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 = -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