Statement 2: The equation $x - \cos x = 0$ has a unique solution in the interval $\left[ 0 ; \frac{\pi}{2} \right]$. Indicate whether this statement is true or false, justifying your answer.
The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ It is desired that the rate of $\mathrm{CO}_2$ in the room returns to a value $V$ less than or equal to $3.5\%$. a. Justify that there exists a unique instant $T$ satisfying this condition. b. Consider the following algorithm: \begin{verbatim} $t \leftarrow 1,75$ $p \leftarrow 0,1$ $V \leftarrow 0,7$ While $V > 0,035$ $t \leftarrow t + p$ $V \leftarrow ( 0,8 t + 0,2 ) \mathrm { e } ^ { - 0,5 t } + 0,03$ End While \end{verbatim} What is the value of the variable $t$ at the end of the algorithm? What does this value represent in the context of the exercise?
Let two real numbers $a$ and $b$ with $a < b$. Consider a function $f$ defined, continuous, strictly increasing on the interval $[a; b]$ and which vanishes at a real number $\alpha$. Among the following propositions, the function in Python language that allows giving an approximate value of $\alpha$ to 0.001 is: a. \begin{verbatim} def racine(a, b): while abs(b - a) >= 0.001: m = (a + b) / 2 if f(m) < 0: b = m else: a = m return m \end{verbatim} c. \begin{verbatim} def racine(a, b): m = (a + b) / 2 while abs(b - a) <= 0.001: if f(m) < 0: a = m else: b = m return m \end{verbatim} b. \begin{verbatim} def racine(a, b): m = (a + b) / 2 while abs(b - a) >= 0.001: if f(m) < 0: a = m else: b = m return m \end{verbatim} d. \begin{verbatim} def racine(a, b): while abs(b - a) >= 0.001: m = (a + b) / 2 if f(m) < 0: a = m else: b = m return m \end{verbatim}
Statements (13) $\lim _ { x \rightarrow 0 } e ^ { \frac { 1 } { x } } = + \infty$. (14) The following inequality is true. $$\lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { 100 } } < \lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { \frac { 1 } { 100 } } }$$ (15) For any positive integer $n$, $$\int _ { - n } ^ { n } x ^ { 2023 } \cos ( n x ) \, dx < \frac { n } { 2023 }$$ (16) There is no polynomial $p ( x )$ for which there is a single line that is tangent to the graph of $p ( x )$ at exactly 100 points.
2. Which of the following statements are true? (a) Let $f ( x ) = x ^ { 3 } + x - 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ - 1,0 )$. (b) Let $f ( x ) = x ^ { 3 } + x - 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ 0,1 )$. (c) Let $f ( x ) = x ^ { 3 } + x + 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ 0,1 )$. (d) Let $f ( x ) = x ^ { 3 } + x + 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ - 1,0 )$. 3. 26 children participated in a chess tournament. A child got two points for winning, zero for losing and one point for a draw. Each child played against every other child. After the tournament was over no child had an odd score. There were no draws in the entire tournament and no two players had the same score. For convenience assume that the children are named $A , B , C , \ldots , Z$, by the 26 letters of the English alphabet in increasing order of scores, so $A$ has lowest score, $Z$ has highest score. Thus we have $A < B < C < \cdots < X < Y < Z$. Pick all statements which are true. (a) K lost to Q (b) K lost to B (c) M lost to N (d) If L lost to M then N lost to M. 4. 14 teams participate in a volleyball tournament. Each team plays the other exactly once. There are no draws. Assume the teams are labeled by $j , 1 \leq j \leq 14$. Let $x _ { j }$ denote the number of games team $j$ wins and $y _ { j }$ the number of games that team $j$ lost. Pick the correct alternative(s): (a) $\sum _ { j } x _ { j } ^ { 2 } = \sum _ { j } y _ { j } ^ { 2 }$. (b) $\sum _ { j } x _ { j } ^ { 2 } > \sum _ { j } y _ { j } ^ { 2 }$. (c) $\sum _ { j } x _ { j } = \sum _ { j } y _ { j }$. (d) $\sum _ { j } \left| x _ { j } \right| = \sum _ { j } \left| y _ { j } \right|$.
Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Find an integer $n$ with the least possible absolute value such that $p(x)$ has a real root between $n$ and $n+1$. Write this number along with your reason as per the given instruction. [2 points] Instruction for (6): Write two numbers separated by a comma: value of $n$, number of the theorem below that justifies this answer. E.g., if you think that $n=5$ because of the factor theorem, then type $\mathbf{5,1}$ as your answer with no space, full stop or any other punctuation.
