# QUESTIONS ON CUBE ROOT OF UNITY FOR CLASS 12

Questions on Cube Root of Unity for Class 12

Here we are going to see some example problems to understand properties of modulus of complex numbers.

## Questions on Cube Root of Unity for Class 12 - Questions

Question 1 :

If ω ≠ 1 is a cube root of unity, show that

[(a + b ω + cω2)/(b + c ω + a ω2)] + [(a + b ω + cω2)/(c + a ω + b ω2)]  =  -1

Solution :

L.H.S :

Multiply both numerator and denominator of the first fraction by ω2

=   (ω2/ω2[(a + b ω + cω2)/(b + c ω + a ω2)]

=  [ω2(a + b ω + cω2)/ω2(b + c ω + a ω2)]

=  (aω2 + ω3 + cω4)/ω2(b + c ω + a ω2)

=  (aω2 + + cω)/ω2(b + c ω + a ω2)

=  1/ω ----(1)

Multiply both numerator and denominator of the second fraction by ω

=  (ω/ω[(a + b ω + cω2)/(c + a ω + b ω2)]

=  [ω(a + b ω + cω2)/ω(c + a ω + b ω2)]

=  (aω + bω2 + cω3)/ω(c + a ω + b ω2)

=  (c + a ω + b ω2)/ω(c + a ω + b ω2)

=  1/ω----(2)

(1) + (2)

=   (1/ω2) + (1/ω)

=  (ω + ω2)/ω3

=  -1/1

=  -1  R.H.S

Hence proved.

Question 2 :

Show that Solution :

First let us try to write the given complex numbers in polar form.

[(√3 + i)/2]5

r  =  √[(√3/2)2 + (1/2)2]

r  =  √[(3 + 1)/4]

r  =   1

Argument :

α  = tan-1|y/x|

α  = tan-1|(1/2) / (√3/2)|

=  tan-1|(1/√3)|

α  = π/6

Polar form of the first part,

[(√3 + i)/2] =  1(cosπ/6 + i sin π/6)5

By applying De moiver's theorem, we get

=  (cos 5π/6 + i sin 5π/6)  -----(1)

Similarly, polar form of the second part

[(√3 - i)/2] =  1(cosπ/6 - i sin π/6)5

By applying De moiver's theorem, we get

=  (cos 5π/6 - i sin 5π/6)  -----(2)

(1) + (2)

=  2 cos 5π/6

=  2 cos (150)

=  2 cos (180 - 30) (lies in 2nd quadrant)

=  -2 (√3/2)

=  - √3

Hence proved.

Question 3 :

Find the value of Solution :

Let z  =  sin π/10 + i cos π/10

z bar  =  1/z  =  sin π/10 - i cos π/10

=  [(1 + z) /  (1 + (1/z))]10

=  z10

=  [sin π/10 + i cos π/10]10

=  [cos [(π/2) - (π/10)] + i sin [(π/2) - (π/10)]]10

=  [cos (4π/10) + i sin (4π/10)]10

=  [cos 4π + i sin 4π]

=  1 + i(0)

=  1 After having gone through the stuff given above, we hope that the students would have understood, "Questions on Cube Root of Unity for Class 12".

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