Question

1. $\frac{5+i}{-2i}$

a. $\frac{1}{2}+2\frac{1}{2}i$
b. $-\frac{1}{2}+2\frac{1}{2}i$
c. $\frac{1}{2}-2\frac{1}{2}i$
d. $-\frac{1}{2}+2\frac{1}{2}i$

Explanation:

Step1: Multiply by conjugate of denominator

To rationalize the denominator, multiply numerator and denominator by \(2i\) (the conjugate of \(-2i\)):
$$\frac{5+i}{-2i} \times \frac{2i}{2i} = \frac{(5+i)(2i)}{-2i \times 2i}$$

Step2: Expand numerator and denominator

Calculate the products:
Numerator: \((5+i)(2i) = 10i + 2i^2 = 10i + 2(-1) = -2 + 10i\)
Denominator: \(-2i \times 2i = -4i^2 = -4(-1) = 4\)
Expression becomes: \(\frac{-2 + 10i}{4}\)

Step3: Simplify the fraction

Split and reduce the terms:
$$\frac{-2}{4} + \frac{10i}{4} = -\frac{1}{2} + \frac{5}{2}i = -\frac{1}{2} + 2\frac{1}{2}i$$

Answer:

b. $-\frac{1}{2}+2\frac{1}{2}i$