1. Have you lived or studied for one month or more in a country where the language of the exam you are now taking is spoken? O Yes O Ho

2. Do you regularly speak or hear the language at home? ◯ No \end{tabular}} & \hline & & \multirow[t]{5}{*}{

Fee Reduction Granted?

0

} \hline 63. Gov. \& Pol. & & \hline \multirow[b]{3}{*}{

64. Essay Choices

Fill in the ovals under the numbers of the essay questions you answered in this examination.[Figure]

} & & \hline & & \hline & & \hline \end{tabular}
\begin{displayquote} SOLUTIONS AND SCORING GUIDES SECTION II 1988 AP CALCULUS AB EXAMINATION \end{displayquote}
The solutions to the free-response questions that follow are intended to illustrate the general level of detail expected of candidates, and the scoring standards indicate how points were awarded to the main aspects of the answers. However, approaches to a problem may vary and, even with the same approach, the written solutions may vary in the way that the different steps are presented. Within the framework of points indicated, the Readers exercise latitude in interpreting the correctness of the solutions.
For example, credit is sometimes awarded for steps not written down, if it is clear that the candidate must have taken those steps to arrive at the result indicated. However, candidates are advised to show their work in order to minimize the risk of not receiving credit for it. Further, if one part of a problem depends on a value obtained in a previous part, and that value is incorrect, full credit is awarded for the latter part if it is done correctly, even though an incorrect result may have been obtained.
Readers often worked with scoring guides more detailed than those presented here in order to award points for correct approaches or to subtract points for calculus or other mathematical errors. In addition, Readers sometimes had scoring guides available for common incorrect approaches that nevertheless revealed some understanding of the calculus involved. In this way, a high degree of consistency in grading is obtained.
1988 Calculus AB

1. Let $f$ be the function given by $f ( x ) = \sqrt { x ^ { 4 } - 16 x ^ { 2 } }$.
   (a) Find the domain of $f$.
   (b) Describe the symmetry, if any, of the graph of $f$.
   (c) Find $f ^ { \prime } ( x )$.
   (d) Find the slope of the line normal to the graph of $f$ at $x = 5$.
   (a) $x ^ { 4 } - 16 x ^ { 2 } \geq 0$

$$\begin{aligned} & x ^ { 2 } \left( x ^ { 2 } - 16 \right) \geq 0 \\ & x ^ { 2 } \geq 16 \text { or } x = 0 \\ & | x | \geq 4 \text { or } x = 0 \end{aligned}$$
(b) Symmetric about the $y$-axis
(c) $f ^ { \prime } ( x ) = \frac { 1 } { 2 } \left( x ^ { 4 } - 16 x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } } \left( 4 x ^ { 3 } - 32 x \right)$
$$= \frac { 2 x \left( x ^ { 2 } - 8 \right) } { | x | \sqrt { x ^ { 2 } - 16 } }$$
(d) $f ^ { \prime } ( 5 ) = \frac { 2 ( 5 ) ( 25 - 8 ) } { 5 \sqrt { 25 - 16 } }$
$$= \frac { 10 ( 17 ) } { 5 \sqrt { 9 } } = \frac { 170 } { 15 } = \frac { 34 } { 3 }$$
. slope of normal line is $- \frac { 3 } { 34 }$
(a) $\left\{ \begin{array} { l } 1 : \text { for radicand } \geq 0 \\ 1 : \text { for } | x | \geq 4 \\ 1 : \text { for } x = 0 \end{array} \right.$
(b) 1: for correct answer
(c) 3: for correct derivative
(d) $\quad \left\{ \begin{aligned} 1 : & \begin{array} { l } \text { for evaluating } f ^ { \prime } ( x ) \\ \\ \text { found in (c) at } x = 5 \end{array} \\ 1 : & \text { for slope of normal } \end{aligned} \right.$
2. A particle moves along the $x$-axis so that its velocity at any time $t \geqq 0$ is given by $v ( t ) = 1 - \sin ( 2 \pi t )$.
(a) Find the acceleration $a ( t )$ of the particle at any time $t$.
(b) Find all values of $t , 0 \leqq t \leqq 2$, for which the particle is at rest.
(c) Find the position $x ( t )$ of the particle at any time $t$ if $x ( 0 ) = 0$.
Solution
(a) $a ( t ) = v ^ { \prime } ( t )$
$$= - 2 \pi \cos ( 2 \pi t )$$
(b) $v ( t ) = 0$ $1 - \sin ( 2 \pi t ) = 0$ or $1 = \sin ( 2 \pi t )$ $2 \pi t = \frac { \pi } { 2 } + 2 k \pi$, where $k = 0 , \pm 1 , \pm 2 , \ldots$ and $0 \leqq t \leqq 2$ $\therefore t = \frac { 1 } { 4 }$ and $t = \frac { 5 } { 4 }$
(c) $x ( t ) = \int v ( t ) d t$
$$\begin{aligned} & = \int [ 1 - \sin ( 2 \pi t ) ] d t \\ & = t + \frac { 1 } { 2 \pi } \cos ( 2 \pi t ) + C \end{aligned}$$
$$\begin{aligned} & x ( 0 ) = 0 = 0 + \frac { \cos ( 0 ) } { 2 \pi } + C \\ & \therefore x ( t ) = t + \frac { 1 } { 2 \pi } \cos ( 2 \pi t ) - \frac { 1 } { 2 \pi } \end{aligned}$$
Distribution of Points
(a) 2: for correct differentiation of velocity
(b) $3 : \left\{ \begin{array} { l } 1 : \text { for } 1 - \sin ( 2 \pi t ) = 0 \\ 1 : \text { for } t = \frac { 1 } { 4 } \\ 1 : \text { for } t = \frac { 5 } { 4 } \end{array} \right.$
(c) $\quad$ 2: for correct antiderivative of $v ( t )$ 4: 1: for $x ( 0 ) = 0$ 1: for finding value of $C$

