Let $y = f ( t )$ be a solution to the differential equation $\frac { d y } { d t } = k y$, where $k$ is a constant. Values of $f$ for selected values of $t$ are given in the table below:

$t$	0	2
$f ( t )$	4	12

Which of the following is an expression for $f ( t )$ ?
(A) $4 e ^ { \frac { t } { 2 } \ln 3 }$
(B) $e ^ { \frac { t } { 2 } \ln 9 } + 3$
(C) $2 t ^ { 2 } + 4$
(D) $4 t + 4$

We are interested in the evolution of the height of a corn plant as a function of time.
We decide to model this growth by a logistic function of the type: $$h ( t ) = \frac { a } { 1 + b \mathrm { e } ^ { - 0,04 t } }$$ where $a$ and $b$ are positive real constants, $t$ is the time variable expressed in days and $h ( t )$ denotes the height of the plant, expressed in metres.
We know that initially, for $t = 0$, the plant measures $0,1 \mathrm{~m}$ and that its height tends towards a limiting height of 2 m.
Part 1. Determine the constants $a$ and $b$ so that the function $h$ corresponds to the growth of the corn plant studied.
Part 2. We now consider that the growth of the corn plant is given by the function $f$ defined on $[0;250]$ by $$f ( t ) = \frac { 2 } { 1 + 19 \mathrm { e } ^ { - 0,04 t } }$$

1. Determine $f ^ { \prime } ( t )$ as a function of $t$ ($f ^ { \prime }$ denoting the derivative function of the function $f$). Deduce the variations of the function $f$ on the interval $[ 0 ; 250 ]$.
2. Calculate the time required for the corn plant to reach a height greater than $1,5 \mathrm{~m}$.
3. a. Verify that for all real $t$ belonging to the interval $[ 0 ; 250 ]$ we have $f ( t ) = \frac { 2 \mathrm { e } ^ { 0,04 t } } { \mathrm { e } ^ { 0,04 t } + 19 }$.
   Show that the function $F$ defined on the interval $[ 0 ; 250]$ by $F ( t ) = 50 \ln \left( \mathrm { e } ^ { 0,04 t } + 19 \right)$ is an antiderivative of the function $f$.
   b. Determine the average value of $f$ on the interval $[ 50 ; 100 ]$. Give an approximate value to $10 ^ { - 2 }$ and interpret this result.
4. We are interested in the growth rate of the corn plant; it is given by the derivative function of the function $f$. The growth rate is maximum for a value of $t$. Using the graph given in the appendix, determine an approximate value of this. Then estimate the height of the plant.

A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
where $C$ denotes the concentration of the medication in the blood (in micromoles per liter), $t$ the time elapsed since the start of the infusion (in hours), $d$ the infusion rate (in micromoles per hour), $a$ a strictly positive real parameter called clearance (in liters per hour).
Part C: determination of appropriate treatment
The purpose of this part is to determine, for a given patient, the value of the infusion rate that allows the treatment to be effective, that is, the plateau to equal 15. The infusion rate $d$ is provisionally set to 105.

1. We seek to determine the clearance $a$ of a patient. The infusion rate is provisionally set to 105. a. Express as a function of $a$ the concentration of the medication 6 hours after the start of the infusion. b. After 6 hours, analyses allow us to know the concentration of the medication in the blood; it is equal to 5.9 micromoles per liter. Determine an approximate value, to the nearest tenth of a liter per hour, of the clearance of this patient.
2. Determine the value of the infusion rate $d$ guaranteeing the effectiveness of the treatment.

A statistical study was conducted in a large city in France between January $1^{\text{st}}$ 2000 and January $1^{\text{st}}$ 2010 to assess the proportion of households with a fixed internet connection. On January $1^{\text{st}}$ 2000, one household in eight had a fixed internet connection and, on January $1^{\text{st}}$ 2010, $64\%$ of households did. Following this study, this proportion was modelled by the function $g$ defined on the interval $[0; +\infty[$ by: $$g(t) = \frac{1}{1 + k\mathrm{e}^{-at}},$$ where $k$ and $a$ are two positive real constants and the variable $t$ denotes the time, measured in years, elapsed since January $1^{\text{st}}$ 2000.

