Let $f$ and $g$ be the functions defined on the set $\mathbb { R }$ of real numbers by
$$f ( x ) = \mathrm { e } ^ { x } \quad \text { and } \quad g ( x ) = \mathrm { e } ^ { - x } .$$
We denote by $\mathscr { C } _ { f }$ the representative curve of function $f$ and $\mathscr { C } _ { g }$ that of function $g$ in an orthonormal coordinate system of the plane.
For every real number $a$, we denote by $M$ the point of $\mathscr { C } _ { f }$ with abscissa $a$ and $N$ the point of $\mathscr { C } _ { g }$ with abscissa $a$.
The tangent line to $\mathscr { C } _ { f }$ at $M$ intersects the $x$-axis at $P$, the tangent line to $\mathscr { C } _ { g }$ at $N$ intersects the $x$-axis at $Q$.
Questions 1 and 2 can be treated independently.

1. Prove that the tangent line to $\mathscr { C } _ { f }$ at $M$ is perpendicular to the tangent line to $\mathscr { C } _ { g }$ at $N$.
2. a. What can be conjectured about the length $PQ$? b. Prove this conjecture.

We consider the function $g$ defined on the interval $] 0 ; + \infty [$, by
$$g ( x ) = \frac { \ln ( x ) } { 1 + x ^ { 2 } }$$
We admit that $g$ is differentiable on the interval $] 0 ; + \infty \left[ \right.$ and we denote $g ^ { \prime }$ its derivative function. We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in the plane with respect to a coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } )$.
We also consider the function $f$ defined on $]0;+\infty[$ by $f(x) = 1 + x^2 - 2x^2\ln(x)$, and $\alpha$ denotes the unique solution of $f(x)=0$ in $[1;+\infty[$. We admit that $g(\alpha) = \frac{1}{2\alpha^2}$.

1. Prove that for all real $x$ in the interval $] 0 ; + \infty \left[ , \quad g ^ { \prime } ( x ) = \frac { f ( x ) } { x \left( 1 + x ^ { 2 } \right) ^ { 2 } } \right.$.
2. Prove that the function $g$ admits a maximum at $x = \alpha$.
3. We denote $T _ { 1 }$ the tangent line to $\mathscr { C } _ { g }$ at the point with abscissa 1 and we denote $T _ { \alpha }$ the tangent line to $\mathscr { C } _ { g }$ at the point with abscissa $\alpha$. Determine, as a function of $\alpha$, the coordinates of the intersection point of the lines $T _ { 1 }$ and $T _ { \alpha }$.

12. Let $f ( x ) = \sqrt { x }$. We draw a tangent to the curve $y = f ( x )$ at the point on the curve whose $x$ coordinate is equal to 4 . Where does this tangent intersect the $X$-axis?
(a) $x = 4$
(b) $x = - 2$
(c) $x = - 4$
(d) $x = 2$

As shown in the figure, let $\mathrm { Q } _ { 1 }$ be the point where the tangent line to the circle $x ^ { 2 } + y ^ { 2 } = 1$ at point $\mathrm { P } _ { 1 }$ meets the $x$-axis. The area of triangle $\mathrm { P } _ { 1 } \mathrm { OQ } _ { 1 }$ is $\frac { 1 } { 4 }$. Let $\mathrm { P } _ { 2 }$ be the point obtained by rotating $\mathrm { P } _ { 1 }$ about the origin O by $\frac { \pi } { 4 }$, and let $\mathrm { Q } _ { 2 }$ be the point where the tangent line at $\mathrm { P } _ { 2 }$ meets the $x$-axis. What is the area of triangle $\mathrm { P } _ { 2 } \mathrm { OQ } _ { 2 }$? (Here, point $\mathrm { P } _ { 1 }$ is in the first quadrant.) [3 points]
(1) 1
(2) $\frac { 5 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 7 } { 4 }$
(5) 2

For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$.
Let A be the point where the tangent line to the curve $y = 3 ^ { x }$ at point P meets the $x$-axis, and let B be the point where the tangent line to the curve $y = a ^ { x - 1 }$ at point P meets the $x$-axis. For point $\mathrm { H } ( k , 0 )$, when $\overline { \mathrm { AH } } = 2 \overline { \mathrm { BH } }$, what is the value of $a$? [4 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For a real number $a > \sqrt{2}$, define the function $f(x)$ as $$f(x) = -x^3 + ax^2 + 2x$$ The tangent line to the curve $y = f(x)$ at the point $\mathrm{O}(0,0)$ intersects the curve $y = f(x)$ at another point A. The tangent line to the curve $y = f(x)$ at point A intersects the $x$-axis at point B. If point A lies on the circle with diameter OB, find the value of $\overline{\mathrm{OA}} \times \overline{\mathrm{AB}}$. [4 points]

