12. If $f ( x ) = \left\{ \begin{aligned} \ln x & \text { for } 0 < x \leq 2 \\ x ^ { 2 } \ln 2 & \text { for } 2 < x \leq 4 , \end{aligned} \right.$ then $\lim _ { x \rightarrow 2 } f ( x )$ is
(A) $\ln 2$
(B) $\ln 8$
(C) $\ln 16$
(D) 4
(E) nonexistent [Figure]

Let $f$ be the function defined above. $$f ( x ) = \begin{cases} \frac { ( 2 x + 1 ) ( x - 2 ) } { x - 2 } & \text { for } x \neq 2 \\ k & \text { for } x = 2 \end{cases}$$ For what value of $k$ is $f$ continuous at $x = 2$ ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5

Let $f ( x ) = ( 2 x + 1 ) ^ { 3 }$ and let $g$ be the inverse function of $f$. Given that $f ( 0 ) = 1$, what is the value of $g ^ { \prime } ( 1 )$ ?
(A) $- \frac { 2 } { 27 }$
(B) $\frac { 1 } { 54 }$
(C) $\frac { 1 } { 27 }$
(D) $\frac { 1 } { 6 }$
(E) 6

If the function $f$ is continuous at $x = 3$, which of the following must be true?
(A) $f ( 3 ) < \lim _ { x \rightarrow 3 } f ( x )$
(B) $\lim _ { x \rightarrow 3 ^ { - } } f ( x ) \neq \lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(C) $f ( 3 ) = \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = \lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(D) The derivative of $f$ at $x = 3$ exists.
(E) The derivative of $f$ is positive for $x < 3$ and negative for $x > 3$.

Let $f$ be the function defined on the interval $]0; +\infty[$ by $f(x) = \ln x$. For every strictly positive real number $a$, we define on $]0; +\infty[$ the function $g_a$ by $g_a(x) = ax^2$. We denote by $\mathscr{C}$ the curve representing the function $f$ and $\Gamma_a$ that of the function $g_a$ in a coordinate system of the plane. The purpose of the exercise is to study the intersection of the curves $\mathscr{C}$ and $\Gamma_a$ according to the values of the strictly positive real number $a$.
Part A
We have constructed in appendix 1 the curves $\mathscr{C}, \Gamma_{0,05}, \Gamma_{0,1}, \Gamma_{0,19}$ and $\Gamma_{0,4}$.

1. Name the different curves on the graph. No justification is required.
2. Use the graph to make a conjecture about the number of intersection points of $\mathscr{C}$ and $\Gamma_a$ according to the values (to be specified) of the real number $a$.

Part B
For a strictly positive real number $a$, we consider the function $h_a$ defined on the interval $]0; +\infty[$ by $$h_a(x) = \ln x - ax^2.$$

1. Justify that $x$ is the abscissa of a point $M$ belonging to the intersection of $\mathscr{C}$ and $\Gamma_a$ if and only if $h_a(x) = 0$.
2. a. We admit that the function $h_a$ is differentiable on $]0; +\infty[$ and we denote by $h_a'$ the derivative of the function $h_a$ on this interval. The variation table of the function $h_a$ is given below. Justify, by calculation, the sign of $h_a'(x)$ for $x$ belonging to $]0; +\infty[$.
   

   $x$	0		$\frac{1}{\sqrt{2a}}$
   $h_a'(x)$		+	0	-
   			$\frac{-1 - \ln(2a)}{2}$
   $h_a(x)$

   
   b. Recall the limit of $\frac{\ln x}{x}$ as $x \to +\infty$. Deduce the limit of the function $h_a$ as $x \to +\infty$. We do not ask you to justify the limit of $h_a$ at 0.
   
3. In this question and only in this question, we assume that $a = 0,1$. a. Justify that, in the interval $\left.]0; \frac{1}{\sqrt{0,2}}\right]$, the equation $h_{0,1}(x) = 0$ admits a unique solution. We admit that this equation also has only one solution in the interval $]\frac{1}{\sqrt{0,2}}; +\infty[$. b. What is the number of intersection points of $\mathscr{C}$ and $\Gamma_{0,1}$?
   
4. In this question and only in this question, we assume that $a = \frac{1}{2\mathrm{e}}$. a. Determine the value of the maximum of $h_{\frac{1}{2\mathrm{e}}}$. b. Deduce the number of intersection points of the curves $\mathscr{C}$ and $\Gamma_{\frac{1}{2\mathrm{e}}}$. Justify.
   
