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Sagot :
Given two complex numbers, [tex]\( z_1 = -i \)[/tex] and [tex]\( z_2 = 2 + i \sqrt{3} \)[/tex].
### Part (a)
First, we will express [tex]\( z_1^2 \)[/tex] and [tex]\( \overline{z_2} \)[/tex] in the form [tex]\( a + bi \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are real numbers.
#### 1. Calculate [tex]\( z_1^2 \)[/tex]:
[tex]\[ z_1 = -i \][/tex]
[tex]\[ z_1^2 = (-i)^2 \][/tex]
Knowing that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ z_1^2 = (-i)^2 = (-i)(-i) = i^2 = -1 \][/tex]
Therefore, in the form [tex]\( a + bi \)[/tex], we have:
[tex]\[ z_1^2 = -1 + 0i \][/tex]
Thus, [tex]\( a = -1 \)[/tex] and [tex]\( b = 0 \)[/tex].
#### 2. Calculate the conjugate of [tex]\( z_2 \)[/tex]:
[tex]\[ z_2 = 2 + i \sqrt{3} \][/tex]
The conjugate of [tex]\( z_2 \)[/tex] is obtained by changing the sign of the imaginary part:
[tex]\[ \overline{z_2} = 2 - i \sqrt{3} \][/tex]
Expressing this in the form [tex]\( a + bi \)[/tex]:
[tex]\[ \overline{z_2} = 2 + (-\sqrt{3})i \][/tex]
Thus, [tex]\( a = 2 \)[/tex] and [tex]\( b = -\sqrt{3} \)[/tex].
So, for part (a), we have:
- [tex]\( z_1^2 \)[/tex] in the form [tex]\( a + bi \)[/tex] is [tex]\( -1 + 0i \)[/tex].
- [tex]\( \overline{z_2} \)[/tex] in the form [tex]\( a + bi \)[/tex] is [tex]\( 2 - \sqrt{3}i \)[/tex].
### Part (b)
Using the results from part (a), we need to find [tex]\( w \)[/tex] given by:
[tex]\[ w = \frac{z_1^2 + \overline{z_2}}{z_1} \][/tex]
Substitute [tex]\( z_1^2 \)[/tex] and [tex]\( \overline{z_2} \)[/tex] into the equation:
[tex]\[ w = \frac{(-1 + 0i) + (2 - \sqrt{3}i)}{-i} \][/tex]
First, combine the numerator:
[tex]\[ z_1^2 + \overline{z_2} = (-1 + 0i) + (2 - \sqrt{3}i) = 1 - \sqrt{3}i \][/tex]
Next, divide by [tex]\( z_1 \)[/tex]:
[tex]\[ w = \frac{1 - \sqrt{3}i}{-i} \][/tex]
To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ w = \frac{(1 - \sqrt{3}i)(i)}{-i(i)} \][/tex]
[tex]\[ w = \frac{i - i^2 \sqrt{3}}{-i^2} \][/tex]
[tex]\[ w = \frac{i - (-1) \sqrt{3}}{1} \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ w = i + \sqrt{3} \][/tex]
Express [tex]\( w \)[/tex] as:
[tex]\[ w = \sqrt{3} + i \][/tex]
#### Calculate the magnitude [tex]\( |w| \)[/tex]:
The magnitude of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[ |w| = \sqrt{a^2 + b^2} \][/tex]
For [tex]\( w = \sqrt{3} + i \)[/tex]:
[tex]\[ |w| = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \][/tex]
#### Calculate the argument [tex]\( \arg(w) \)[/tex]:
The argument of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[ \arg(w) = \tan^{-1} \left( \frac{b}{a} \right) \][/tex]
For [tex]\( w = \sqrt{3} + i \)[/tex]:
[tex]\[ \arg(w) = \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) \][/tex]
[tex]\[ \arg(w) = \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) = \frac{\pi}{6} \text{ radians} \][/tex]
In conclusion:
- [tex]\( w = \sqrt{3} + i \)[/tex]
- The magnitude [tex]\( |w| = 2 \)[/tex]
- The argument [tex]\( \arg(w) = \frac{\pi}{6} \)[/tex] radians
### Part (a)
First, we will express [tex]\( z_1^2 \)[/tex] and [tex]\( \overline{z_2} \)[/tex] in the form [tex]\( a + bi \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are real numbers.
