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Home Question Bank Math Archive 2024 December 24

Math Questions from Dec 24,2024

Browse the Math Q&A Archive for Dec 24,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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а) \( \operatorname{tg} \frac{\alpha}{2}=\frac{1-\cos \alpha}{\sin \alpha} \); Toždestwony subut edin: a) \( 1+2 \cos 2 \alpha+\cos 4 \alpha=4 \cos ^{2} \alpha \cos 2 \alpha \) Solve the equation: \( \sqrt{x + 4} = 6 \) Determine if the functions \( f(x) = 3x + 1 \) and \( g(x) = \frac{x - 1}{3} \) are inverse functions. 3. \( y=\frac{x-x^{2}}{\sqrt{x}} \) If \( X^{-1}=\left[\begin{array}{ccc}1 & -2 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 2\end{array}\right] \), then the sum of elements in the \( 3^{r d} \) column of \( X \) is Exercice 2. On souhaite étudier la fonction \( f \) définic par \[ f(x)=\frac{e^{x}}{1+e^{x}}-\ln \left(1+e^{x}\right) \] On notera \( \mathcal{C}_{f} \) la courbe représentative de \( f \). a) Déterminer l'ensemble de définition ct de dérivabilité de \( f \). b) Calculer \( \lim _{x \rightarrow-\infty} f(x) \). Que peut-on en déduire concernant une asymptote (éventuelle) à \( \mathcal{C}_{f} \) en c) Caleuler \( \lim _{x \rightarrow+\infty} f(x) \). d) On veut dans cette question déterminer l'asymptote cn \( +\infty \) (i) Montrer que, pour tout \( x \in \mathbb{R} \), on a \( f(x)-1+x=\frac{-1}{1+e^{x}}-\ln \left(1+e^{-x}\right) \). (iii) En déduire \( \lim _{x \rightarrow+\infty}(f(x)-1+x) \). La regla de una sucesión es: el primer término de la sucesión es tres y los siguientes términos se obtienen del doble de cada término anterior. ¿Cuál es la sucesión que se obtiene de la regla anterior? \( \begin{array}{ll}\text { A) } 3,9,27,81,243 \ldots & \text { C) } 3,9,15,21,27 \ldots \\ \text { B) } 3,6,12,24,48 \ldots & \text { D) } 3,5,7,9,11 \ldots\end{array} \) \( \begin{array}{l}\text { Индивидуальное задание "Комбинаторика и } \\ \text { теория вероятнстей" } \\ \text {-. Во время спортивной игры по команде ведущего } \\ \text { «становись!» } 10 \text { учеников в случайном порядке } \\ \text { образовали строй в одну шеренгу. Какова вероятность } \\ \text { того, что ученики А и В будут отделенными друг от } \\ \text { друга тремя учениками? }\end{array} \). Maria has 32 postcards. Henry has \( h \) postcards. Maria has 4 times as many postcards as Henry. Which statement is true? A The number of postcards Henry has can be represented by the expression \( 32-4 \). B Henry has 6 postcards. C The number of postcards Henry has can be found by solving the equation \( 32=4 \times h \). D Henry has 128 postcards.
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