20. The statement " $\lim _ { x \rightarrow a } f ( x ) = L$ " means that for each $\varepsilon > 0$, there exists a $\delta > 0$ such that
(A) if $0 < | x - a | < \varepsilon$, then $| f ( x ) - L | < \delta$
(B) if $0 < | f ( x ) - L | < \varepsilon$, then $| x - a | < \delta$
(C) if $| f ( x ) - L | < \delta$, then $0 < | x - a | < \varepsilon$
(D) $\quad 0 < | x - a | < \delta$ and $| f ( x ) - L | < \varepsilon$
(E) if $0 < | x - a | < \delta$, then $| f ( x ) - L | < \varepsilon$

1. Have you lived or studied for one month or more in a country where the language of the exam you are now taking is spoken? ◯ Yes

2. Do you regularly speak or hear the language at home? ◯ Yes \end{tabular} \hline \end{tabular}

\multicolumn{17}{|l|}{INDICATE YOUR ANSWERS TO THE EXAM QUESTIONS IN THIS SECTION. IF A QUESTION HAS ONLY FOUR ANSWER OPTIONS, DO NOT MARK OPTION (E). YOUR ANSWER SHEET WILL BE SCORED BY MACHINE. USE ONLY NO. 2 PENCILS TO MARK YOUR ANSWERS ON PAGES 2 AND 3 (ONE RESPONSE PER QUESTION). AFTER YOU HAVE DETERMINED YOUR RESPONSE, BE SURE TO COMPLETELY FILL IN THE OVAL CORRESPONDING TO THE NUMBER OF THE QUESTION YOU ARE ANSWERING. STRAY MARKS AND SMUDGES COULD BE READ AS ANSWERS, SO ERASE CAREFULLY AND COMPLETELY. ANY IMPROPER GRIDDING MAY AFFECT YOUR GRADE.}

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FOR QUESTIONS 76-151, SEE PAGE 3.

PAGE 3

\multicolumn{17}{|r|}{BE SURE EACH MARK IS DARK AND COMPLETELY FILLS THE OVAL. IF A QUESTION HAS ONLY FOUR ANSWER OPTIONS, DO NOT MARK OPTION E.}

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\multicolumn{4}{|c|}{ETS USE ONLY}
	R	W	FS
PT1
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DO NOT WRITE IN THIS AREA.

CALCULUS AB

A CALCULATOR CANNOT BE USED ON PART A OF SECTION I. A GRAPHING CALCULATOR FROM THE APPROVED LIST IS REQUIRED ON PART B OF SECTION I AND FOR SECTION II OF THE EXAMINATION. CALCULATOR MEMORIES NEED NOT BE CLEARED. COMPUTERS, NONGRAPHING SCIENTIFIC CALCULATORS, CALCULATORS WITH QWERTY KEYBOARDS, AND ELECTRONIC WRITING PADS ARE NOT ALLOWED. CALCULATORS MAY NOT BE SHARED AND COMMUNICATION BETWEEN CALCULATORS IS PROHIBITED DURING THE EXAMINATION. ATTEMPTS TO REMOVE TEST MATERIALS FROM THE ROOM BY ANY METHOD WILL RESULT IN THE INVALIDATION OF TEST SCORES.

SECTION I

Time - 1 hour and 45 minutes All questions are given equal weight. Percent of total grade - 50 Part A: 55 minutes, 28 multiple-choice questions
A calculator is NOT allowed. Part B: 50 minutes, 17 multiple-choice questions
A graphing calculator is required. Parts A and B of Section I are printed in this examination booklet; Section II, which consists of longer problems, is printed in a separate booklet.

General Instructions

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE INSTRUCTED TO DO SO. INDICATE YOUR ANSWERS TO QUESTIONS IN PART A ON PAGE 2 OF THE SEPARATE ANSWER SHEET. THE ANSWERS TO QUESTIONS IN PART B SHOULD BE INDICATED ON PAGE 3 OF THE ANSWER SHEET. No credit will be given for anything written in this examination booklet, but you may use the booklet for notes or scratchwork. After you have decided which of the suggested answers is best, COMPLETELY fill in the corresponding oval on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely.
Example: What is the arithmetic mean of the numbers 1,3 , and 6 ? Sample Answer
(A) 1
(B) $\frac { 7 } { 3 }$
(C) 3
(D) $\frac { 10 } { 3 }$
(E) $\frac { 7 } { 2 }$
Many candidates wonder whether or not to guess the answers to questions about which they are not certain. In this section of the examination, as a correction for haphazard guessing, one-fourth of the number of questions you answer incorrectly will be subtracted from the number of questions you answer correctly. It is improbable, therefore, that mere guessing will improve your score significantly; it may even lower your score, and it does take time. If, however, you are not sure of the best answer but have some knowledge of the question and are able to eliminate one or more of the answer choices as wrong, your chance of answering correctly is improved, and it may be to your advantage to answer such a question. Use your time effectively, working as rapidly as you can without losing accuracy. Do not spend too much time on questions that are too difficult. Go on to other questions and come back to the difficult ones later if you have time. It is not expected that everyone will be able to answer all the multiple-choice questions.

A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number.

1. What is the $x$-coordinate of the point of inflection on the graph of $y = \frac { 1 } { 3 } x ^ { 3 } + 5 x ^ { 2 } + 24$ ?
   (A) 5
   (B) 0
   (C) $- \frac { 10 } { 3 }$
   (D) - 5
   (E) - 10 [Figure]
2. The graph of a piecewise-linear function $f$, for $- 1 \leq x \leq 4$, is shown above. What is the value of $\int _ { - 1 } ^ { 4 } f ( x ) d x ?$
   (A) 1
   (B) 2.5
   (C) 4
   (D) 5.5
   (E) 8
3. $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x =$
   (A) $- \frac { 1 } { 2 }$
   (B) $\frac { 7 } { 24 }$
   (C) $\frac { 1 } { 2 }$
   (D) 1
   (E) $2 \ln 2$
4. If $f$ is continuous for $a \leq x \leq b$ and differentiable for $a < x < b$, which of the following could be false?
   (A) $f ^ { \prime } ( c ) = \frac { f ( b ) - f ( a ) } { b - a }$ for some $c$ such that $a < c < b$.
   (B) $f ^ { \prime } ( c ) = 0$ for some $c$ such that $a < c < b$.
   (C) $f$ has a minimum value on $a \leq x \leq b$.
   (D) $f$ has a maximum value on $a \leq x \leq b$.
   (E) $\int _ { a } ^ { b } f ( x ) d x$ exists.
5. $\int _ { 0 } ^ { x } \sin t d t =$
   (A) $\sin x$
   (B) $- \cos x$
   (C) $\cos x$
   (D) $\cos x - 1$
   (E) $1 - \cos x$
6. If $x ^ { 2 } + x y = 10$, then when $x = 2 , \frac { d y } { d x } =$
   (A) $- \frac { 7 } { 2 }$
   (B) - 2
   (C) $\frac { 2 } { 7 }$
   (D) $\frac { 3 } { 2 }$
   (E) $\frac { 7 } { 2 }$
7. $\int _ { 1 } ^ { e } \left( \frac { x ^ { 2 } - 1 } { x } \right) d x =$
   (A) $e - \frac { 1 } { e }$
   (B) $e ^ { 2 } - e$
   (C) $\frac { e ^ { 2 } } { 2 } - e + \frac { 1 } { 2 }$
   (D) $e ^ { 2 } - 2$
   (E) $\frac { e ^ { 2 } } { 2 } - \frac { 3 } { 2 }$
8. Let $f$ and $g$ be differentiable functions with the following properties:
   (i) $g ( x ) > 0$ for all $x$
   (ii) $f ( 0 ) = 1$

If $h ( x ) = f ( x ) g ( x )$ and $h ^ { \prime } ( x ) = f ( x ) g ^ { \prime } ( x )$, then $f ( x ) =$
(A) $f ^ { \prime } ( x )$
(B) $g ( x )$
(C) $e ^ { x }$
(D) 0
(E) 1 [Figure]

The rate at which people enter an amusement park on a given day is modeled by the function $E$ defined by $$E ( t ) = \frac { 15600 } { \left( t ^ { 2 } - 24 t + 160 \right) }$$ The rate at which people leave the same amusement park on the same day is modeled by the function $L$ defined by $$L ( t ) = \frac { 9890 } { \left( t ^ { 2 } - 38 t + 370 \right) }$$ Both $E ( t )$ and $L ( t )$ are measured in people per hour and time $t$ is measured in hours after midnight. These functions are valid for $9 \leq t \leq 23$, the hours during which the park is open. At time $t = 9$, there are no people in the park.
(a) How many people have entered the park by 5:00 P.M. ( $t = 17$ )? Round your answer to the nearest whole number.
(b) The price of admission to the park is $\$15$ until 5:00 P.M. ( $t = 17$ ). After 5:00 P.M., the price of admission to the park is $\$11$. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number.
(c) Let $H ( t ) = \int _ { 9 } ^ { t } ( E ( x ) - L ( x ) ) d x$ for $9 \leq t \leq 23$. The value of $H ( 17 )$ to the nearest whole number is 3725. Find the value of $H ^ { \prime } ( 17 )$, and explain the meaning of $H ( 17 )$ and $H ^ { \prime } ( 17 )$ in the context of the amusement park.
(d) At what time $t$, for $9 \leq t \leq 23$, does the model predict that the number of people in the park is a maximum?

The graph of the function $f$ is shown above. Which of the following statements is false?
(A) $\lim _ { x \rightarrow 2 } f ( x )$ exists.
(B) $\lim _ { x \rightarrow 3 } f ( x )$ exists.
(C) $\lim _ { x \rightarrow 4 } f ( x )$ exists.
(D) $\lim _ { x \rightarrow 5 } f ( x )$ exists.
(E) The function $f$ is continuous at $x = 3$.

The temperature outside a house during a 24-hour period is given by $$F(t) = 80 - 10\cos\left(\frac{\pi t}{12}\right), \quad 0 \leq t \leq 24,$$ where $F(t)$ is measured in degrees Fahrenheit and $t$ is measured in hours.
(a) Sketch the graph of $F$ on the grid provided.
(b) Find the average temperature, to the nearest degree Fahrenheit, between $t = 6$ and $t = 14$.
(c) An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit. For what values of $t$ was the air conditioner cooling the house?
(d) The cost of cooling the house accumulates at the rate of $\$0.05$ per hour for each degree the outside temperature exceeds 78 degrees Fahrenheit. What was the total cost, to the nearest cent, to cool the house for this 24-hour period?

The Maclaurin series for the function $f$ is given by $$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } = 2 x + \frac { 4 x ^ { 2 } } { 2 } + \frac { 8 x ^ { 3 } } { 3 } + \frac { 16 x ^ { 4 } } { 4 } + \cdots + \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } + \cdots$$ on its interval of convergence.
(a) Find the interval of convergence of the Maclaurin series for $f$. Justify your answer.
(b) Find the first four terms and the general term for the Maclaurin series for $f ^ { \prime } ( x )$.
(c) Use the Maclaurin series you found in part (b) to find the value of $f ^ { \prime } \left( - \frac { 1 } { 3 } \right)$.

The graphs of the circles $x^2 + y^2 = 2$ and $(x-1)^2 + y^2 = 1$ intersect at the points $(1,1)$ and $(1,-1)$. Let $R$ be the shaded region in the first quadrant bounded by the two circles and the $x$-axis.
(a) Set up an expression involving one or more integrals with respect to $x$ that represents the area of $R$.
(b) Set up an expression involving one or more integrals with respect to $y$ that represents the area of $R$.
(c) The polar equations of the circles are $r = \sqrt{2}$ and $r = 2\cos\theta$, respectively. Set up an expression involving one or more integrals with respect to the polar angle $\theta$ that represents the area of $R$.

Consider the logistic differential equation $\frac { d y } { d t } = \frac { y } { 8 } ( 6 - y )$. Let $y = f ( t )$ be the particular solution to the differential equation with $f ( 0 ) = 8$.
(a) A slope field for this differential equation is given below. Sketch possible solution curves through the points $( 3,2 )$ and $( 0,8 )$.
(b) Use Euler's method, starting at $t = 0$ with two steps of equal size, to approximate $f ( 1 )$.
(c) Write the second-degree Taylor polynomial for $f$ about $t = 0$, and use it to approximate $f ( 1 )$.
(d) What is the range of $f$ for $t \geq 0$ ?

At time $t$, a particle moving in the $xy$-plane is at position $(x(t), y(t))$, where $x(t)$ and $y(t)$ are not explicitly given. For $t \geq 0$, $\frac{dx}{dt} = 4t + 1$ and $\frac{dy}{dt} = \sin\left(t^2\right)$. At time $t = 0$, $x(0) = 0$ and $y(0) = -4$.
(a) Find the speed of the particle at time $t = 3$, and find the acceleration vector of the particle at time $t = 3$.
(b) Find the slope of the line tangent to the path of the particle at time $t = 3$.
(c) Find the position of the particle at time $t = 3$.
(d) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$.

Let $f$ be the function defined by $f ( x ) = \sqrt { | x - 2 | }$ for all $x$. Which of the following statements is true?
(A) $f$ is continuous but not differentiable at $x = 2$.
(B) $f$ is differentiable at $x = 2$.
(C) $f$ is not continuous at $x = 2$.
(D) $\lim _ { x \rightarrow 2 } f ( x ) \neq 0$
(E) $x = 2$ is a vertical asymptote of the graph of $f$.

Let $f$ be a differentiable function such that $\int f ( x ) \sin x \, d x = - f ( x ) \cos x + \int 4 x ^ { 3 } \cos x \, d x$. Which of the following could be $f ( x )$ ?
(A) $\cos x$
(B) $\sin x$
(C) $4 x ^ { 3 }$
(D) $- x ^ { 4 }$
(E) $x ^ { 4 }$

Train $A$ runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function $v _ { A } ( t )$, where time $t$ is measured in minutes. Selected values for $v _ { A } ( t )$ are given in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 2 & 5 & 8 & 12 \hline

$v _ { A } ( t )$

(meters/minute)

& 0 & 100 & 40 & - 120 & - 150 \hline \end{tabular}
(a) Find the average acceleration of train $A$ over the interval $2 \leq t \leq 8$.
(b) Do the data in the table support the conclusion that train $A$'s velocity is $-100$ meters per minute at some time $t$ with $5 < t < 8$? Give a reason for your answer.
(c) At time $t = 2$, train $A$'s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train $A$, in meters from the Origin Station, at time $t = 12$. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time $t = 12$.
(d) A second train, train $B$, travels north from the Origin Station. At time $t$ the velocity of train $B$ is given by $v _ { B } ( t ) = - 5 t ^ { 2 } + 60 t + 25$, and at time $t = 2$ the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train $A$ and train $B$ is changing at time $t = 2$.

Figures 1 and 2 illustrate regions in the first quadrant associated with the graphs of $y = \frac { 1 } { x }$ and $y = \frac { 1 } { x ^ { 2 } }$, respectively. In Figure 1, let $R$ be the region bounded by the graph of $y = \frac { 1 } { x }$, the $x$-axis, and the vertical lines $x = 1$ and $x = 5$. In Figure 2, let $W$ be the unbounded region between the graph of $y = \frac { 1 } { x ^ { 2 } }$ and the $x$-axis that lies to the right of the vertical line $x = 3$.
(a) Find the area of region $R$.
(b) Region $R$ is the base of a solid. For the solid, at each $x$ the cross section perpendicular to the $x$-axis is a rectangle with area given by $x e ^ { x / 5 }$. Find the volume of the solid.
(c) Find the volume of the solid generated when the unbounded region $W$ is revolved about the $x$-axis.

This exercise is a multiple choice questionnaire. For each question, three answers are proposed and only one of them is correct. The candidate will write on the answer sheet the number of the question followed by the chosen answer and will justify their choice. One point is awarded for each correct and properly justified answer. An unjustified answer will not be taken into account. No points are deducted in the absence of an answer or in case of an incorrect answer.
For questions 1 and 2, space is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The line $\mathscr{D}$ is defined by the parametric representation $\left\{\begin{array}{rl} x &= 5-2t \\ y &= 1+3t \\ z &= 4 \end{array},\, t \in \mathbb{R}\right.$.

1. We denote by $\mathscr{P}$ the plane with Cartesian equation $3x + 2y + z - 6 = 0$. a. The line $\mathscr{D}$ is perpendicular to the plane $\mathscr{P}$. b. The line $\mathscr{D}$ is parallel to the plane $\mathscr{P}$. c. The line $\mathscr{D}$ is contained in the plane $\mathscr{P}$.
2. We denote by $\mathscr{D}'$ the line that passes through point A with coordinates $(3;1;1)$ and has direction vector $\vec{u} = 2\vec{i} - \vec{j} + 2\vec{k}$. a. The lines $\mathscr{D}$ and $\mathscr{D}'$ are parallel. b. The lines $\mathscr{D}$ and $\mathscr{D}'$ are secant. c. The lines $\mathscr{D}$ and $\mathscr{D}'$ are not coplanar.

For questions 3 and 4, the plane is equipped with a direct orthonormal coordinate system with origin O.

1. Let $\mathscr{E}$ be the set of points $M$ with affix $z$ satisfying $|z + \mathrm{i}| = |z - \mathrm{i}|$. a. $\mathscr{E}$ is the $x$-axis. b. $\mathscr{E}$ is the $y$-axis. c. $\mathscr{E}$ is the circle with center O and radius 1.
2. We denote by B and C two points in the plane whose respective affixes $b$ and $c$ satisfy the equality $\dfrac{c}{b} = \sqrt{2}\,\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$. a. The triangle OBC is isosceles with apex O. b. The points O, B, C are collinear. c. The triangle OBC is isosceles and right-angled at B.

Exercise 4 — For candidates who have followed the specialization course
Let $E$ denote the set of twenty-seven integers between 0 and 26.
Let $A$ denote the set whose elements are the twenty-six letters of the alphabet and a separator between two words, denoted ``$\star$'' and considered as a character.
To encode the elements of $A$, we proceed as follows:

• First: We associate to each of the letters of the alphabet, arranged in alphabetical order, a natural integer between 0 and 25, arranged in increasing order. We thus have $a \rightarrow 0 , b \rightarrow 1 , \ldots z \rightarrow 25$. We associate to the separator ``$\star$'' the number 26.
• Second: to each element $x$ of $E$, the function $g$ associates the remainder of the Euclidean division of $4 x + 3$ by 27. Note that for every $x$ in $E$, $g ( x )$ belongs to $E$.
• Third: The initial character is then replaced by the character of rank $g ( x )$.

Example: $s \rightarrow 18 , \quad g ( 18 ) = 21$ and $21 \rightarrow v$. So the letter $s$ is replaced during encoding by the letter $v$.
1. Find all integers $x$ of $E$ such that $g ( x ) = x$, that is, invariant under $g$.
Deduce the invariant characters in this encoding.
2. Prove that, for every natural number $x$ belonging to $E$ and every natural number $y$ belonging to $E$, if $y \equiv 4 x + 3$ modulo 27 then $x \equiv 7 y + 6$ modulo 27.
Deduce that two distinct characters are encoded by two distinct characters.
3. Propose a decoding method.
4. Decode the word ``vfv''.

Part A - Hill Cipher
Here are the different encryption steps for a word with an even number of letters:

Step 1	The word is divided into blocks of two consecutive letters, then for each block, each of the following steps is performed.
Step 2	To the two letters of the block are associated two integers $x_1$ and $x_2$ both between 0 and 25, which correspond to the two letters in the same order, in the following table: \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} A	B	C	D	E	F	G	H	I	J	K	L	M
0	1	2	3	4	5	6	7	8	9	10	11	12
N	O	P	Q	R	S	T	U	V	W	X	Y	Z
13	14	15	16	17	18	19	20	21	22	23	24	25

\hline Step 3 & The matrix $X = \binom{x_1}{x_2}$ is transformed into the matrix $Y = \binom{y_1}{y_2}$ satisfying $Y = AX$, where $A = \left(\begin{array}{ll} 5 & 2 \\ 7 & 7 \end{array}\right)$. \hline Step 4 & The matrix $Y = \binom{y_1}{y_2}$ is transformed into the matrix $R = \binom{r_1}{r_2}$, where $r_1$ is the remainder of the Euclidean division of $y_1$ by 26 and $r_2$ is the remainder of the Euclidean division of $y_2$ by 26. \hline Step 5 & To the integers $r_1$ and $r_2$ are associated the two corresponding letters from the table in step 2. The encrypted block is the block obtained by concatenating these two letters. \hline \end{tabular}
Use the encryption method presented to encrypt the word ``HILL''.
Part B - Some mathematical tools necessary for decryption

1. Let $a$ be an integer relatively prime to 26. Prove that there exists an integer $u$ such that $u \times a \equiv 1$ modulo 26.
   
2. Consider the following algorithm:
   

   VARIABLES : PROCESSING :	\begin{tabular}{l} $a, u$, and $r$ are numbers ($a$ is a natural number and relatively prime to 26)
   Read $a$
   $u$ takes the value 0, and $r$ takes the value 0
   While $r \neq 1$
   $u$ takes the value $u + 1$
   $r$ takes the value of the remainder of the Euclidean division of $u \times a$ by 26
   End While
   Display $u$

   \hline \end{tabular}
   The value $a = 21$ is entered into this algorithm. a. Reproduce on your paper and complete the following table, until the algorithm stops.
   

   $u$	0	1	2	$\ldots$
   $r$	0	21	$\ldots$	$\ldots$

   
   b. Deduce that $5 \times 21 \equiv 1$ modulo 26.
   
3. Recall that $A$ is the matrix $A = \left(\begin{array}{ll} 5 & 2 \\ 7 & 7 \end{array}\right)$ and denote by $I$ the matrix: $I = \left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)$. a. Calculate the matrix $12A - A^2$. b. Deduce the matrix $B$ such that $BA = 21I$. c. Prove that if $AX = Y$, then $21X = BY$.

Part C - Decryption
We want to decrypt the word VLUP. We denote by $X = \binom{x_1}{x_2}$ the matrix associated, according to the correspondence table, to a block of two letters before encryption, and $Y = \binom{y_1}{y_2}$ the matrix defined by the equality: $Y = AX = \left(\begin{array}{ll} 5 & 2 \\ 7 & 7 \end{array}\right) X$. If $r_1$ and $r_2$ are the respective remainders of $y_1$ and $y_2$ in the Euclidean division by 26, the block of two letters after encryption is associated with the matrix $R = \binom{r_1}{r_2}$.

1. Prove that: $\left\{ \begin{aligned} 21x_1 &= 7y_1 - 2y_2 \\ 21x_2 &= -7y_1 + 5y_2 \end{aligned} \right.$
   
2. Using question B.2., establish that: $\begin{cases} x_1 \equiv 9r_1 + 16r_2 & \text{modulo } 26 \\ x_2 \equiv 17r_1 + 25r_2 & \text{modulo } 26 \end{cases}$
   
3. Decrypt the word VLUP, associated with the matrices $\binom{21}{11}$ and $\binom{20}{15}$.

Exercise 4 (Candidates who have followed the specialization course)
A bank card number is of the form: $$a _ { 1 } a _ { 2 } a _ { 3 } a _ { 4 } a _ { 5 } a _ { 6 } a _ { 7 } a _ { 8 } a _ { 9 } a _ { 10 } a _ { 11 } a _ { 12 } a _ { 13 } a _ { 14 } a _ { 15 } c$$ where $a _ { 1 } , a _ { 2 } , \ldots , a _ { 15 }$ and $c$ are digits between 0 and 9. The first fifteen digits contain information about the card type, the bank, and the bank account number. $c$ is the validation key for the number. This digit is calculated from the other fifteen. The following algorithm allows validation of the conformity of a given card number.
Initialization: $I$ takes the value 0 $P$ takes the value 0 $R$ takes the value 0 Processing: For $k$ going from 0 to 7: $R$ takes the value of the remainder of the Euclidean division of $2 a _ { 2 k + 1 }$ by 9 $I$ takes the value $I + R$ End For For $k$ going from 1 to 7: $P$ takes the value $P + a _ { 2 k }$ End For $S$ takes the value $I + P + c$ Output: If $S$ is a multiple of 10

A company packages white sugar from two farms $U$ and $V$ in 1 kg packets of different qualities. Extra fine sugar is packaged separately in packets bearing the label ``extra fine''. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.
To calibrate the sugar according to the size of its crystals, it is passed through a series of three sieves. Sugar crystals with a size less than $0.2\,\mathrm{mm}$ are found in the sealed bottom container at the end of calibration. They will be packaged in packets bearing the label ``extra fine sugar''.

1. A sugar crystal is randomly selected from farm U. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{U}}$ which follows the normal distribution with mean $\mu_{\mathrm{U}} = 0.58\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{U}} = 0.21\,\mathrm{mm}$. a. Calculate the probabilities of the following events: $X_{\mathrm{U}} < 0.2$ and $0.5 \leqslant X_{\mathrm{U}} < 0.8$. b. 1800 grams of sugar from farm $U$ is passed through the series of sieves. Deduce from the previous question an estimate of the mass of sugar recovered in the sealed bottom container and an estimate of the mass of sugar recovered in sieve 2.
2. A sugar crystal is randomly selected from farm V. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{V}}$ which follows the normal distribution with mean $\mu_{\mathrm{V}} = 0.65\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{V}}$ to be determined. During the calibration of a large quantity of sugar crystals from farm V, it is observed that $40\%$ of these crystals end up in sieve 2. What is the value of the standard deviation $\sigma_{\mathrm{V}}$ of the random variable $X_{\mathrm{V}}$?

Exercise 3

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. The candidate will indicate on their answer sheet the number of the question and the chosen answer. No justification is required.
A wrong answer, multiple answers, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent. Throughout the exercise, $\mathbb { R }$ denotes the set of real numbers.

1. A primitive of the function $f$, defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$, is the function $F$, defined on $\mathbb { R }$, by: a. $F ( x ) = \frac { x ^ { 2 } } { 2 } \mathrm { e } ^ { x }$ b. $F ( x ) = ( x - 1 ) \mathrm { e } ^ { x }$ c. $F ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ d. $F ( x ) = x ^ { 2 } \mathrm { e } ^ { x ^ { 2 } }$
   
2. We consider the function $g$ defined by $g ( x ) = \ln \left( \frac { x - 1 } { 2 x + 4 } \right)$. The function $g$ is defined on: a. $\mathbb { R }$ b. $] - \infty ; - 2 [ \cup ] 1 ; + \infty [$ c. $] - 2 ; + \infty [$ d. $] - 2 ; 1 [$
   
3. The function $h$ defined on $\mathbb { R }$ by $h ( x ) = ( x + 1 ) \mathrm { e } ^ { x }$ is: a. concave on $\mathbb { R }$ b. convex on $\mathbb { R }$ c. convex on $] - \infty ; - 3 ]$ and concave on $[-3; + \infty [$ d. concave on $] - \infty ; - 3 ]$ and convex on $[-3; + \infty [$
   
4. A sequence ( $u _ { n }$ ) is bounded below by 3 and converges to a real number $\ell$. We can affirm that: a. $\ell = 3$ b. $\ell \geqslant 3$ c. The sequence ( $u _ { n }$ ) is decreasing. d. The sequence ( $u _ { n }$ ) is constant from a certain rank onwards.
   
5. The sequence ( $w _ { n }$ ) is defined by $w _ { 1 } = 2$ and for every strictly positive natural number $n$, $w _ { n + 1 } = \frac { 1 } { n } w _ { n }$. a. The sequence ( $w _ { n }$ ) is geometric b. The sequence ( $w _ { n }$ ) does not have a limit c. $w _ { 5 } = \frac { 1 } { 15 }$ d. The sequence ( $w _ { n }$ ) converges to 0.