#### 1. Calculate [tex]\( z_1^2 \)[/tex]:
[tex]\[ z_1 = -i \][/tex]
[tex]\[ z_1^2 = (-i)^2 \][/tex]
Knowing that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ z_1^2 = (-i)^2 = (-i)(-i) = i^2 = -1 \][/tex]
Therefore, in the form [tex]\( a + bi \)[/tex], we have:
[tex]\[ z_1^2 = -1 + 0i \][/tex]
Thus, [tex]\( a = -1 \)[/tex] and [tex]\( b = 0 \)[/tex].
#### 2. Calculate the conjugate of [tex]\( z_2 \)[/tex]:
[tex]\[ z_2 = 2 + i \sqrt{3} \][/tex]
The conjugate of [tex]\( z_2 \)[/tex] is obtained by changing the sign of the imaginary part:
[tex]\[ \overline{z_2} = 2 - i \sqrt{3} \][/tex]
Expressing this in the form [tex]\( a + bi \)[/tex]:
[tex]\[ \overline{z_2} = 2 + (-\sqrt{3})i \][/tex]
Thus, [tex]\( a = 2 \)[/tex] and [tex]\( b = -\sqrt{3} \)[/tex].
So, for part (a), we have:
- [tex]\( z_1^2 \)[/tex] in the form [tex]\( a + bi \)[/tex] is [tex]\( -1 + 0i \)[/tex].
- [tex]\( \overline{z_2} \)[/tex] in the form [tex]\( a + bi \)[/tex] is [tex]\( 2 - \sqrt{3}i \)[/tex].
### Part (b)
Using the results from part (a), we need to find [tex]\( w \)[/tex] given by:
[tex]\[ w = \frac{z_1^2 + \overline{z_2}}{z_1} \][/tex]
Substitute [tex]\( z_1^2 \)[/tex] and [tex]\( \overline{z_2} \)[/tex] into the equation:
[tex]\[ w = \frac{(-1 + 0i) + (2 - \sqrt{3}i)}{-i} \][/tex]
First, combine the numerator:
[tex]\[ z_1^2 + \overline{z_2} = (-1 + 0i) + (2 - \sqrt{3}i) = 1 - \sqrt{3}i \][/tex]
Next, divide by [tex]\( z_1 \)[/tex]:
[tex]\[ w = \frac{1 - \sqrt{3}i}{-i} \][/tex]
To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[ w = \frac{(1 - \sqrt{3}i)(i)}{-i(i)} \][/tex]
[tex]\[ w = \frac{i - i^2 \sqrt{3}}{-i^2} \][/tex]
[tex]\[ w = \frac{i - (-1) \sqrt{3}}{1} \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ w = i + \sqrt{3} \][/tex]
Express [tex]\( w \)[/tex] as:
[tex]\[ w = \sqrt{3} + i \][/tex]
#### Calculate the magnitude [tex]\( |w| \)[/tex]:
The magnitude of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[ |w| = \sqrt{a^2 + b^2} \][/tex]
For [tex]\( w = \sqrt{3} + i \)[/tex]:
[tex]\[ |w| = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \][/tex]
#### Calculate the argument [tex]\( \arg(w) \)[/tex]:
The argument of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[ \arg(w) = \tan^{-1} \left( \frac{b}{a} \right) \][/tex]
For [tex]\( w = \sqrt{3} + i \)[/tex]:
[tex]\[ \arg(w) = \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) \][/tex]
[tex]\[ \arg(w) = \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) = \frac{\pi}{6} \text{ radians} \][/tex]
In conclusion:
- [tex]\( w = \sqrt{3} + i \)[/tex]
- The magnitude [tex]\( |w| = 2 \)[/tex]
- The argument [tex]\( \arg(w) = \frac{\pi}{6} \)[/tex] radians
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