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: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ In this question, round both results to the nearest thousandth. a. Calculate $f(20)$. b. Determine the maximum rate of $\mathrm{CO}_2$ present in the room during the experiment.
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?
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: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ Let $V_m$ denote the average rate (as a percentage) of $\mathrm{CO}_2$ present in the room during the first 11 minutes of operation of the extractor hood. a. Let $F$ be the function defined on the interval $[0;11]$ by: $$F(t) = (-1{,}6t - 3{,}6)\mathrm{e}^{-0{,}5t} + 0{,}03t.$$ Show that the function $F$ is an antiderivative of the function $f$ on the interval $[0;11]$. b. Deduce the average rate $V_m$, the average value of the function $f$ on the interval $[0;11]$. Round the result to the nearest thousandth, that is to $0.1\%$.
A type of oscilloscope has a lifespan, expressed in years, which can be modelled by a random variable $D$ that follows an exponential distribution with parameter $\lambda$. It is known that the average lifespan of this type of oscilloscope is 8 years. Statement 1: for an oscilloscope of this type chosen at random and having already operated for 3 years, the probability that the lifespan is greater than or equal to 10 years, rounded to the nearest hundredth, is equal to 0.42. Recall that if $X$ is a random variable that follows an exponential distribution with parameter $\lambda$, then for all positive real $t$: $P(X \leqslant t) = 1 - \mathrm{e}^{-\lambda t}$. Indicate whether Statement 1 is true or false, justifying your answer.
In 2016, in France, law enforcement carried out 9.8 million alcohol screening tests with motorists, and $3.1\%$ of these tests were positive. In a given region, on 15 June 2016, a gendarmerie unit conducted screening on 200 motorists. Statement 2: rounding to the nearest hundredth, the probability that, out of the 200 tests, there were strictly more than 5 positive tests, is equal to 0.59. Indicate whether Statement 2 is true or false, justifying your answer.
Consider in $\mathbb{R}$ the equation: $$\ln(6x - 2) + \ln(2x - 1) = \ln(x)$$ Statement 3: the equation has two solutions in the interval $]\frac{1}{2}; +\infty[$. Indicate whether Statement 3 is true or false, justifying your answer.
Consider in $\mathbb{C}$ the equation: $$\left(4z^2 - 20z + 37\right)(2z - 7 + 2i) = 0$$ Statement 4: the solutions of the equation are the affixes of points belonging to the same circle with centre the point P with affix 2. Indicate whether Statement 4 is true or false, justifying your answer.
A fruit and vegetable retailer buys melons from market gardener A. The mass in grams of melons from market gardener A is modelled by a random variable $M_\mathrm{A}$ that follows a uniform distribution on the interval $[850; x]$, where $x$ is a real number greater than 1200. Melons are described as ``compliant'' if their mass is between 900 g and 1200 g. The retailer observes that $75\%$ of melons from market gardener A are compliant. Determine $x$.
The mass in grams of melons from market gardener B is modelled by a random variable $M_\mathrm{B}$ that follows a normal distribution with mean 1050 and unknown standard deviation $\sigma$. Melons are described as ``compliant'' if their mass is between 900 g and 1200 g. The retailer observes that $85\%$ of melons supplied by market gardener B are compliant. Determine the standard deviation $\sigma$ of the random variable $M_\mathrm{B}$. Give the value rounded to the nearest integer.
QIII.A.3
Hypothesis test of binomial distributionsView
Market gardener C claims that $80\%$ of the melons in his production are compliant (mass between 900 g and 1200 g). The retailer doubts this claim. He observes that out of 400 melons delivered by this market gardener during one week, only 294 are compliant. Is the retailer right to doubt the claim of market gardener C?
A customer is chosen at random from those who bought a melon during week 1. Among customers who buy a melon in a given week, $90\%$ of them buy a melon the following week; among customers who do not buy a melon in a given week, $60\%$ of them do not buy a melon the following week. For $n \geqslant 1$, we denote by $A_n$ the event: ``the customer buys a melon during week $n$''. Thus $p(A_1) = 1$. a. Reproduce and complete the probability tree below, relating to the first three weeks. b. Prove that $p(A_3) = 0{,}85$. c. Given that the customer buys a melon during week 3, what is the probability that he bought one during week 2? Round to the nearest hundredth.
A customer is chosen at random from those who bought a melon during week 1. Among customers who buy a melon in a given week, $90\%$ of them buy a melon the following week; among customers who do not buy a melon in a given week, $60\%$ of them do not buy a melon the following week. For $n \geqslant 1$, we set $p_n = P(A_n)$, where $A_n$ is the event ``the customer buys a melon during week $n$''. Thus $p_1 = 1$. Prove that, for all integer $n \geqslant 1$: $p_{n+1} = 0{,}5\, p_n + 0{,}4$.
For $n \geqslant 1$, we set $p_n = P(A_n)$, where $A_n$ is the event ``the customer buys a melon during week $n$'', with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$ for all $n \geqslant 1$. a. Prove by induction that, for all integer $n \geqslant 1$: $p_n > 0{,}8$. b. Prove that the sequence $(p_n)$ is decreasing. c. Is the sequence $(p_n)$ convergent?
For $n \geqslant 1$, we set $p_n = P(A_n)$ with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$. We set for all integer $n \geqslant 1$: $v_n = p_n - 0{,}8$. a. Prove that $(v_n)$ is a geometric sequence and give its first term $v_1$ and common ratio. b. Express $v_n$ as a function of $n$. Deduce that, for all $n \geqslant 1$, $p_n = 0{,}8 + 0{,}2 \times 0{,}5^{n-1}$. c. Determine the limit of the sequence $(p_n)$.
The figure below represents a cube ABCDEFGH. The three points I, J, K are defined by the following conditions:
I is the midpoint of segment [AD];
J is such that $\overrightarrow{\mathrm{AJ}} = \frac{3}{4}\overrightarrow{\mathrm{AE}}$;
K is the midpoint of segment [FG].
On the figure provided in the appendix, construct without justification the point of intersection P of the plane (IJK) and the line (EH). Leave the construction lines on the figure.
Deduce from this, by justifying, the intersection of the plane (IJK) and the plane (EFG).
The figure below represents a cube ABCDEFGH. The three points I, J, K are defined by the following conditions:
I is the midpoint of segment [AD];
J is such that $\overrightarrow{\mathrm{AJ}} = \frac{3}{4}\overrightarrow{\mathrm{AE}}$;
K is the midpoint of segment [FG].
We place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.
a. Give without justification the coordinates of points I, J and K. b. Determine the real numbers $a$ and $b$ such that the vector $\vec{n}(4; a; b)$ is orthogonal to the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{IK}}$. c. Deduce that a Cartesian equation of the plane (IJK) is: $4x - 6y - 4z + 3 = 0$.
a. Give a parametric representation of the line (CG). b. Calculate the coordinates of point N, the intersection of the plane (IJK) and the line (CG). c. Place point N on the figure and construct in colour the cross-section of the cube by the plane (IJK).
The figure below represents a cube ABCDEFGH with the plane (IJK) having Cartesian equation $4x - 6y - 4z + 3 = 0$ in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. We denote by R the orthogonal projection of point F onto the plane (IJK). Point R is therefore the unique point of the plane (IJK) such that the line (FR) is orthogonal to the plane (IJK). We define the interior of the cube as the set of points $M(x; y; z)$ such that $\left\{\begin{array}{l} 0 < x < 1 \\ 0 < y < 1 \\ 0 < z < 1 \end{array}\right.$ Is point R inside the cube?