Let $S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{100}$. Among the Python scripts below, the one that allows calculating the sum $S$ is:
a. \begin{verbatim} def somme_a() : S = 0 for k in range(100) : S =1/( k+1) return S \end{verbatim}
b. \begin{verbatim} def somme_b() : S = 0 for k in range(100) : S = S + 1/(k + 1) return S \end{verbatim}
c. \begin{verbatim} def somme_c() : k = 0 while S < 100 : S = S+1 /(k+1) return S \end{verbatim}
d. \begin{verbatim} def somme_d() : k = 0 while k < 100: S = S + 1/(k + 1) return S \end{verbatim}

A certification body is commissioned to evaluate two heating devices, one from brand A and the other from brand B.
Parts 1 and 2 are independent.
Part 1: device from brand A Using a probe, the temperature inside the combustion chamber of a brand A device was measured. Below is a representation of the temperature curve in degrees Celsius inside the combustion chamber as a function of time elapsed, expressed in minutes, since the combustion chamber was ignited.
By reading the graph:

1. Give the time at which the maximum temperature is reached inside the combustion chamber.
2. Give an approximate value, in minutes, of the duration during which the temperature inside the combustion chamber exceeds $300 ^ { \circ } \mathrm { C }$.
3. We denote by $f$ the function represented on the graph. Estimate the value of $\frac { 1 } { 600 } \int _ { 0 } ^ { 600 } f ( t ) \mathrm { d } t$. Interpret the result.

Part 2: study of a function Let the function $g$ be defined on the interval $[0 ; + \infty [$ by: $$g ( t ) = 10 t \mathrm { e } ^ { - 0.01 t } + 20 .$$

1. Determine the limit of $g$ at $+ \infty$.
2. a. Show that for all $t \in \left[ 0 ; + \infty \left[ , \quad g ^ { \prime } ( t ) = ( - 0.1 t + 10 ) \mathrm { e } ^ { - 0.01 t } \right. \right.$. b. Study the variations of the function $g$ on $[0 ; + \infty [$ and construct its variation table.
3. Prove that the equation $g ( t ) = 300$ has exactly two distinct solutions on $[0 ; + \infty [$. Give approximate values to the nearest integer.
4. Using integration by parts, calculate $\int _ { 0 } ^ { 600 } g ( t ) \mathrm { d } t$.

Part 3: evaluation For a brand B device, the temperature in degrees Celsius inside the combustion chamber $t$ minutes after ignition is modelled on $[0 ; 600]$ by the function $g$.
The certification body awards one star for each criterion validated among the following four:

• Criterion 1: the maximum temperature is greater than $320 ^ { \circ } \mathrm { C }$.
• Criterion 2: the maximum temperature is reached in less than 2 hours.
• Criterion 3: the average temperature during the first 10 hours after ignition exceeds $250 ^ { \circ } \mathrm { C }$.
• Criterion 4: the temperature inside the combustion chamber must not exceed $300 ^ { \circ } \mathrm { C }$ for more than 5 hours.

Does each device obtain exactly three stars? Justify your answer.

Part B - Second model
After studying a larger collection of data over the last 50 years, another modelling appears more relevant:

• if the El Niño phenomenon is dominant in one year, then the probability that it is still dominant the following year is 0.5
• on the other hand, if the El Niño phenomenon is not dominant in one year, then the probability that it is dominant the following year is 0.3.

We consider that the reference year is 2023. We denote for every natural integer $n$:

• $E _ { n }$ the event ``the El Niño phenomenon is dominant in the year $2023 + n$ '';
• $p _ { n }$ the probability of the event $E _ { n }$.

In 2023, El Niño was not dominant. We thus have $p _ { 0 } = 0$.

1. Let $n$ be a natural integer. Copy and complete the following weighted tree.
2. Justify that $p _ { 1 } = 0.3$.
3. Using the tree, show that, for every natural integer $n$, we have: $$p _ { n + 1 } = 0.2 p _ { n } + 0.3$$
4. a. Conjecture the variations and the possible limit of the sequence $( p _ { n } )$. b. Show by induction that, for every natural integer $n$, we have: $p _ { n } \leqslant \frac { 3 } { 8 }$. c. Determine the direction of variation of the sequence $\left( p _ { n } \right)$. d. Deduce the convergence of the sequence $\left( p _ { n } \right)$.
5. Let $\left( u _ { n } \right)$ be the sequence defined by $u _ { n } = p _ { n } - \frac { 3 } { 8 }$ for every natural integer $n$. a. Show that the sequence $( u _ { n } )$ is geometric with ratio 0.2 and specify its first term. b. Show that, for every natural integer $n$, we have: $$p _ { n } = \frac { 3 } { 8 } \left( 1 - 0.2 ^ { n } \right) .$$ c. Calculate the limit of the sequence $\left( p _ { n } \right)$. d. Interpret this result in the context of the exercise.