सबसे छोटा धनात्मक पूर्णांक n क्या है जिसके लिए \(\left(\dfrac{1+i}{1-i}\right)^n=1\) है?

  1. 1
  2. 2
  3. 3
  4. 4

Answer (Detailed Solution Below)

Option 4 : 4
Free
Agniveer Navy SSR Full Test - 01
4.1 K Users
100 Questions 100 Marks 60 Mins

Detailed Solution

Download Solution PDF

संकल्पना:

  • संयुक्त इकाई i को \(\rm i=\sqrt{-1}\) के रूप में परिभाषित किया गया है। 
  • i2 = -1, i3 = - i, i4 = 1 इत्यादि 
  • (a + b)2 = a2 + 2ab + b2

 

गणना:

Given that,

\(\left(\frac{1+i}{1 - i}\right)^n = 1\)    ---(1)

Let, y = \(\left(\frac{1+i}{1 - i}\right)\)

\(\Rightarrow \ y=\dfrac{1+i}{1-i}\times \dfrac{1+i}{1+i}\).

\(\Rightarrow\ y\ =\dfrac{(1+i)^2}{1^2-i^2}\)

∵ (a + b)2 = a2 + 2ab + b2

\(\Rightarrow\ y\ =\dfrac{1+2i+i^2}{1-i^2}\)

\(\Rightarrow y\ =\dfrac{2i}{2}=i\)    (∵ i2 = -1)

Therefore, from equation (1)

\( \left(\dfrac{1+i}{1-i}\right)^n=\ i^n\ =\ 1\)

सबसे छोटा धनात्मक पूर्णांक n = 4 है, जिसके लिए in = 1 है। 

Latest Indian Navy Agniveer SSR Updates

Last updated on May 31, 2025

-> The Indian Navy SSR Agniveeer Merit List will be released soon on the official website.

-> The Indian Navy Agniveer SSR CBT Exam was conducted from 25th to 26th May 2025.

->The application window was open from 29th March 2025 to 16th April 2025.  

-> The Indian Navy SSR Agniveer Application Link is active on the official website of the Indian Navy.

.->Only unmarried males and females can apply for Indian Navy SSR Recruitment 2025.

-> The selection process includes a Computer Based Test, Written Exam, Physical Fitness Test (PFT), and  Medical Examination.

->Candidates Qualified in 10+2 with Mathematics & Physics from a recognized Board with a minimum 50% marks in aggregate can apply for the vacancy.

-> With a salary of Rs. 30,000, it is a golden opportunity for defence job seekers.

-> Candidates must go through the Indian Navy SSR Agniveer Previous Year Papers and Agniveer Navy SSR mock tests to practice a variety of questions for the exam.

More Complex Numbers Questions

Get Free Access Now
Hot Links: teen patti flush teen patti win teen patti app teen patti 3a teen patti plus