Q3
6 marks
Vectors 3D & Lines
Line-Plane Intersection
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This exercise is a multiple choice questionnaire. For each question, four answers are proposed and only one of them is correct. No justification is required. 1.5 points are awarded for each correct answer. No points are deducted for no answer or an incorrect answer.
Question 1 In space with an orthonormal coordinate system ($\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k}$), we consider the line ($D$) with parametric representation $\left\{\begin{array}{l} x = 2 + t \\ y = 1 - 3t \\ z = 2t \end{array} \quad (t \in \mathbb{R})\right.$, and the plane $(P)$ with Cartesian equation $x + y + z - 3 = 0$.
We can assert that: Answer A: the line ($D$) and the plane ($P$) are strictly parallel. Answer B: the line ($D$) is contained in the plane ($P$). Answer C: the line ($D$) and the plane ($P$) intersect at the point with coordinates $(4; -5; 4)$. Answer D: the line ($D$) and the plane ($P$) are orthogonal.
Question 2 In the computer department of a large store, only one salesperson is present and there are many customers. We assume that the random variable $T$, which associates to each customer the waiting time in minutes for the salesperson to be available, follows an exponential distribution with parameter $\lambda$. The average waiting time is 20 minutes. Given that a customer has already waited 20 minutes, the probability that their total waiting time exceeds half an hour is: Answer A: $\mathrm{e}^{-\frac{1}{2}}$ Answer B: $\mathrm{e}^{-\frac{3}{2}}$ Answer C: $1 - \mathrm{e}^{-\frac{1}{2}}$ Answer D: $1 - \mathrm{e}^{-10\lambda}$
Question 3
A factory manufactures tennis balls in large quantities. To comply with international competition regulations, the diameter of a ball must be between $63.5\text{ mm}$ and $66.7\text{ mm}$. We denote by $D$ the random variable which associates to each ball produced its diameter measured in millimetres. We assume that $D$ follows a normal distribution with mean 65.1 and standard deviation $\sigma$. We call $P$ the probability that a ball chosen at random from the total production is compliant. The factory decides to adjust the machines so that $P$ equals 0.99. The value of $\sigma$, rounded to the nearest hundredth, allowing this objective to be achieved is: Answer A: 0.69 \quad Answer B: 2.58 \quad Answer C: 0.62 \quad Answer D: 0.80
Question 4 The curve below is the graph, in an orthonormal coordinate system, of the function $f$ defined by: $$f(x) = \frac{4x}{x^{2} + 1}$$ The exact value of the positive real number $a$ such that the line with equation $x = a$ divides the shaded region into two regions of equal area is: Answer A: $\sqrt{\sqrt{\frac{3}{2}}}$ \quad Answer B: $\sqrt{\sqrt{5} - 1}$ \quad Answer C: $\ln 5 - 0.5$ \quad Answer D: $\frac{10}{9}$