Nombres complexes : point de vue algébrique

QCM Bilan Numéro 1

Exercice 1

Cet exercice est un questionnaire à choix multiples (Q. C. M.) . Pour chacune des questions, une seule des quatre réponses est exacte. Justifier la réponse choisie.
1

La partie imaginaire du nombre complexe z=62iz=6-2i est :
a.\bf{a.} 22                                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 2i2i

c.\bf{c.} 2i-2i                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 2-2

Correction
2

La partie réelle du nombre complexe z=13i2+4iz=\frac{1-3i}{2+4i} est :
a.\bf{a.} 12\frac{1}{2}                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 34-\frac{3}{4}

c.\bf{c.} 34\frac{3}{4}                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 12-\frac{1}{2}

Correction
3

Soit zz un nombre complexe différent de 1-1. La partie imaginaire du nombre complexe Z=1z+1Z=\frac{1}{z+1} est :
a.\bf{a.} 11                                                                                                                                             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} y(x+1)2+y2\frac{y}{\left(x+1\right)^{2} +y^{2} }

c.\bf{c.} y(x+1)2+y2-\frac{y}{\left(x+1\right)^{2} +y^{2} }                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} x+1(x+1)2+y2\frac{x+1}{\left(x+1\right)^{2} +y^{2} }

Correction
4

La forme algébrique de Z=(1i)2(2+i)Z=\left(1-i\right)^{2} \left(2+i\right) est :
a.\bf{a.} 24i2-4i                                                                                                       \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 2+4i2+4i

c.\bf{c.} 2+4i-2+4i                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 24i-2-4i

Correction
5

La forme algébrique du nombre complexe (23i)4\left(2-3i\right)^{4} est :
a.\bf{a.} 119+120i119+120i                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 119+120i-119+120i

c.\bf{c.} 119120i119-120i                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 119120i-119-120i

Correction
6

L'inverse de i7i^{7} est :
a.\bf{a.} ii                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} i-i

c.\bf{c.} 11                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 1-1

Correction
7

Soit zCz\in \mathbb{C} . La solution de l'équation 6z3z=57i6z-3\overline{z}=5-7i est :
a.\bf{a.} 5379i\frac{5}{3}-\frac{7}{9}i                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} z=5379iz=-\frac{5}{3}-\frac{7}{9}i

c.\bf{c.} 53+79i\frac{5}{3}+\frac{7}{9}i                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 53+79i-\frac{5}{3}+\frac{7}{9}i

Correction
8

k=0n(kn)(1)k=\sum _{k=0}^{n}\left(\begin{array}{c} {k} \\ {n} \end{array}\right)\left(-1\right)^{k} =
a.\bf{a.} nn                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 11

c.\bf{c.} 00                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 1-1

Correction
9

On considère le nombre complexe j=12+i32j=-\frac{1}{2} +i\frac{\sqrt{3} }{2} .
Le nombre 1+j+j21+j+j^{2} est égale à :
a.\bf{a.} jj                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 11

c.\bf{c.} 00                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 1-1

Correction
10

Le nombre complexe Z=(13i)2+5(2i)+2Z=\left(1-3i\right)^{2} +5\left(2-i\right)+2 est égale à :
a.\bf{a.} 411i4-11i                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 4+11i4+11i

c.\bf{c.} 4+11i-4+11i                                                                                              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 411i-4-11i

Correction
11

Soit le nombre complexe z=12+12iz=-\frac{1}{2}+\frac{1}{2}i . La valeur de z4z^{4} est :
a.\bf{a.} 14-\frac{1}{4}                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 14\frac{1}{4}

c.\bf{c.} 4-4                                                                                              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 44

Correction
12

Soit z=x+iyz = x + iy avec xx et yy réels. On note ZZ le nombre complexe : Z=2zz2(1+i)z2(1i)z8Z=2z\overline{z}-2\left(1+i\right)z-2\left(1-i\right)\overline{z}-8 . On peut affimer que Z=Z=
a.\bf{a.} 2x2+2y24x+3y+82x^{2} +2y^{2} -4x+3y+8                                                                                           \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 2x2+2y24x+3y82x^{2} +2y^{2} -4x+3y-8

c.\bf{c.} 2x2+2y2+4x+3y82x^{2} +2y^{2} +4x+3y-8                                                                                              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 2x2+2y24x3y82x^{2} +2y^{2} -4x-3y-8

Correction
13

(1+i)8\left(1+i\right)^{8} est égale à :
a.\bf{a.} 16-16                                                                                         \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 16i-16i

c.\bf{c.} 1616                                                                                              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 16i16i

Correction
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