Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7

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Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7

Question 1.
Write in polar form of the following complex numbers.
(i) 2 + i2√3
(ii) 3 – i√3
(iii) -2 – i2
(iv) \(\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}\)
Solution:

Question 2.
Find the rectangular form of the following complex numbers.

Solution:

Question 3.

Solution:
(i) (x₁ + iy₁) (x₂ + iy₂) (x₃ + iy₃) …….. (x_n + iy_n) = a + ib …… (1)
Taking modulus on both sides,
|(x₁ + iy₁) (x₂ + iy₂) (x₃ + iy₃) …….. (x_n + iy_n)| = |a + ib|
|x₁ + iy₁| |x₂ + iy₂| |x₃ + iy₃| ….. |x_n + iy_n| = |a + ib|

Question 4.
If \(\frac{1+z}{1-z}\) = cos 2θ + i sin 2θ, show that z = i tan θ.
Solution:

Question 5.
If cos α + cos β + cos γ = sin α + sin β + sin γ = 0, then show that
(i) cos 3α + cos 3β + cos 3γ = 3 cos (α + β + γ)
(ii) sin 3α + sin 3β + sin 3γ = 3 sin (α + β + γ)
Solution:
Let a = cos α + i sin α = e^iα
b = cos β + i sin β = e^iβ
c = cos γ + i sin γ = e^iγ
a + b + c = (cos α + cos β + cos γ) + i (sin α + sin β + sin γ)
⇒ a + b + c = 0 + i 0
⇒ a + b + c = 0
If a + b + c = 0 then a³ + b³ + c³ = 3abc

(cos 3α + i sin 3α + cos 3β + i sin 3β + cos 3γ + i sin 3γ) = 3 [cos (α + β + γ) + i sin (α + β + γ)]
(cos 3α + cos 3β + cos 3γ) + i (sin 3α + sin 3β + sin 3γ) = 3 cos (α + β + γ) + i 3sin(α + β + γ)
Equating real and Imaginary parts
(i) cos 3α + cos 3β + cos 3γ = 3 cos (α + β + γ)
(ii) sin 3α + sin 3β + sin 3γ = 3 sin (α + β + γ)

Question 6.
If z = x + iy and arg \(\left(\frac{z-i}{z+2}\right)=\frac{\pi}{4}\), then show that x² + y² + 3x – 3y + 2 = 0.
Solution:
arg \(\left(\frac{z-i}{z+2}\right)=\frac{\pi}{4}\)
We have arg (\(\frac{z_{1}}{z_{2}}\)) = arg(z₁) – arg(z₂)
arg (z – i) – arg (z + 2) = \(\frac{\pi}{4}\)
Let z = x + iy
arg (x + iy – i) – arg (x + iy + 2) = \(\frac{\pi}{4}\)
arg(x + i(y – 1)) – arg(x + 2 + iy) = \(\frac{\pi}{4}\)

2y – x – 2 = x² + y² + 2x – y
x² + y² + 3x – 3y + 2 = 0
Hence proved.

Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Additional Problems

Question 1.
Write the following complex numbers in the polar form:

Solution:

Question 2.
Find the modulus and principal argument of (1 + i) and hence express it in the polar form.
Solution:

Question 3.
Express the following complex numbers in the polar form.

Solution:

Question 4.
Express the following complex numbers in the polar form: \(2+2 \sqrt{3} i\)
Solution:

Question 5.
Express the following complex numbers in the polar form: \(-1+i \sqrt{3}\)
Solution:

Question 6.
Express the following complex numbers in the polar form: -1 – i
Solution:

Question 7.
Express the following complex numbers in the polar form: 1 – i
Solution: