Exercise 1 (5 points)
The Delmas chocolate factory decides to market new confectionery: chocolate drops in the shape of a water droplet. To do this, it must manufacture custom moulds that must meet the following constraint: for this range of sweets to be profitable, the chocolate factory must be able to produce at least 80 with 1 litre of liquid chocolate paste.

Part A: modelling by a function

The half-perimeter of the upper face of the drop will be modelled by a portion of the curve of the function $f$ defined on $]0;+\infty[$ by: $$f(x) = \frac{x^2 - 2x - 2 - 3\ln x}{x}.$$

1. Let $\varphi$ be the function defined on $]0;+\infty[$ by: $$\varphi(x) = x^2 - 1 + 3\ln x.$$ a. Calculate $\varphi(1)$ and the limit of $\varphi$ at 0. b. Study the variations of $\varphi$ on $]0;+\infty[$. Deduce the sign of $\varphi(x)$ according to the values of $x$.
   
2. a. Calculate the limits of $f$ at the boundaries of its domain of definition. b. Show that on $]0;+\infty[$: $f'(x) = \dfrac{\varphi(x)}{x^2}$. Deduce the variation table of $f$. c. Prove that the equation $f(x) = 0$ has a unique solution $\alpha$ on $]0;1]$. Determine using a calculator an approximate value of $\alpha$ to $10^{-2}$ near. It is admitted that the equation $f(x) = 0$ also has a unique solution $\beta$ on $[1;+\infty[$ with $\beta \approx 3.61$ to $10^{-2}$ near. d. Let $F$ be the function defined on $]0;+\infty[$ by: $$F(x) = \frac{1}{2}x^2 - 2x - 2\ln x - \frac{3}{2}(\ln x)^2.$$ Show that $F$ is an antiderivative of $f$ on $]0;+\infty[$.

Part B: solving the problem

In this part, calculations will be performed with the approximate values to $10^{-2}$ near of $\alpha$ and $\beta$ from Part A. To obtain the shape of the droplet, we consider the representative curve $C$ of the function $f$ restricted to the interval $[\alpha;\beta]$ as well as its reflection $C'$ with respect to the horizontal axis. The two curves $C$ and $C'$ delimit the upper face of the drop. For aesthetic reasons, the chocolatier would like his drops to have a thickness of $0.5$ cm. Under these conditions, would the profitability constraint be respected?

Let $k$ be a strictly positive real number. Consider the functions $f _ { k }$ defined on $\mathbb { R }$ by: $$f _ { k } ( x ) = x + k \mathrm { e } ^ { - x } .$$ We denote by $\mathscr { C } _ { k }$ the representative curve of function $f _ { k }$ in a plane with an orthonormal coordinate system.
For every strictly positive real number $k$, the function $f _ { k }$ admits a minimum on $\mathbb { R }$. The value at which this minimum is attained is the abscissa of the point denoted $A _ { k }$ on the curve $\mathscr { C } _ { k }$. It would seem that, for every strictly positive real number $k$, the points $A _ { k }$ are collinear. Is this the case?

In a cardboard disk of radius $R$, we cut out an angular sector corresponding to an angle of measure $\alpha$ radians. We overlap the edges to create a cone of revolution. We wish to choose the angle $\alpha$ to obtain a cone of maximum volume.
We call $\ell$ the radius of the circular base of this cone and $h$ its height. We recall that:

• the volume of a cone of revolution with base a disk of area $\mathscr{A}$ and height $h$ is $\frac{1}{3}\mathscr{A}h$.
• the length of an arc of a circle of radius $r$ and angle $\theta$, expressed in radians, is $r\theta$.

1. We choose $R = 20\mathrm{~cm}$. a. Show that the volume of the cone, as a function of its height $h$, is $$V(h) = \frac{1}{3}\pi\left(400 - h^2\right)h.$$ b. Justify that there exists a value of $h$ that makes the volume of the cone maximum. Give this value. c. How should we cut the cardboard disk to have maximum volume? Give an approximation of $\alpha$ to the nearest degree.
2. Does the angle $\alpha$ depend on the radius $R$ of the cardboard disk?

Exercise 2

During a laboratory experiment, a projectile is launched into a fluid medium. The objective is to determine for which firing angle $\theta$ with respect to the horizontal the height of the projectile does not exceed 1.6 meters. Since the projectile does not move through air but through a fluid, the usual parabolic model is not adopted. Here we model the projectile as a point that moves, in a vertical plane, on the curve representing the function $f$ defined on the interval $[0; 1[$ by: $$f(x) = bx + 2\ln(1-x)$$ where $b$ is a real parameter greater than or equal to 2, $x$ is the abscissa of the projectile, $f(x)$ its ordinate, both expressed in meters.

1. The function $f$ is differentiable on the interval $[0; 1[$. We denote $f'$ its derivative function.
   We admit that the function $f$ has a maximum on the interval $[0; 1[$ and that, for every real $x$ in the interval $[0; 1[$: $$f'(x) = \frac{-bx + b - 2}{1 - x}$$ Show that the maximum of the function $f$ is equal to $b - 2 + 2\ln\left(\frac{2}{b}\right)$.
2. Determine for which values of the parameter $b$ the maximum height of the projectile does not exceed 1.6 meters.
3. In this question, we choose $b = 5.69$.
   The firing angle $\theta$ corresponds to the angle between the abscissa axis and the tangent to the curve of the function $f$ at the point with abscissa 0. Determine an approximate value of the angle $\theta$ to the nearest tenth of a degree.

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.

Having a large piece of land, an entertainment company intends to build a rectangular space for shows and events, as shown in the figure.
The area for the public will be fenced with two types of materials:

• on the sides parallel to the stage, a type A screen will be used, more resistant, whose value per linear meter is $\mathrm{R}\$ 20.00$;
• on the other two sides, a type B screen will be used, common, whose linear meter costs $\mathrm{R}\$ 5.00$.

The company has $\mathrm{R}\$ 5000.00$ to buy all the screens, but wants to do it in such a way that it obtains the largest possible area for the public.
The quantity of each type of screen that the company should buy is
(A) $50.0 \mathrm{~m}$ of type A screen and $800.0 \mathrm{~m}$ of type B screen.
(B) $62.5 \mathrm{~m}$ of type A screen and $250.0 \mathrm{~m}$ of type B screen.
(C) $100.0 \mathrm{~m}$ of type A screen and $600.0 \mathrm{~m}$ of type B screen.
(D) $125.0 \mathrm{~m}$ of type A screen and $500.0 \mathrm{~m}$ of type B screen.
(E) $200.0 \mathrm{~m}$ of type A screen and $200.0 \mathrm{~m}$ of type B screen.

Lobster hatcheries are built, by local fishing cooperatives, in the shape of right-rectangular prisms, fixed to the ground and with flexible nets of the same height, capable of withstanding marine corrosion. For each hatchery to be built, the cooperative uses entirely 100 linear meters of this net, which is used only on the sides.
What should be the values of $X$ and $Y$, in meters, so that the area of the base of the hatchery is maximum?
(A) 1 and 49
(B) 1 and 99
(C) 10 and 10
(D) 25 and 25
(E) 50 and 50

By definition the region inside the parabola $y = x^{2}$ is the set of points $(a,b)$ such that $b \geq a^{2}$. We are interested in those circles all of whose points are in this region. A bubble at a point $P$ on the graph of $y = x^{2}$ is the largest such circle that contains $P$. (You may assume the fact that there is a unique such circle at any given point on the parabola.)
(a) A bubble at some point on the parabola has radius 1. Find the center of this bubble.
(b) Find the radius of the smallest possible bubble at some point on the parabola. Justify.

You have a piece of land close to a river, running straight. You are required to cut off a rectangular portion of the land, with the river forming one of the sides of the rectangle so, your fence will have three sides to it. You only have 60 meters of fencing. The maximum area that you can enclose is \_\_\_\_

For a natural number $n$, when two points $\mathrm { P } _ { n - 1 } , \mathrm { P } _ { n }$ are on the graph of the function $y = x ^ { 2 }$, the point $\mathrm { P } _ { n + 1 }$ is determined according to the following rule.
(a) The coordinates of the two points $\mathrm { P } _ { 0 } , \mathrm { P } _ { 1 }$ are $(0,0)$ and $(1,1)$, respectively.
(b) The point $\mathrm { P } _ { n + 1 }$ is the intersection of the line passing through point $\mathrm { P } _ { n }$ and perpendicular to the line $\mathrm { P } _ { n - 1 } \mathrm { P } _ { n }$ and the graph of the function $y = x ^ { 2 }$. (Here, $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ are distinct points.) Let $l _ { n } = \overline { \mathrm { P } _ { n - 1 } \mathrm { P } _ { n } }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { l _ { n } } { n }$? [3 points]
(1) $2 \sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) $\sqrt { 3 }$
(5) $\sqrt { 2 }$

As shown in the figure, there are two points $\mathrm { A } ( - 1,0 )$ and $\mathrm { P } ( t , t + 1 )$ on the line $y = x + 1$. Let Q be the point where the line passing through P and perpendicular to the line $y = x + 1$ meets the $y$-axis. What is the value of $\lim _ { t \rightarrow \infty } \frac { \overline { \mathrm { AQ } } ^ { 2 } } { \overline { \mathrm { AP } } ^ { 2 } }$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