For the function $f ( x ) = x ^ { 2 } - 4 x - 3$, let $l$ be the tangent line to the curve $y = f ( x )$ at the point $( 1 , - 6 )$, and for the function $g ( x ) = \left( x ^ { 3 } - 2 x \right) f ( x )$, let $m$ be the tangent line to the curve $y = g ( x )$ at the point $( 1,6 )$. What is the area of the figure enclosed by the two lines $l , m$ and the $y$-axis? [4 points]
(1) 21
(2) 28
(3) 35
(4) 42
(5) 49

10. The system of inequalities $\left\{ \begin{array} { l } x - 1 \geq 0 , \\ k x - y \leq 0 , \\ x + \sqrt { 3 } y - 3 \sqrt { 3 } \leq 0 \end{array} \right.$ represents a planar region that is an equilateral triangle. The minimum value of $z = x + 3 y$ is
A. $2 + 3 \sqrt { 3 }$ B. $1 + 3 \sqrt { 3 }$ C. $2 + \sqrt { 3 }$ D. $1 + \sqrt { 3 }$

118- The tangent line to the graph of $f(x) = (x+2)e^{1-x}$ at the point $x = 1$ meets the line connecting this point to the origin. What is $\tan\alpha$?
(1) $0.5$ (2) $1$ (3) $1.5$ (4) $2$

121. The tangent line to the curve $f(x) = \dfrac{5x - 4}{\sqrt{x}}$ at the point $x = 4$. At which values does it intersect the $y$-axis?
(1) $-4$ (2) $-1$ (3) $2$ (4) $3$

122. If $\tan\alpha$ and $\tan\beta$ are the roots of the equation $2x^2 + 3x - 1 = 0$, what is $\tan(\alpha + \beta)$?
(1) $1$ (2) $\dfrac{3}{2}$ (3) $-3$ (4) $-1$

120- At the intersection points of the curves $f(x) = \sin x + \dfrac{1}{2}\cos x$ and $g(x) = \dfrac{3}{2}\sin x$ on the interval $[0, \pi]$, a tangent line to the curve $f(x)$ is drawn. This tangent line intersects the $x$-axis at which interval?
(1) $\dfrac{\pi}{4} - 1$ (2) $\dfrac{\pi}{4} - 2$ (3) $\dfrac{\pi}{4} + \dfrac{1}{\lambda}$ (4) $\dfrac{\pi}{4} + \dfrac{3}{\lambda}$
%% Page 6 121-- Function $f$ is differentiable and periodic with period 5. If $f'(-1)=\dfrac{3}{2}$ and $g(x)=f(x+1)+f(3x+10)$, then $g'(-2)$ is which of the following?
(1) $3$ (2) $\dfrac{7}{2}$ (3) $6$ (4) $\dfrac{13}{2}$

If $x^{2/3} + y^{2/3} = a^{1/3}$, find the equation of the tangent to the curve at a point, and show that the length of the tangent intercepted between the axes is constant.

3) Considering the similarity of the right triangles ACL and ALM in Figure 4, and recalling the geometric meaning of the derivative, verify that the value of the ordinate $d$ of the centre of the wheel remains constant during motion. Therefore, it seems to the cyclist that they are moving on a flat surface.
[Figure]
Figure 4
\footnotetext{${ } ^ { 1 }$ In general, the length of the arc of a curve with equation $y = \varphi ( x )$ between the abscissae $x _ { 1 }$ and $x _ { 2 }$ is given by $\int _ { x _ { 1 } } ^ { x _ { 2 } } \sqrt { 1 + \left( \varphi ^ { \prime } ( x ) \right) ^ { 2 } } d x$. }

Ministry of Education, University and Research

The graph of the function:
$$f ( x ) = \frac { 2 } { \sqrt { 3 } } - \frac { e ^ { x } + e ^ { - x } } { 2 } , \quad \text { for } x \in \left[ - \frac { \ln ( 3 ) } { 2 } ; \frac { \ln ( 3 ) } { 2 } \right]$$
if replicated several times, can also represent the profile of a platform suitable for being traversed by a bicycle with very particular wheels, having the shape of a regular polygon.

The tangent to the curve $y = e^x$ drawn at the point $(c, e^c)$ intersects the line joining the points $(c-1, e^{c-1})$ and $(c+1, e^{c+1})$
(A) on the left of $x = c$
(B) on the right of $x = c$
(C) at no point
(D) at all points

The tangent at the point $( 2 , - 2 )$ to the curve, $x ^ { 2 } y ^ { 2 } - 2 x = 4 ( 1 - y )$ does not pass through the point:
(1) $( - 2 , - 7 )$
(2) $( 8,5 )$
(3) $( - 4 , - 9 )$
(4) $\left( 4 , \frac { 1 } { 3 } \right)$

At what point does the tangent line to the curve $\mathbf { y } = \sin ( \pi \mathrm { x } ) + \mathrm { e } ^ { \mathrm { x } }$ at $\mathrm { x } = 1$ intersect the y-axis?
A) $- \pi$
B) - 1
C) 0
D) $e - 1$
E) $\pi$