1. Determine the exact values of $k$ and $a$ so that $g(0) = \frac{1}{8}$ and $g(10) = \frac{64}{100}$.
2. In the following, we take $k = 7$ and $a = 0.25$. The function $g$ is therefore defined by: $$g(t) = \frac{1}{1 + 7\mathrm{e}^{-\left(\frac{t}{4}\right)}}$$ a. Show that the function $g$ is increasing on the interval $[0; +\infty[$. b. According to this model, can we assert that one day, at least $99\%$ of households in this city will have a fixed internet connection? Justify your answer.
3. a. Give, to the nearest hundredth, the proportion of households, predicted by the model, equipped with a fixed internet connection on January $1^{\text{st}}$ 2018. b. Given the development of mobile telephony, some statisticians believe that the modelling by the function $g$ of the evolution of the proportion of households with a fixed internet connection should be reconsidered. At the beginning of 2018 a survey was conducted. Out of 1000 households, 880 had a fixed connection. Does this survey support these sceptical statisticians? (You may use an asymptotic confidence interval at the $95\%$ level.)

Um capital é aplicado a juros compostos de 10\% ao ano. Após 2 anos, o montante é R\$\,12\,100,00. O capital inicial é
(A) R\$\,9\,000,00 (B) R\$\,9\,500,00 (C) R\$\,10\,000,00 (D) R\$\,10\,500,00 (E) R\$\,11\,000,00

A solução da equação $e^{2x} = e^5$ é
(A) $x = 1$ (B) $x = \dfrac{5}{2}$ (C) $x = 3$ (D) $x = 4$ (E) $x = 5$

Organochlorine pesticides were widely used in agriculture; however, due to their high toxicity and persistence in the environment, they were banned. Consider the application of 500 g of an organochlorine pesticide to a crop and that, under certain conditions, the half-life of the pesticide in the soil is 5 years.
The mass of pesticide over 35 years will be closest to
(A) $3.9 \mathrm{~g}$.
(B) $31.2 \mathrm{~g}$.
(C) $62.5 \mathrm{~g}$.
(D) $125.0 \mathrm{~g}$.
(E) $250.0 \mathrm{~g}$.

A researcher analyzed the data on the number of new cases of a disease in a city over a period of 5 consecutive years and organized them in the table below.

Year	New cases
1	200
2	400
3	800
4	1600
5	3200

Based on this data, the researcher modeled the number of new cases as a function of the year $x$ by the expression $f(x) = 100 \cdot 2^x$.
If this trend continues, in which year will the number of new cases first exceed 100,000?
(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

The compound interest on R\$\,1{,}000.00 at a rate of 10\% per year for 2 years is:
(A) R\$\,100.00
(B) R\$\,200.00
(C) R\$\,210.00
(D) R\$\,220.00
(E) R\$\,230.00

The graph of the function $y = k \cdot 3 ^ { x } ( 0 < k < 1 )$ intersects the graphs of the two functions $y = 3 ^ { - x }$ and $y = - 4 \cdot 3 ^ { x } + 8$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When the ratio of the $x$-coordinates of points P and Q is $1 : 2$, find the value of $35 k$. [4 points]

The graph of the function $y = k \cdot 3 ^ { x }$ ($0 < k < 1$) meets the graphs of the two functions $y = 3 ^ { - x }$ and $y = - 4 \cdot 3 ^ { x } + 8$ at points P and Q, respectively. When the ratio of the $x$-coordinates of points P and Q is $1 : 2$, find the value of $35k$. [4 points]

In a certain region, the average number of earthquakes $N$ with magnitude $M$ or greater occurring in 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]

What is the sum of all real roots of the exponential equation $2 ^ { x } + 2 ^ { 2 - x } = 5$? [3 points]
(1) - 2
(2) - 1
(3) 0
(4) 1
(5) 2

What is the sum of all natural numbers $x$ that satisfy the exponential inequality $\left( 3 ^ { x } - 5 \right) \left( 3 ^ { x } - 100 \right) < 0$? [3 points]
(1) 5
(2) 7
(3) 9
(4) 11
(5) 13

For a certain financial product, when an initial asset $W _ { 0 }$ is invested, the expected asset $W$ after $t$ years is given as follows. $$\begin{aligned} & W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right) \\ & \text { (where } W _ { 0 } > 0 , t \geq 0 \text {, and } a \text { is a constant.) } \end{aligned}$$ When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [3 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

For a certain financial product, the expected asset $W$ after $t$ years of investing an initial asset $W _ { 0 }$ is given as follows: $$W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right)$$ (where $W _ { 0 } > 0 , t \geq 0$, and $a$ is a constant.) When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [4 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

When the graph of the function $y = 2 ^ { x } + 2$ is translated in the $x$-direction by $m$ units, and this graph is symmetric to the graph of the function $y = \log _ { 2 } 8 x$ translated in the $x$-direction by 2 units with respect to the line $y = x$, what is the value of the constant $m$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

