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.

i^{1947} + i^{1950}

Solution:

i^{1947} + i^{1950 }= (i^{4})^{486} i³ + (i^{4})^{487} i²

[∵ i² = -1; i³ = i; i^{4 }= 1]

= (1)^{486 }(-i) + (1)^{487} (-1)

= – i – 1 = -(1 + i)

Question 2.

i^{1948} – i^{-1869}

Solution:

= (i^{4})^{487} – [(i^{4})^{467} i^{1}]^{-1}

= (1)^{487} – (i)^{-1} = 1 – \(\frac{1}{i}\) = 1 + i

Question 3.

\(\sum_{n=1}^{12} i^{n}\)

Solution:

= i^{1} + i² + i³ + i^{4} + …. + i^{12}

= (i – 1 – i + 1) + (i – 1 – i + 1) + (i – 1 – i + 1)

= 0

Question 4.

i^{59} + \(\frac{1}{i^{59}}\)

Solution:

(-i) + \(\frac{1}{(-i)}\)

[∵ i^{59} = i^{56} i³ = (i^{4})^{14} (-i) = 1(-i) = -i]

= -i + i = 0

Question 5.

i i² i³ …… i^{2000}

Solution:

= i^{1+2+3+}^{2000}

[∵ 1 + 2 + 3 + …….. n = \(\frac{n(n+1)}{2}\)]

= i^{(\(\frac{2000×2001}{2}\))}

= i^{(1000×2001)}

= [i^{1000}]^{2001
}= i^{2001}^{000}

= 1

Question 6.

\(\sum_{n=1}^{10} i^{n+50}\)

solution:

= i^{51} + i^{52} + i^{53} + ……. + i^{60
}= i^{3} + i^{4} +i + i^{2} + ……. + i^{4}

= -i + 1 + i – 1 – i + 1 + i – 1 – i + 1

= 1 – i