We consider the function $f$ defined on $]0; 8]$ by $$f ( x ) = \frac { 10 \ln \left( - x ^ { 2 } + 7 x + 9 \right) } { x }$$ Let $C _ { f }$ be the graphical representation of the function $f$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

Part A

1. Solve in $\mathbb { R }$ the inequality $- x ^ { 2 } + 7 x + 8 \geqslant 0$.
2. Deduce that for all $x \in ] 0 ; 8 ]$, we have $f ( x ) \geqslant 0$.
3. Interpret this result graphically.

Part B

The curve $C _ { f }$ is represented below. Let $M$ be the point of $C _ { f }$ with abscissa $x$ where $x \in ] 0; 8]$. We call $N$ and $P$ the orthogonal projections of the point $M$ respectively on the abscissa axis and on the ordinate axis. In this part, we are interested in the area $\mathscr { A } ( x )$ of the rectangle $\mathrm{O}NMP$.

1. Give the coordinates of points $N$ and $P$ as a function of $x$.
2. Show that for all $x$ belonging to the interval $] 0 ; 8 ]$, $$\mathscr { A } ( x ) = 10 \ln \left( - x ^ { 2 } + 7 x + 9 \right)$$
3. Does there exist a position of the point $M$ for which the area of the rectangle $\mathrm{O}NMP$ is maximum? If it exists, determine this position.

Part C

We consider a strictly positive real number $k$. We wish to determine the smallest value of $x$, approximated to the nearest tenth, belonging to $[ 3.5; 8 ]$ for which the area $\mathscr { A } ( x )$ becomes less than or equal to $k$. To do this, we consider the algorithm below. As a reminder, in Python language, $\ln ( x )$ is written log$(x)$.
\begin{verbatim} from math import * def A(x) : return 10*log (- 1* x**2 + 7*x + 9) def pluspetitevaleur(k) : x = 3.5 while A(x).........: x = x + 0.1 return ........... \end{verbatim}

1. Copy and complete lines 8 and 10 of the algorithm.
2. What number does the instruction \texttt{pluspetitevaleur(30)} then return?
3. What happens when $k = 35$? Justify.

Question 178
O conjunto solução da inequação $2x - 3 > 7$ é
(A) $\{x \in \mathbb{R} \mid x < 5\}$ (B) $\{x \in \mathbb{R} \mid x > 5\}$ (C) $\{x \in \mathbb{R} \mid x < 2\}$ (D) $\{x \in \mathbb{R} \mid x > 2\}$ (E) $\{x \in \mathbb{R} \mid x > 10\}$

O conjunto solução da inequação $2x - 5 > 3$ no conjunto dos números reais é
(A) $\{x \in \mathbb{R} \mid x < 4\}$ (B) $\{x \in \mathbb{R} \mid x > 4\}$ (C) $\{x \in \mathbb{R} \mid x < -4\}$ (D) $\{x \in \mathbb{R} \mid x > -4\}$ (E) $\{x \in \mathbb{R} \mid x = 4\}$

In the calibration of a new traffic light, the times are adjusted so that, in each complete cycle (green-yellow-red), the yellow light remains on for 5 seconds, and the time in which the green light remains on is equal to $\frac{2}{3}$ of the time in which the red light stays on. The green light is on, in each cycle, for $X$ seconds and each cycle lasts $Y$ seconds.
Which expression represents the relationship between $X$ and $Y$?
(A) $5X - 3Y + 15 = 0$ (B) $5X - 2Y + 10 = 0$ (C) $3X - 3Y + 15 = 0$ (D) $3X - 2Y + 15 = 0$ (E) $3X - 2Y + 10 = 0$

QUESTION 148
The solution set of the inequality $2x - 5 > 3$ is
(A) $x > 1$
(B) $x > 2$
(C) $x > 3$
(D) $x > 4$
(E) $x > 5$

A region of a factory must be isolated, as employees are exposed to accident risks there. This region is represented by the gray portion (quadrilateral with area S) in the figure.
So that employees are informed about the location of the isolated area, informational posters will be posted throughout the factory. To create them, a programmer will use software that allows drawing this region from a set of algebraic inequalities.
The inequalities that should be used in the said software for drawing the isolation region are
(A) $3y - x \leq 0 ; 2y - x \geq 0 ; y \leq 8 ; x \leq 9$
(B) $3y - x \leq 0 ; 2y - x \geq 0 ; y \leq 9 ; x \leq 8$
(C) $3y - x \geq 0 ; 2y - x \leq 0 ; y \leq 9 ; x \leq 8$
(D) $4y - 9x \leq 0 ; 8y - 3x \geq 0 ; y \leq 8 ; x \leq 9$
(E) $4y - 9x \leq 0 ; 8y - 3x \geq 0 ; y \leq 9 ; x \leq 8$

Some electronic equipment can ``burn out'' during operation when its internal temperature reaches a maximum value $\mathrm{T}_{\mathrm{M}}$. For greater durability of its products, the electronics industry connects temperature sensors to this equipment, which activate an internal cooling system, turning it on when the temperature of the electronic device exceeds a critical level $\mathrm{T}_{\mathrm{c}}$, and turning it off only when the temperature drops to values below $\mathrm{T}_{\mathrm{m}}$. The graph illustrates the oscillation of the internal temperature of an electronic device during the first six hours of operation, showing that its internal cooling system was activated several times.
How many times did the temperature sensor activate the system, turning it on or off?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 9

