording method satisfy $l = 5 + \log_v$. A student's vision data in the five-point recording method is 4.4. Then their vision data in the decimal recording method is approximately ($\sqrt[10]{10} \approx 1.259$)
A. 1.5
B. 1.2
C. 0.8
D. 0.6

Let the water quality index be $d = \frac { S - 1 } { \ln n }$, and the larger $d$ is, the better the water quality. If $S$ remains constant and $d _ { 1 } = 2.1 , d _ { 2 } = 2.2$, then the relationship between $n _ { 1 }$ and $n_2$ is \_\_\_\_