5. What are the values of $a$ for which $\mathscr{C}$ and $\Gamma_a$ have no intersection points? Justify.

The manufacturer of padlocks of the brand ``K'' wishes to print a logo for his company. This logo has the shape of a stylized capital letter K, inscribed in a square ABCD, with side length one unit of length. We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD})$.
Part B: study of Proposal B
This proposal is characterized by the following two conditions:

• the line with endpoints A and E is a portion of the graph of the function $f$ defined for all real $x \geqslant 0$ by: $f(x) = \ln(2x + 1)$;
• the line with endpoints B and G is a portion of the graph of the function $g$ defined for all real $x > 0$ by: $g(x) = k\left(\frac{1 - x}{x}\right)$, where $k$ is a positive real number to be determined.

1. a) Determine the abscissa of point E. b) Determine the value of the real number $k$, knowing that the abscissa of point G is equal to 0.5.
   
2. a) Prove that the function $f$ has as a primitive the function $F$ defined for all real $x \geqslant 0$ by: $$F(x) = (x + 0.5) \times \ln(2x + 1) - x.$$ b) Prove that $r = \frac{\mathrm{e}}{2} - 1$.
   
3. Determine a primitive $G$ of the function $g$ on the interval $]0; +\infty[$.
   
4. It is admitted that the previous results allow us to establish that $s = [\ln(2)]^2 + \frac{\ln(2) - 1}{2}$. Does Proposal B satisfy the conditions imposed by the manufacturer?

Question 156
Um estudante realizou um experimento e obteve os seguintes dados:

$x$	$y$
1	3
2	5
3	7
4	9

A função que melhor representa a relação entre $x$ e $y$ é
(A) $y = x + 2$ (B) $y = 2x + 1$ (C) $y = 3x$ (D) $y = x^2 + 2$ (E) $y = 2x^2 - 1$

Uma função $f: \mathbb{R} \to \mathbb{R}$ é definida por $f(x) = 2x^2 - 3x + 1$. O valor de $f(2)$ é
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

A function $f$ is defined by $f(x) = 2x + 5$. What is the value of $f^{-1}(11)$?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Given a real number $x$, define $g ( x ) = x ^ { 2 } e ^ { x }$ if $x \geq 0$ and $g ( x ) = x e ^ { - x }$ if $x < 0$.
(A) The function $g$ is continuous everywhere.
(B) The function $g$ is differentiable everywhere.
(C) The function $g$ is one-to-one.
(D) The range of $g$ is the set of all real numbers.

Solve the following two independent problems.
(i) Let $f$ be a function from domain $S$ to codomain $T$. Let $g$ be another function from domain $T$ to codomain $U$. For each of the blanks below choose a single letter corresponding to one of the four options listed underneath. (It is not necessary that each choice is used exactly once.) Write your answers as a sequence of four letters in correct order. Do NOT explain your answers.
If $g \circ f$ is one-to-one then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ . If $g \circ f$ is onto then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ .
Option A: must be one-to-one and must be onto. Option B: must be one-to-one but need not be onto. Option C: need not be one-to-one but must be onto. Option D: need not be one-to-one and need not be onto. Recall: $g \circ f$ is the function defined by $g \circ f ( a ) = g ( f ( a ) )$. The function $f$ is said to be one-to-one if, for any $a _ { 1 }$ and any $a _ { 2 }$ in $S , f \left( a _ { 1 } \right) = f \left( a _ { 2 } \right)$ implies $a _ { 1 } = a _ { 2 }$. The function $f$ is said to be onto if, for any $b$ in $T$, there is an $a$ in $S$ such that $f ( a ) = b$.
(ii) In the given figure $ABCD$ is a square. Points $X$ and $Y$, respectively on sides $BC$ and $CD$, are such that $X$ lies on the circle with diameter $AY$. What is the area of the square $ABCD$ if $AX = 4$ and $AY = 5$? (Figure is schematic and not to scale.)

18. Let $f$ be a function on the positive real numbers such that $f ( x y ) = f ( x ) + f ( y )$. If $f ( 2024 ) = 2$ then which of the following statement(s) is/ are true?
(a) $f \left( \frac { 1 } { 2024 } \right) = 1$
(b) $f \left( \frac { 1 } { 2024 } \right) = - 1$
(c) $f \left( \frac { 1 } { 2024 } \right) = - 2$
(d) $f \left( \frac { 1 } { 2024 } \right) = 2$

The following description is for questions 19 and 20.

