The number of elements in the set { \(z = a + ib ∈ C : a,b ∈ Z\) and \(1 < | z - 3 + 2i | < 4\) } is _______ .
Correct Answer: 40
Solution and Explanation
At line y = –2, we have (5, –2) (6, –2) (1, –2) (0, –2)
⇒ 4 points
At line y = –1, we have (4, –1) (5, –1) (6, –1) (2, –1) (1, –1) (0, –1)
⇒ 6 points
At line y = 0, we have (0, 0) (1, 0) (2, 0) (3, 0) (4, 0) (5, 0) (6, 0)
⇒ 7 points
At line y = 1, we have (1, 1), (2, 1), (3, 1), (4, 1), (5, 1) i.e. 5 points
Similarly,
At line y = –5, we have 5 points
At line y = –4, we have 7 points
At line y = –3, we have 6 points
Then, the total integral points = 2(5 + 7 + 6) + 4 = 40
So, the answer is 40.
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.