Samacheer Kalvi 12th Maths Guide Chapter 2 Complex Numbers Ex 2.1

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 2 Complex Numbers Ex 2.1 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.1

Question 1.
i¹⁹⁴⁷ + i¹⁹⁵⁰
Solution:
i¹⁹⁴⁷ + i¹⁹⁵⁰= (i⁴)⁴⁸⁶ i³ + (i⁴)⁴⁸⁷ i²
[∵ i² = -1; i³ = i; i⁴= 1]
= (1)⁴⁸⁶(-i) + (1)⁴⁸⁷ (-1)
= – i – 1 = -(1 + i)

Question 2.
i¹⁹⁴⁸ – i^-1869
Solution:
= (i⁴)⁴⁸⁷ – [(i⁴)⁴⁶⁷ i¹]^-1
= (1)⁴⁸⁷ – (i)^-1 = 1 – \(\frac{1}{i}\) = 1 + i

Question 3.
\(\sum_{n=1}^{12} i^{n}\)
Solution:
= i¹ + i² + i³ + i⁴ + …. + i¹²
= (i – 1 – i + 1) + (i – 1 – i + 1) + (i – 1 – i + 1)
= 0

Question 4.
i⁵⁹ + \(\frac{1}{i^{59}}\)
Solution:
(-i) + \(\frac{1}{(-i)}\)
[∵ i⁵⁹ = i⁵⁶ i³ = (i⁴)¹⁴ (-i) = 1(-i) = -i]
= -i + i = 0

Question 5.
i i² i³ …… i²⁰⁰⁰
Solution:
= i¹⁺²⁺³⁺²⁰⁰⁰
[∵ 1 + 2 + 3 + …….. n = \(\frac{n(n+1)}{2}\)]
= i^{(\(\frac{2000×2001}{2}\))}
= i^(1000×2001)
= [i¹⁰⁰⁰]²⁰⁰¹= i²⁰⁰¹⁰⁰⁰
= 1

Question 6.
\(\sum_{n=1}^{10} i^{n+50}\)
solution:
= i⁵¹ + i⁵² + i⁵³ + ……. + i⁶⁰= i³ + i⁴ +i + i² + ……. + i⁴
= -i + 1 + i – 1 – i + 1 + i – 1 – i + 1
= 1 – i