If $Z = \frac {(\sqrt {3}+ i)^3 (3i+4)^2}{{(8+6i)^2}}$ then $|Z|$ is equal to
- 0
- 1
- 2
- 3
The Correct Option is C
Solution and Explanation
Z = \(\frac{(√3 + 1)^2(3i+4)^2 }{(8+6i)^2}\) ,
Then using identity |a+ib| = √a2 +b2
and \(\frac{|Z_1Z_2|}{|Z_3|}=\frac{|Z_!||Z_2|}{|Z_3|}\)
Therefore,
|Z| = \(|\frac{(√3 + 1)^2(3i+4)^2 }{(8+6i)^2}|\)
= \(\frac{|(√3 + 1)^2||(3i+4)^2| }{(8+6i)^2}\)
= \(\frac{2^3.5^3}{10^2}\)
= 2
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
