Exercise 4 (Candidates who have not followed the specialization course)
The common spruce is a species of coniferous tree that can measure up to 40 meters in height and live more than 150 years. The objective of this exercise is to estimate the age and height of a spruce based on the diameter of its trunk measured at $1.30 \mathrm {~m}$ from the ground.

Part A - Modeling the age of a spruce

For a spruce whose age is between 20 and 120 years, the relationship between its age (in years) and the diameter of its trunk (in meters) measured at $1.30 \mathrm {~m}$ from the ground is modeled by the function $f$ defined on the interval $] 0 ; 1 [$ by: $$f ( x ) = 30 \ln \left( \frac { 20 x } { 1 - x } \right)$$ where $x$ denotes the diameter expressed in meters and $f ( x )$ the age in years.

1. Prove that the function $f$ is strictly increasing on the interval $] 0 ; 1 [$.
2. Determine the values of the trunk diameter $x$ such that the age calculated in this model remains consistent with its validity conditions, that is, between 20 and 120 years.

Part B

The average height of spruces in representative samples of trees aged 50 to 150 years was measured. The following table, created using a spreadsheet, groups these results and allows calculation of the average growth rate of a spruce.

	A	B	C	D	E	F	G	H	I	J	K	L	M
1	Ages (in years)	50	70	80	85	90	95	100	105	110	120	130	150
2	Heights (in meters)	11.2	15.6	18.05	19.3	20.55	21.8	23	24.2	25.4	27.6	29.65	33
3	Growth rate (in meters per year)		0.22	0.245	0.25

1. a. Interpret the number 0.245 in cell D3. b. What formula should be entered in cell C3 to complete line 3 by copying cell C3 to the right?
2. Determine the expected height of a spruce whose trunk diameter measured at $1.30 \mathrm {~m}$ from the ground is 27 cm.
3. The quality of the wood is better when the growth rate is maximal. a. Determine an age interval during which the wood quality is best by explaining the approach. b. Is it consistent to ask loggers to cut trees when their diameter measures approximately 70 cm?

EXERCISE A - Natural logarithm function
Part A:
In a country, a disease affects the population with a probability of 0.05. There is a screening test for this disease. We consider a sample of $n$ people ($n \geqslant 20$) taken at random from the population, assimilated to a draw with replacement. The sample is tested using this method: the blood of these $n$ individuals is mixed, the mixture is tested. If the test is positive, an individual analysis of each person is performed. Let $X_n$ be the random variable that gives the number of analyses performed.

1. Show that $X_n$ takes the values 1 and $(n+1)$.
2. Prove that $P(X_n = 1) = 0.95^n$.

Establish the distribution of $X_n$ by copying on the answer sheet and completing the following table:

$x_i$	1	$n+1$
$P(X_n = x_i)$

1. What does the expectation of $X_n$ represent in the context of the experiment?

Show that $E(X_n) = n + 1 - n \times 0.95^n$.
Part B:

1. Consider the function $f$ defined on $[20;+\infty[$ by $f(x) = \ln(x) + x\ln(0.95)$.

Show that $f$ is decreasing on $[20;+\infty[$.

1. We recall that $\lim_{x\rightarrow+\infty} \frac{\ln x}{x} = 0$. Show that $\lim_{x\rightarrow+\infty} f(x) = -\infty$.
2. Show that $f(x) = 0$ has a unique solution $a$ on $[20;+\infty[$. Give an approximation to 0.1 of this solution.
3. Deduce the sign of $f$ on $[20;+\infty[$.

Part C:
We seek to compare two types of screening. The first method is described in Part A, the second, more classical, consists of testing all individuals. The first method makes it possible to reduce the number of analyses as soon as $E(X_n) < n$. Using Part B, show that the first method reduces the number of analyses for samples containing at most 87 people.

Exercise 1 — Multiple Choice Questionnaire (Logarithmic function)
For each of the following questions, only one of the four proposed answers is correct. The six questions are independent.

1. Consider the function $f$ defined for all real $x$ by $f(x) = \ln\left(1 + x^2\right)$.
   On $\mathbb{R}$, the equation $f(x) = 2022$ a. has no solution. b. has exactly one solution. c. has exactly two solutions. d. has infinitely many solutions.
   
