**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.

**Question 1 :**

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

[(a + b ω + cω^{2})/(b + c ω + a ω^{2})] + [(a + b ω + cω^{2})/(c + a ω + b ω^{2})] = -1

**Solution :**

L.H.S :

Multiply both numerator and denominator of the first fraction by ω^{2}

= (ω^{2}/ω^{2}) [(a + b ω + cω^{2})/(b + c ω + a ω^{2})]

= [ω^{2}(a + b ω + cω^{2})/ω^{2}(b + c ω + a ω^{2})]

= (aω^{2 + }b ω^{3 }+ cω^{4})/ω^{2}(b + c ω + a ω^{2})

= (aω^{2 + }b + cω)/ω^{2}(b + c ω + a ω^{2})

= 1/ω^{2 } ----(1)

Multiply both numerator and denominator of the second fraction by ω

= (ω/ω) [(a + b ω + cω^{2})/(c + a ω + b ω^{2})]

= [ω(a + b ω + cω^{2})/ω(c + a ω + b ω^{2})]

= (aω + bω^{2 }+ cω^{3})/ω(c + a ω + b ω^{2})

= (c + a ω + b ω^{2})/ω(c + a ω + b ω^{2})

= 1/ω----(2)

(1) + (2)

= (1/ω^{2}) + (1/ω)

= (ω + ω^{2})/ω^{3}

= -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]^{5 } = 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]^{5 } = 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 2^{nd} 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}

= z^{10}

= [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

Hence the answer is 1.

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