If $z_{1} = 1 -2i ; z_{2} = 1 + i$ and $z_{3 } = 3 + 4i,$ then $ \left( \frac{1}{z_{1}} + \frac{3}{z_{2}}\right) \frac{z_{3}}{z_{2}} = $
- $13 - 6i $
- $13 - 3i $
- $6 - \frac{13}{2} i $
- $\frac{13}{2} - 3 i $
The Correct Option is D
Solution and Explanation
Z1 = 1-2i, Z2 = 1+i, and Z3 = 3+4i
(1/Z1 + 3/Z2) x (Z3/Z2) = (1/1-2i + 3/1+i) x (3+4i/1+i)
= [(1+2i)/5 + 3(1-i)/2] [(3+4i)x(1-i)/2]
Solving this equation one gets,
= (17-11i)/10 + (7+i)/2
= (119+17i-77i+11)/20
= 130-60i/20
= (13/2) - 3i
Any integer that may be expressed as a+ib is referred to be a complex number. Complex numbers, such as +3i and 7+8i, are an example. Here i = -1. This allows us to state that i2 = 1. Therefore, we may use i = -1 for any equation that does not have a true solution.
A polynomial with two roots or one of degree two is referred to as a quadratic equation. A quadratic equation has the generic form y=ax2+bx+c. The real numbers a ≠ 0, b, and c are present here.
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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.