Discussion
The modulus and argument of \(\dfrac{1}{-1 + i}\) are :
Answer: A
The modulus is \(1/\sqrt{2}\) and the argument is \(5\pi/4\), option \(\frac{1}{\sqrt{2}}, \frac{5\pi}{4}\).
Multiply numerator and denominator by the conjugate \(-1 - i\):
\[ \begin{aligned} \frac{1}{-1 + i} &= \frac{-1 - i}{(-1)^{2} + 1^{2}} \\ &= -\frac{1}{2} - \frac{1}{2}i \end{aligned} \]
Its modulus is \(\sqrt{1/4 + 1/4} = 1/\sqrt{2}\). The point \((-1/2, -1/2)\) lies in the third quadrant on the line \(y = x\), so its argument is \(\pi + \pi/4 = 5\pi/4\) (equivalently \(-3\pi/4\)).
\(3\pi/4\) is the argument of \(-1 + i\) itself, \(7\pi/4\) points into the fourth quadrant, and \(\sqrt{2}\) is the modulus of \(-1 + i\), not of its reciprocal.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development