When two constants $a , b$ satisfy $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - ( a + 2 ) x + 2 a } { x ^ { 2 } - b } = 3$, what is the value of $a + b$? [2 points] (1) $- 6$ (2) $- 4$ (3) $- 2$ (4) 0 (5) 2
On the coordinate plane, for natural numbers $n$, consider the region $$\left\{ (x, y) \mid 2^x - n \leq y \leq \log_2(x + n) \right\}$$ Let $a_n$ be the number of points in this region satisfying the following conditions. (가) The $x$-coordinate and $y$-coordinate are equal. (나) Both the $x$-coordinate and $y$-coordinate are integers. For example, $a_1 = 2, a_2 = 4$. Find the value of $\sum_{n=1}^{30} a_n$. [4 points]
17. If $x _ { 0 }$ is a solution to the equation $\lg x + x = 2$ , then $x _ { 0 }$ belongs to the interval A. $( 0,1 )$ B. $(1, 1.25)$ C. $(1.25, 1.75)$ D. $( 1.75,2 )$
10. Let $x \in \mathbf{R}$ and $[x]$ denote the greatest integer not exceeding $x$. If there exists a real number $t$ such that $[t] = 1$, $[t^2] = 2$, $\ldots$, $[t^n] = n$ all hold simultaneously, then the maximum value of the positive integer $n$ is A. 3 B. 4 C. 5 D. 6 II. Fill-in-the-Blank Questions: This section has 6 questions. Candidates must answer 5 of them, each worth 5 points, for a total of 25 points. Write your answers in the corresponding positions on the answer sheet. Answers in wrong positions, illegible writing, or ambiguous answers will receive no credit.
Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Show that there exists a unique $n_{0} \in \mathbf{N}$ such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.
Let $x \in \mathbb{R} \backslash \mathbb{Z}$. Let $m \in \mathbb{N}$ such that $m > |x|$. We set, for $n \in \mathbb{N}$ such that $n > m$: $$u_{m,n}(x) = (2n+1)\sin\left(\frac{\pi x}{2n+1}\right) \prod_{k=1}^{m}\left(1 - \frac{\sin^2\left(\frac{\pi x}{2n+1}\right)}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right)$$ and $$v_{m,n}(x) = \prod_{k=m+1}^{n}\left(1 - \frac{\sin^2\left(\frac{\pi x}{2n+1}\right)}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right).$$ Show that the sequences, indexed by $n$, $\left(u_{m,n}(x)\right)_{n > m}$ and $\left(v_{m,n}(x)\right)_{n > m}$ are convergent in $\mathbb{R}^*$.
Let $[ a , b ]$ be a closed bounded interval of $\mathbb { R }$. If $\phi : [ a , b ] \rightarrow [ a , b ]$ is continuous, show that $\phi$ has at least one fixed point.
Let $g : [ 0,1 ] \rightarrow [ 0,1 ]$ be an increasing function (but not necessarily continuous). Show that $g$ has at least one fixed point. Hint: one may consider the set $$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$
Illustrate the construction of the secant method by means of a figure. When $f ^ { \prime } > 0$ on $I$, express $x _ { n + 1 }$ as a function of $x _ { n - 1 } , x _ { n }$ by means of the function $H _ { f }$ defined in question 3 of the third part. (Recall: the secant method initializes with $x_0, x_1 \in I$, and at each step considers the line $L_n$ passing through $(x_{n-1}, f(x_{n-1}))$ and $(x_n, f(x_n))$, defining $x_{n+1}$ as the $x$-intercept of $L_n$.)
We return to the general case, $f$ being any function of class $\mathcal { C } ^ { 3 }$. We assume that $f$ vanishes at a point $x ^ { * } \in I$, for which $f ^ { \prime } \left( x ^ { * } \right) > 0$. (a) Show that there exists $\epsilon > 0$ such that $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] \subset I$ and $f ^ { \prime } > 0$ on the interval $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. We fix such an $\epsilon$ for the rest and we define $$M = \sup _ { ( x , y ) \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] ^ { 2 } } \left| \frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } ( x , y ) \right| .$$ (b) We assume that $x _ { n - 1 } , x _ { n } \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. Show that $$\left| x _ { n + 1 } - x ^ { * } \right| \leqslant M \left| x _ { n - 1 } - x ^ { * } \right| \cdot \left| x _ { n } - x ^ { * } \right| .$$ (c) We fix $\left. \epsilon ^ { \prime } \in \right] 0 , \epsilon ]$ such that $M \epsilon ^ { \prime } < 1$. Show that if $x _ { 0 } , x _ { 1 }$ belong to $\left[ x ^ { * } - \epsilon ^ { \prime } , x ^ { * } + \epsilon ^ { \prime } \right]$ then the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined and converges to $x ^ { * }$.