As shown in the figure, let Q be the foot of the perpendicular from point P on a circle with center O and diameter AB of length 2 to the line segment AB, let R be the foot of the perpendicular from point Q to the line segment OP, and let S be the foot of the perpendicular from point O to the line segment AP. When $\angle \mathrm { PAQ } = \theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$, let $f ( \theta )$ be the area of triangle AOS and $g ( \theta )$ be the area of triangle PRQ. When $\lim _ { \theta \rightarrow +0 } \frac { \theta ^ { 2 } f ( \theta ) } { g ( \theta ) } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

As shown in the figure, there is an isosceles triangle ABC with $\angle \mathrm { CAB } = \angle \mathrm { BCA } = \theta$ that is externally tangent to a circle of radius 1. On the extension of segment AB, a point D (not equal to A) is chosen such that $\angle \mathrm { DCB } = \theta$. Let the area of triangle BDC be $S ( \theta )$. What is the value of $\lim _ { \theta \rightarrow + 0 } \{ \theta \times S ( \theta ) \}$? (Here, $0 < \theta < \frac { \pi } { 4 }$) [4 points]
(1) $\frac { 2 } { 3 }$
(2) $\frac { 8 } { 9 }$
(3) $\frac { 10 } { 9 }$
(4) $\frac { 4 } { 3 }$
(5) $\frac { 14 } { 9 }$

In the coordinate plane, point A has coordinates $( 1,0 )$, and for $\theta$ with $0 < \theta < \frac { \pi } { 2 }$, point B has coordinates $( \cos \theta , \sin \theta )$. For point C in the first quadrant such that quadrilateral OACB is a parallelogram, let $f ( \theta )$ be the area of quadrilateral OACB and $g ( \theta )$ be the square of the length of segment OC. What is the maximum value of $f ( \theta ) + g ( \theta )$? (Here, O is the origin.) [4 points]
(1) $2 + \sqrt { 5 }$
(2) $2 + \sqrt { 6 }$
(3) $2 + \sqrt { 7 }$
(4) $2 + 2 \sqrt { 2 }$
(5) 5

A cubic function $f ( x )$ with leading coefficient 1 and a quadratic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions. (가) The tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = g ( x )$ at the point $( 2,0 )$ are both the $x$-axis. (나) The number of tangent lines to the curve $y = f ( x )$ drawn from the point $( 2,0 )$ is 2. (다) The equation $f ( x ) = g ( x )$ has exactly one real root. For all real numbers $x > 0$, $$g ( x ) \leq k x - 2 \leq f ( x )$$ Let $\alpha$ and $\beta$ be the maximum and minimum values of the real number $k$ satisfying the above inequality, respectively. When $\alpha - \beta = a + b \sqrt { 2 }$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$ and $b$ are rational numbers.) [4 points]

A cone has a base radius of 1 and slant height of 3. The volume of the largest sphere that can be inscribed in this cone is $\_\_\_\_$ .

A sphere $O$ has radius 1. A pyramid has its apex at $O$ and the four vertices of its base all on the surface of sphere $O$. When the volume of this pyramid is maximized, its height is
A. $\frac { 1 } { 3 }$
B. $\frac { 1 } { 2 }$
C. $\frac { \sqrt { 3 } } { 3 }$
D. $\frac { \sqrt { 2 } } { 2 }$

Given that sphere $O$ has radius $1$, and a quadrangular pyramid has vertex at $O$ with the four vertices of its base all on the surface of sphere $O$. When the volume of this quadrangular pyramid is maximum, its height is
A. $\frac{1}{3}$
B. $\frac{1}{2}$
C. $\frac{\sqrt{3}}{3}$
D. $\frac{\sqrt{2}}{2}$

Let $a$, $b$, $h$ be the three edges meeting at a particular vertex of a triangular prism, such that $a$, $b$ are sides of a base triangle with angle $\theta$ between them and $h$ is the height of the prism. Given that the total surface area is $K$, show that the volume $V$ satisfies $V \leq \sqrt{K^3/54}$, and find the dimensions of the prism of maximum volume.

Find the maximum volume of a rectangular box (with a lid) that can be inscribed in a cylinder of radius $30$ cm and height $60$ cm.

Consider the diagram below where $ABZP$ is a rectangle and $ABCD$ and $CXYZ$ are squares whose areas add up to 1. The maximum possible area of the rectangle $ABZP$ is
(a) $1 + 1 / \sqrt{2}$
(b) $2 - \sqrt{2}$
(c) $1 + \sqrt{2}$
(d) $( 1 + \sqrt{2} ) / 2$

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e^{-x}$ is:
(A) $1/e$
(B) 1
(C) $1/2$
(D) $e$

The minimum area of the triangle formed by any tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the coordinate axes is
(A) $a b$
(B) $\frac { a ^ { 2 } + b ^ { 2 } } { 2 }$
(C) $\frac { ( a + b ) ^ { 2 } } { 2 }$
(D) $\frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 }$

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$