The complex number $z_1 = a + bi$, where $i^2 = -1$, is shown on the Argand Diagram below - Leaving Cert Mathematics - Question 2 - 2015
Question 2
The complex number $z_1 = a + bi$, where $i^2 = -1$, is shown on the Argand Diagram below.
(i) Write down the value of $a$ and the value of $b$.
a = _______ b = _... show full transcript
Worked Solution & Example Answer:The complex number $z_1 = a + bi$, where $i^2 = -1$, is shown on the Argand Diagram below - Leaving Cert Mathematics - Question 2 - 2015
Step 1
Write down the value of $a$ and the value of $b$.
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Answer
From the Argand diagram for z1=a+bi, we can see that:
The x-coordinate (real part) corresponds to a, which is 3.
The y-coordinate (imaginary part) corresponds to b, which is −1.
Thus, we have:
a = 3
b = -1.
Step 2
Plot $z_2$ on the Argand Diagram.
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Answer
To plot z2=−1+2i on the Argand diagram:
Start at the origin (0,0).
Move left to the value −1 on the real axis.
From there, move up to 2 on the imaginary axis.
Mark the point (−1,2).
Step 3
Write $z_3$ in the form $x + yi$.
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Answer
To calculate z3=z2z1, we substitute z1=3−i and z2=−1+2i:
First, multiply the numerator and the denominator by the conjugate of the denominator:
z3=(−1+2i)(−1−2i)(3−i)(−1−2i)
Calculate the denominator:
(−1+2i)(−1−2i)=1+4=5
Calculate the numerator:
(3−i)(−1−2i)=−3−6i+i+2=−1−5i
Then, we have:
z3=5−1−5i=−51−i
Therefore, in the form x+yi:
z3=−51−i.
Step 4
Solve for $z$:
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Answer
To solve 2z−6(4−6i)=(−1+i)(4−2i):
Simplifying the left side:
2z−24+36i
Expanding the right side:
(−1+i)(4−2i)=−4+2i+4i−2i2=−4+6i+2=−2+6i