When the graphs of the quadratic function $y = f ( x )$ and the linear function $y = g ( x )$ are as shown in the figure, the sum of all natural numbers $x$ satisfying the inequality $$\left( \frac { 1 } { 2 } \right) ^ { f ( x ) g ( x ) } \geq \left( \frac { 1 } { 8 } \right) ^ { g ( x ) }$$ is? [4 points] [Figure]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

The line $y = 2 x + k$ meets the graphs of the two functions $$y = \left( \frac { 2 } { 3 } \right) ^ { x + 3 } + 1 , \quad y = \left( \frac { 2 } { 3 } \right) ^ { x + 1 } + \frac { 8 } { 3 }$$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When $\overline { \mathrm { PQ } } = \sqrt { 5 }$, what is the value of the constant $k$? [4 points]
(1) $\frac { 31 } { 6 }$
(2) $\frac { 16 } { 3 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 17 } { 3 }$
(5) $\frac { 35 } { 6 }$

Solve the equation $3^{x-8} = \left(\frac{1}{27}\right)^x$ for the real number $x$. [3 points]

Let $k$ be the $x$-coordinate of the intersection point of the curve $y = \left(\frac{1}{5}\right)^{x-3}$ and the line $y = x$. A function $f(x)$ defined on the set of all real numbers satisfies the following conditions. For all real numbers $x > k$, $f(x) = \left(\frac{1}{5}\right)^{x-3}$ and $f(f(x)) = 3x$. What is the value of $f\left(\frac{1}{k^{3} \times 5^{3k}}\right)$? [4 points]

Point A$(a, b)$ is on the curve $y = \log _ { 16 } ( 8 x + 2 )$ and point B is on the curve $y = 4 ^ { x - 1 } - \frac { 1 } { 2 }$, both in the first quadrant. The point obtained by reflecting A across the line $y = x$ lies on the line OB, and the midpoint of segment AB has coordinates $\left( \frac { 77 } { 8 } , \frac { 133 } { 8 } \right)$. When $a \times b = \frac { q } { p }$, find the value of $p + q$. (Here, O is the origin, and $p$ and $q$ are coprime natural numbers.) [4 points]

8. The shelf life $y$ (in hours) of a certain food and storage temperature $x$ (in ${ } ^ { \circ } \mathrm { C }$ ) satisfy the functional relationship $y = e ^ { k x + b }$ (where $e = 2.718 \ldots$ is the base of natural logarithm, and $k , b$ are constants). If the shelf life of this food at $0 { } ^ { \circ } \mathrm { C }$ is 192 hours and at $22 { } ^ { \circ } \mathrm { C }$ is 48 hours, then the shelf life at $33 { } ^ { \circ } \mathrm { C }$ is
(A) 16 hours
(B) 20 hours
(C) 24 hours
(D) 28 hours

13. The shelf life $y$ (in hours) of a certain food and the storage temperature $x$ (in ${}^{\circ} \mathrm { C }$) satisfy the functional relationship $y = e ^ { k x + b }$ ($e = 2.718 \cdots$ is the base of the natural logarithm, $k$ and $b$ are constants). If the shelf life of this food at $0 ^ { \circ } \mathrm{C}$ is designed to be $192$ hours, and the shelf life at $22 ^ { \circ } \mathrm{C}$ is $45$ hours, then the shelf life of this food at $33 ^ { \circ } \mathrm{C}$ is $\_\_\_\_$ hours.

17. (This problem is worth 14 points) There are two mutually perpendicular straight-line highways on the periphery of a mountainous area. To further improve the traffic situation in the mountainous area, a plan is made to build a straight-line highway connecting the two highways and the boundary of the mountainous area. Let the two mutually perpendicular highways be $l _ { 1 } , l _ { 2 }$, the boundary curve of the mountainous area be C, and the planned highway be l. As shown in the figure, $\mathrm { M } , \mathrm { N }$ are two endpoints of C. It is measured that the distances from point M to $l _ { 1 } , l _ { 2 }$ are 5 kilometers and 40 kilometers respectively, and the distances from point N to $l _ { 1 } , l _ { 2 }$ are 20 kilometers and 2.5 kilometers respectively. Taking the lines where $l _ { 1 } , l _ { 2 }$ are located as the $\mathrm { x } , \mathrm { y }$ axes respectively, establish a rectangular coordinate system xOy. Assume that the curve C conforms to the function model $y = \frac { a } { x ^ { 2 } + b }$ (where $\mathrm { a } , \mathrm { b }$ are constants). (I) Find the values of $\mathrm { a } , \mathrm { b }$; (II) Let the highway l be tangent to curve C at point P, and the x-coordinate of P is t.
(1) Write out the function expression $f ( t )$ for the length of highway l and its domain;
(2) When t takes what value is the length of highway l shortest? Find the shortest length.