Exercise 4 - Candidates who have NOT followed the specialization course
For each of the five following propositions, indicate whether it is true or false and justify the answer chosen. One point is awarded for each correct answer correctly justified. An unjustified answer is not taken into account. An absence of answer is not penalized.
Proposition 1:
In the complex plane equipped with an orthonormal coordinate system, the points A, B and C with affixes respectively $z _ { \mathrm { A } } = \sqrt { 2 } + 3 \mathrm { i } , z _ { \mathrm { B } } = 1 + \mathrm { i }$ and $z _ { \mathrm { C } } = - 4 \mathrm { i }$ are not collinear.
Proposition 2:
There does not exist a non-zero natural integer $n$ such that $[ \mathrm { i } ( 1 + \mathrm { i } ) ] ^ { 2n }$ is a strictly positive real number.
Proposition 3:
ABCDEFGH is a cube with side 1. The point L is such that $\overrightarrow { \mathrm { EL } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { EF } }$. The section of the cube by the plane (BDL) is a triangle.
Proposition 4:
ABCDEFGH is a cube with side 1. The point L is such that $\overrightarrow { \mathrm { EL } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { EF } }$. The triangle DBL is right-angled at B.
Proposition 5:
We consider the function $f$ defined on the interval [2;5] and whose variation table is given below:
| $x$ | 2 | 3 | 4 | 5 |
| \begin{tabular}{ c } Variations |
| $\operatorname { of } f$ |
& 3 & & & 2 & & & 0 & 1 \end{tabular}
The integral $\int _ { 2 } ^ { 5 } f ( x ) \mathrm { d } x$ is between 1,5 and 6.