The solution set of the inequality $2x - 3 > 7$ is:
(A) $x > 2$
(B) $x > 3$
(C) $x > 4$
(D) $x > 5$
(E) $x > 6$

The number of integers satisfying $|x - 3| \leq 2$ is:
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

5. Which of the following statements is/are true for real numbers $x , y$ ?
(a) If $x ^ { 2 } = y ^ { 2 }$ then $x = y$.
(b) If $x ^ { 3 } = y ^ { 3 }$ then $x = y$.
(c) If $x < y$ then $x ^ { 2 } < y ^ { 2 }$.
(d) If $x < y$ then $x ^ { 3 } < y ^ { 3 }$.

7. Which of the following inequalities are correct?
(a) $1 + x \leq e ^ { x }$ for all $x \in \mathbb { R }$
(b) $2 ^ { n } \leq 2 n$ ! for all positive integers $n$
(c) $\left( 1 + x ^ { 2 } \right) ^ { n } \leq ( 1 + x ) ^ { 2 n }$ for all $x \in ( 0 , \infty )$
(d) $\sin ( x ) \leq \tan ( x )$ for all $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$

System of inequalities $$\left\{ \begin{array} { l } \frac { x + 2 } { x ^ { 2 } - 4 x + 3 } \geqq 0 \\ \frac { 9 } { x - 8 } \leqq - 1 \end{array} \right.$$ What is the number of integers $x$ that satisfy the system? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6

Solve the system of inequalities $$\left\{ \begin{array} { l } \log _ { 3 } | x - 3 | < 4 \\ \log _ { 2 } x + \log _ { 2 } ( x - 2 ) \geqq 3 \end{array} \right.$$ and find the number of integers $x$ that satisfy it. [3 points]

For two natural numbers $a , b ( a < b )$, the fractional inequality
$$\frac { x } { x - a } + \frac { x } { x - b } \leqq 0$$
is satisfied by exactly 2 integers $x$. What is the number of ordered pairs $( a , b )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

System of inequalities $$\left\{ \begin{array} { l } x ( x - 4 ) ( x - 5 ) \geqq 0 \\ \frac { x - 3 } { x ^ { 2 } - 3 x + 2 } \leqq 0 \end{array} \right.$$ What is the number of integers $x$ that satisfy the system? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Two quadratic expressions $f ( x ) , g ( x )$ with leading coefficient 1 have greatest common divisor $x + 3$ and least common multiple $x ( x + 3 ) ( x - 4 )$. How many integers $x$ satisfy the fractional inequality $\frac { 1 } { f ( x ) } + \frac { 1 } { g ( x ) } \leqq 0$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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

For the fractional inequality in $x$ $$1 + \frac { k } { x - k } \leqq \frac { 1 } { x - 1 }$$ Find the value of the natural number $k$ such that the number of integers $x$ satisfying the inequality is 3. [3 points]

For two sets
$$A = \left\{ x \left\lvert \, \frac { ( x - 2 ) ^ { 2 } } { x - 4 } \leq 0 \right. \right\} , \quad B = \left\{ x \mid x ^ { 2 } - 8 x + a \leq 0 \right\}$$
When $A \cup B = \{ x \mid x \leq 5 \}$, what is the value of the constant $a$? [3 points]
(1) 7
(2) 10
(3) 12
(4) 15
(5) 16

As shown in the figure, the graphs of function $f ( x )$ defined on the closed interval $[ - 4,4 ]$ and function $g ( x ) = - \frac { 1 } { 2 } x + 1$ meet at three points, and the $x$-coordinates of these three points are $\alpha , \beta , 2$. The inequality $$\frac { g ( x ) } { f ( x ) } \leq 1$$ is satisfied. How many integers $x$ satisfy this inequality? (Here, $- 4 < \alpha < - 3,0 < \beta < 1$) [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the logarithmic inequality in $x$ $$\log _ { 5 } ( x - 1 ) \leq \log _ { 5 } \left( \frac { 1 } { 2 } x + k \right)$$ when the number of all integers $x$ satisfying this inequality is 3, what is the value of the natural number $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two conditions on the real number $x$: $$\begin{aligned} & p : | x - 1 | \leq 3 , \\ & q : | x | \leq a \end{aligned}$$ What is the minimum value of the natural number $a$ such that $p$ is a sufficient condition for $q$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two conditions on real number $x$: $$\begin{aligned} & p : ( x - 1 ) ( x - 4 ) = 0 , \\ & q : 1 < 2 x \leq a \end{aligned}$$ Find the minimum value of the natural number $a$ such that $p$ is a sufficient condition for $q$. [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8

For two conditions $p$ and $q$ on real numbers $x$: $$\begin{aligned} & p : x ^ { 2 } - 4 x + 3 > 0 , \\ & q : x \leq a \end{aligned}$$ What is the minimum value of the real number $a$ such that $\sim p$ is a sufficient condition for $q$? [3 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

How many natural numbers $x$ satisfy the inequality $\left( \frac { 1 } { 9 } \right) ^ { x } < 3 ^ { 21 - 4 x }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10