2. Let the function $g$ defined for all strictly positive real $x$ by: $$g(x) = x\ln(x) - x^2$$ We denote $\mathscr{C}_g$ its representative curve in a coordinate system of the plane. a. The function $g$ is convex on $]0; +\infty[$. b. The function $g$ is concave on $]0; +\infty[$. c. The curve $\mathscr{C}_g$ has exactly one inflection point on $]0; +\infty[$. d. The curve $\mathscr{C}_g$ has exactly two inflection points on $]0; +\infty[$.
   
3. Consider the function $f$ defined on $]-1; 1[$ by $$f(x) = \frac{x}{1 - x^2}$$ An antiderivative of the function $f$ is the function $g$ defined on the interval $]-1; 1[$ by: a. $g(x) = -\frac{1}{2}\ln\left(1 - x^2\right)$ b. $g(x) = \frac{1 + x^2}{\left(1 - x^2\right)^2}$ c. $g(x) = \frac{x^2}{2\left(x - \frac{x^3}{3}\right)}$ d. $g(x) = \frac{x^2}{2}\ln\left(1 - x^2\right)$
   
4. The function $x \longmapsto \ln\left(-x^2 - x + 6\right)$ is defined on a. $]-3; 2[$ b. $]-\infty; 6]$ c. $]0; +\infty[$ d. $]2; +\infty[$
   
5. Consider the function $f$ defined on $]0.5; +\infty[$ by $$f(x) = x^2 - 4x + 3\ln(2x - 1)$$ An equation of the tangent line to the representative curve of $f$ at the point with abscissa 1 is: a. $y = 4x - 7$ b. $y = 2x - 4$ c. $y = -3(x - 1) + 4$ d. $y = 2x - 1$
   
6. The set $S$ of solutions in $\mathbb{R}$ of the inequality $\ln(x + 3) < 2\ln(x + 1)$ is: a. $S = ]-\infty; -2[ \cup ]1; +\infty[$ b. $S = ]1; +\infty[$ c. $S = \varnothing$ d. $S = ]-1; 1[$

Exercise 2 — 6 points

Theme: Exponential function Main topics covered: Sequences; Functions, Logarithm function. This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer, or no answer to a question earns or loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.

1. A container initially containing 1 litre of water is left in the sun. Every hour, the volume of water decreases by $15 \%$. After how many whole hours does the volume of water become less than a quarter of a litre? a. 2 hours b. 8 hours. c. 9 hours d. 13 hours
   
2. We consider the function $f$ defined on the interval $] 0$; $+ \infty [ \operatorname { by } f ( x ) = 4 \ln ( 3 x )$. For every real $x$ in the interval $] 0$; $+ \infty [$, we have: a. $f ( 2 x ) = f ( x ) + \ln ( 24 )$ b. $f ( 2 x ) = f ( x ) + \ln ( 16 )$ c. $f ( 2 x ) = \ln ( 2 ) + f ( x )$ d. $f ( 2 x ) = 2 f ( x )$
   
3. We consider the function $g$ defined on the interval $] 1 ; + \infty [$ by: $$g ( x ) = \frac { \ln ( x ) } { x - 1 } .$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathscr { C } _ { g }$ has: a. a vertical asymptote and a horizontal asymptote. b. a vertical asymptote and no horizontal asymptote. c. no vertical asymptote and a horizontal asymptote. d. no vertical asymptote and no horizontal asymptote.
   In the rest of the exercise, we consider the function $h$ defined on the interval ]0;2] by: $$h ( x ) = x ^ { 2 } [ 1 + 2 \ln ( x ) ] .$$ We denote $\mathscr { C } _ { h }$ the representative curve of $h$ in a coordinate system of the plane. We assume that $h$ is twice differentiable on the interval ]0; 2]. We denote $h ^ { \prime }$ its derivative and $h ^ { \prime \prime }$ its second derivative. We assume that, for every real $x$ in the interval ] 0 ; 2], we have: $$h ^ { \prime } ( x ) = 4 x ( 1 + \ln ( x ) ) .$$
   
4. On the interval $\left. ] \frac { 1 } { \mathrm { e } } ; 2 \right]$, the function $h$ equals zero: a. exactly 0 times. b. exactly 1 time. c. exactly 2 times. d. exactly 3 times.
   
5. An equation of the tangent line to $\mathscr { C } _ { h }$ at the point with abscissa $\sqrt { \mathrm { e } }$ is: a. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x$ b. $y = ( 6 \sqrt { \mathrm { e } } ) \cdot x + 2 \mathrm { e }$ c. $y = 6 \mathrm { e } ^ { \frac { x } { 2 } }$ d. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x - 4 \mathrm { e }$.
   
