The modulus of the complex number $z=\frac{\left(1-i\sqrt{3}\right)\left(cos\,\theta+i\, sin\,\theta\right)}{2\left(1-i\right)\left(cos\,\theta-i\, sin\,\theta\right)}$ is
- $\frac{1}{2\sqrt{2}}$
- $\frac{1}{\sqrt{3}}$
- $\frac{1}{\sqrt{2}}$
- $2\sqrt{3}$
The Correct Option is C
Solution and Explanation
\(\left|cos\,\theta+i\, sin\,\theta \right|=\left|cos\,\theta-i\, sin\,\theta\right|=1\) \(\therefore\,\left|z\right|=\frac{\left|1-i\sqrt{3}\right|}{2\left|1-i\right|}\) \(=\frac{2}{2\sqrt{2}}\) \(=\frac{1}{\sqrt{2}}\)
Complex numbers have the formula a + ib, where a and b are real numbers and "i" is referred to as the imaginary number. "iota" is another name for it.
If a complex number has the formula z=a+ib, then an is the real component, which is represented by Re z, and b is the imaginary part, represented by Im z.
For instance, Re z=4 and Im z=7 if z=4+i7.
If a = c and b = d, two complex numbers of the form z1 = a + ib and z2 = c + id are said to be equivalent.
a + ib
A number containing both real and imaginary components is referred to as a complex number if it can be represented in the form a + bi, where a and b are real numbers and i is an imaginary unit, satisfying the equation i2 = -1. The complex number's real portion is represented by a, while its imaginary part, b, is used in this equation.
Real numbers and imaginary numbers are combined to create complex numbers.
A complex number is a two-dimensional complex plane that uses a vertical axis for the imaginary component and a horizontal axis for the real component to expand the idea of a one-dimensional number line.
With the aid of complex numbers and the use of the horizontal axis for the real portion and the vertical axis for the imaginary part, the notion of a two-dimensional complex plane is thoroughly described.
- A complex number uses a vertical axis for the imaginary component and a horizontal axis for the real part to extend the idea of a one-dimensional number line to the two-dimensional complex plane.
- The square root of a negative real number is known as an imaginary number, for example, √-2, √-5, etc.
- The quantity √-1 is an imaginary unit and it is represented by the symbol ‘i’.
- They may be used to control alternating currents as well as imitate periodic motions.
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Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)
Then consider:
Statement-I: \( z \) is a real number
Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)
Then:
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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.