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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:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q1
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q1.1
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q1.2
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q1.3

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

Question 2.
Find the rectangular form of the following complex numbers.
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q2
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q2.1
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q2.2

Question 3.
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q3
Solution:
(i) (x1 + iy1) (x2 + iy2) (x3 + iy3) …….. (xn + iyn) = a + ib …… (1)
Taking modulus on both sides,
|(x1 + iy1) (x2 + iy2) (x3 + iy3) …….. (xn + iyn)| = |a + ib|
|x1 + iy1| |x2 + iy2| |x3 + iy3| ….. |xn + iyn| = |a + ib|
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q3.1
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q3.2

Question 4.
If \(\frac{1+z}{1-z}\) = cos 2θ + i sin 2θ, show that z = i tan θ.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q4

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
b = cos β + i sin β = e
c = cos γ + i sin γ = e
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
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q5
(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)

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

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) = \(\frac{\pi}{4}\)
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 Q6
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:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 1
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 2
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 3
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 4

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

Question 2.
Find the modulus and principal argument of (1 + i) and hence express it in the polar form.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 55

Question 3.
Express the following complex numbers in the polar form.
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 5
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 6
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 7

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

Question 4.
Express the following complex numbers in the polar form: \(2+2 \sqrt{3} i\)
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 8

Question 5.
Express the following complex numbers in the polar form: \(-1+i \sqrt{3}\)
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 9

Question 6.
Express the following complex numbers in the polar form: -1 – i
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 10

Question 7.
Express the following complex numbers in the polar form: 1 – i
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.7 11

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