6. On the interval $] 0 ; 2 ]$, the number of inflection points of the curve $\mathscr { C } _ { h }$ is equal to: a. 0 b. 1 c. 2 d. 3
   
7. We consider the sequence $\left( u _ { n } \right)$ defined for every natural number $n$ by $$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 3 \quad \text { and } \quad u _ { 0 } = 6 .$$ We can affirm that: a. the sequence $\left( u _ { n } \right)$ is strictly increasing. b. the sequence $( u _ { n } )$ is strictly decreasing. c. the sequence $( u _ { n } )$ is not monotonic. d. the sequence $( u _ { n } )$ is constant.

This exercise is a multiple choice questionnaire. For each of the six following questions, only one of the four proposed answers is correct.
A wrong answer, multiple answers, or absence of an answer to a question earns neither points nor deducts points.

1. Consider the function $g$ defined and differentiable on $]0;+\infty[$ by: $$g(x) = \ln\left(x^2 + x + 1\right).$$ For every strictly positive real number $x$: a. $g^{\prime}(x) = \frac{1}{2x+1}$ b. $g^{\prime}(x) = \frac{1}{x^2+x+1}$ c. $g^{\prime}(x) = \ln(2x+1)$ d. $g^{\prime}(x) = \frac{2x+1}{x^2+x+1}$
   
2. The function $x \longmapsto \ln(x)$ admits as an antiderivative on $]0;+\infty[$ the function: a. $x \longmapsto \ln(x)$ b. $x \longmapsto \frac{1}{x}$ c. $x \longmapsto x\ln(x) - x$ d. $x \longmapsto \frac{\ln(x)}{x}$
   
3. Consider the sequence $(a_n)$ defined for all $n$ in $\mathbb{N}$ by: $$a_n = \frac{1 - 3^n}{1 + 2^n}.$$ The limit of the sequence $(a_n)$ is equal to: a. $-\infty$ b. $-1$ c. $1$ d. $+\infty$
   
4. Consider a function $f$ defined and differentiable on $[-2;2]$. The variation table of the function $f^{\prime}$ derivative of the function $f$ on the interval $[-2;2]$ is given by:
   

   $x$	$-2$	$-1$	$0$	$2$
   variations of $f^{\prime}$	$1$			$>_{-2}^{-1}$

   
   The function $f$ is: a. convex on $[-2;-1]$ b. concave on $[0;1]$ c. convex on $[-1;2]$ d. concave on $[-2;0]$
   
5. The representative curve of the derivative $f^{\prime}$ of a function $f$ defined on the interval $[-2;4]$ is given above. By graphical reading of the curve of $f^{\prime}$, determine the correct statement for $f$: a. $f$ is decreasing on $[0;2]$ b. $f$ is decreasing on $[-1;0]$ c. $f$ admits a maximum at $1$ on $[0;2]$ d. $f$ admits a maximum at $3$ on $[2;4]$
   
6. A stock is quoted at $57\,€$. Its value increases by $3\%$ every month. The Python function \texttt{seuil()} which returns the number of months to wait for its value to exceed $200\,€$ is: a. \begin{verbatim} def seuil() : m=0 v=57 while v < 200 : m=m+1 v = v*1.03 return m \end{verbatim} b. \begin{verbatim} def seuil() : m=0 v=57 while v > 200 : m=m+1 v = v*1.03 return m \end{verbatim} c. \begin{verbatim} def seuil() : v=57 for i in range (200) : v = v*1.03 return v \end{verbatim} d. \begin{verbatim} def seuil() : m=0 v=57 if v<200: m=m+1 else : v = v*1.03 return m \end{verbatim}

This exercise is a multiple choice questionnaire. For each of the six following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns no points and loses no points.