A perfect shuffle of a deck of cards divides the deck into two equal parts and then interleaves the cards from each half, starting with the first card of the first half.
For instance, if we shuffle a deck of cards containing 10 cards arranged $[ 1,2,3,4,5,6,7,8,9,10 ]$ we first create two equal decks with cards $[ 1,2,3,4,5 ]$ and $[ 6,7,8,9,10 ]$ and then interleave them to get a new deck $[ 1,6,2,7,3,8,4,9,5,10 ]$.

3. What is the domain of the following real valued function?
$$f ( x ) = \log _ { 2 } \left( x ^ { 2 } - 5 x + 6 \right)$$
(a) $( - \infty , 2 )$
(b) $( 3 , \infty )$
(c) $( - \infty , 2 ) \cup ( 3 , \infty )$
(d) $( - \infty , \infty )$

Function $$f ( x ) = \left\{ \begin{array} { c c } \frac { x ^ { 2 } + x - 12 } { x - 3 } & ( x \neq 3 ) \\ a & ( x = 3 ) \end{array} \right.$$ When this function is continuous for all real numbers $x$, what is the value of $a$? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6

For the function $f ( x ) = x ^ { 2 } - 4 x + a$ and the function $g ( x ) = \lim _ { n \rightarrow \infty } \frac { 2 | x - b | ^ { n } + 1 } { | x - b | ^ { n } + 1 }$, let $h ( x ) = f ( x ) g ( x )$. What is the value of $a + b$, the sum of the two constants $a , b$ such that the function $h ( x )$ is continuous for all real numbers $x$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7

For a function $y = f ( x )$ defined on the closed interval $[ 0,5 ]$, define the function $g ( x )$ as $$g ( x ) = \begin{cases} \{ f ( x ) \} ^ { 2 } & ( 0 \leqq x \leqq 3 ) \\ ( f \circ f ) ( x ) & ( 3 < x \leqq 5 ) \end{cases}$$ Which of the following graphs of the function $y = f ( x )$ make the function $g ( x )$ continuous on the closed interval $[ 0,5 ]$? Select all that apply from . [4 points] ㄱ. [graph] ㄴ. [graph] ㄷ. [graph]
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

Find the value of $\lim _ { x \rightarrow 1 } \frac { ( x - 1 ) \left( x ^ { 2 } + 3 x + 7 \right) } { x - 1 }$. [3 points]

The graph of a function $y = f ( x )$ defined on all real numbers is as shown in the figure, and a cubic function $g ( x )$ has leading coefficient 1 and $g ( 0 ) = 3$. When the composite function $( g \circ f ) ( x )$ is continuous on all real numbers, what is the value of $g ( 3 )$? [4 points]
(1) 31
(2) 30
(3) 29
(4) 28
(5) 27

Find the value of $\lim_{x \rightarrow 2} \frac{(x-2)(x+3)}{x-2}$. [3 points]

For a quadratic function $f ( x )$ with leading coefficient 1 and the function $$g ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { \ln ( x + 1 ) } & ( x \neq 0 ) \\ 8 & ( x = 0 ) \end{array} \right.$$ when the function $f ( x ) g ( x )$ is continuous on the interval $( - 1 , \infty )$, what is the value of $f ( 3 )$? [3 points]
(1) 6
(2) 9
(3) 12
(4) 15
(5) 18

Find the value of $\lim _ { x \rightarrow 0 } \frac { x ( x + 7 ) } { x }$. [3 points]

For the function $$f ( x ) = \begin{cases} 2 x + 10 & ( x < 1 ) \\ x + a & ( x \geq 1 ) \end{cases}$$ find the value of the constant $a$ such that $f$ is continuous on the entire set of real numbers. [3 points]

What is the value of $\lim _ { x \rightarrow - 2 } \frac { ( x + 2 ) \left( x ^ { 2 } + 5 \right) } { x + 2 }$? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

Two functions $$f ( x ) = \left\{ \begin{array} { l l } x + 3 & ( x \leq a ) \\ x ^ { 2 } - x & ( x > a ) \end{array} , \quad g ( x ) = x - ( 2 a + 7 ) \right.$$ Find the product of all real values of $a$ such that the function $f ( x ) g ( x )$ is continuous on the entire set of real numbers. [4 points]

The figure shows a function $f : X \rightarrow X$. What is the value of $f ( 2 ) + f ^ { - 1 } ( 2 )$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7