SATHEE: Complex Numbers 4 Question 19

Complex Numbers 4 Question 19

19. For any integer $k$ , let $α_{k} = \cos \frac{k π}{7} + i \sin \frac{k π}{7}$ , where $i = \sqrt{- 1}$ . The value of the expression

$\frac{\sum_{k = 1}^{12} | α_{k + 1} - α_{k} |}{\sum_{k = 1}^{3} | α_{4 k - 1} - α_{4 k - 2} |} is$

(2016 Adv.)

Show Answer

Solution:

Given, $α_{k} = \cos \frac{k π}{7} + i \sin \frac{k π}{7}$

$= \cos \frac{2 k π}{14} + i \sin \frac{2 k π}{14}$

$∴ α_{k}$ are vertices of regular polygon having 14 sides.

Let the side length of regular polygon be $a$ .

$∴ | α_{k + 1} - α_{k} | =$ length of a side of the regular polygon

$= a$

and $| α_{4 k - 1} - α_{4 k - 2} | =$ length of a side of the regular polygon

$∴ \frac{\sum_{k = 1}^{12} | α_{k + 1} - α_{k} |}{\sum_{k = 1}^{3} | α_{4 k - 1} - α_{4 k - 2} |} = \frac{12 (a)}{3 (a)} = 4$

पिछला

अगला

गोपनीयता नीति

द्वारा संचालित Prutor@IITK

sathee@iitk.ac.in

SATHEE का मीडिया कवरेज

खेल स्टोर

ऐप स्टोर

Ask SATHEE

Welcome to SATHEE !
Select from 'Menu' to explore our services, or ask SATHEE to get started. Let's embark on this journey of growth together! 🌐📚🚀🎓

I'm relatively new and can sometimes make mistakes.
If you notice any error, such as an incorrect solution, please use the thumbs down icon to aid my learning.
To begin your journey now, click on "I understand".

Complex Numbers 4 Question 19

19. For any integer k, let αk=cos⁡kπ7+isin⁡kπ7, where i=−1. The value of the expression

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Menu

19. For any integer $k$ , let $α_{k} = \cos \frac{k π}{7} + i \sin \frac{k π}{7}$ , where $i = \sqrt{- 1}$ . The value of the expression