1. Consider the function $f$ defined and differentiable on $] 0 ; + \infty [$ by: $$f ( x ) = x \ln ( x ) - x + 1 .$$ Among the four expressions below, which one is the derivative of $f$?
   

   a. $\ln ( x )$

   b. $\frac { 1 } { x } - 1$

   c. $\ln ( x ) - 2$

   d. $\ln ( x ) - 1$

   
   
2. Consider the function $g$ defined on $] 0$; $+ \infty \left[ \text{ by } g ( x ) = x ^ { 2 } [ 1 - \ln ( x ) ] \right.$. Among the four statements below, which one is correct?
   

   a. $\lim _ { x \rightarrow 0 } g ( x ) = + \infty$	b. $\lim _ { x \rightarrow 0 } g ( x ) = - \infty$	c. $\lim _ { x \rightarrow 0 } g ( x ) = 0$	\begin{tabular}{ l } d. The function $g$
   does not have a li-
   mit at 0.

   \hline \end{tabular}
   
3. Consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 3 } - 0,9 x ^ { 2 } - 0,1 x$. The number of solutions to the equation $f ( x ) = 0$ on $\mathbb { R }$ is:
   

   a. 0

   b. 1

   c. 2

   d. 3

   
   
4. If $H$ is an antiderivative of a function $h$ defined and continuous on $\mathbb { R }$, and if $k$ is the function defined on $\mathbb { R }$ by $k ( x ) = h ( 2 x )$, then an antiderivative $K$ of $k$ is defined on $\mathbb { R }$ by:
   

   a. $K ( x ) = H ( 2 x )$

   b. $K ( x ) = 2 H ( 2 x )$

   c. $K ( x ) = \frac { 1 } { 2 } H ( 2 x )$

   d. $K ( x ) = 2 H ( x )$

   
   
5. The equation of the tangent line at the point with abscissa 1 to the curve of the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$ is:
   

   a. $y = \mathrm { e } x + \mathrm { e }$

   b. $y = 2 \mathrm { e } x - \mathrm { e }$

   c. $y = 2 \mathrm { e } x + \mathrm { e }$

   d. $y = \mathrm { e } x$

   
   
6. The integers $n$ that are solutions to the inequality $( 0,2 ) ^ { n } < 0,001$ are all integers $n$ such that:
   

   a. $n \leqslant 4$

   b. $n \leqslant 5$

   c. $n \geqslant 4$

   d. $n \geqslant 5$

   
   

Given the function $f(x) = \ln x + \ln(2 - x)$, then
A. $f(x)$ is monotonically increasing on $(0, 2)$
B. $f(x)$ is monotonically decreasing on $(0, 2)$
C. $f(x)$ is increasing on $(0, 1)$ and decreasing on $(1, 2)$
D. $f(x)$ is decreasing on $(0, 1)$ and increasing on $(1, 2)$

The monotone increasing interval of the function $f(x) = \ln(x^2 - 2x - 9)$ is
A. $(-\infty, -2)$
B. $(-\infty, 1)$
C. $(1, +\infty)$
D. $(4, +\infty)$

The precise interval on which the function $f(x) = \log_{1/2}\left(x^2 - 2x - 3\right)$ is monotonically decreasing, is
(A) $(-\infty, -1)$
(B) $(-\infty, 1)$
(C) $(1, \infty)$
(D) $(3, \infty)$

(Course 2) Answer the following questions, where log is the natural logarithm.
(1) Let $f ( x ) = x - 1 - \log x$. We are to find the minimum value of $f ( x )$.
First, we have
$$f ^ { \prime } ( x ) = \mathbf { A } - \frac { \mathbf { B } } { x } .$$
Examining the increases and decreases of the value of $f ( x )$, we see that at $x = \mathbf { C }$ the function is minimized and its value is $\mathbf { D }$. From this, we derive the inequality $x - 1 \geqq \log x$.
(2) For $\mathbf { G }$ in the following sentences, choose the correct answer from among choices (0) $\sim$ (3) below. For the other $\square$, enter the correct number.
Let $k$ be a positive real number and $n$ be a positive integer. We denote by $S$ the area of the figure bounded by the three straight lines $y = \frac { x } { n }$, $x = k$ and $y = 0$, and by $T$ the area of the figure bounded by the curve $y = \log x$, the straight line $x = k$ and the $x$ axis.

On the set of real numbers greater than 1, a function $f$ is defined as
$$f ( x ) = 3 \ln \left( x ^ { 2 } - 1 \right) + 2 \ln \left( x ^ { 3 } - 1 \right) - 5 \ln ( x - 1 )$$
Accordingly,
$$\lim _ { x \rightarrow 1 ^ { + } } e ^ { f ( x ) }$$
what is the value of this limit?
A) 30
B) 36
C) 60
D) 64
E) 72