For Complex Lovers

Algebra Level 5

Z 1 , Z 2 , Z 3 ∈ C ∣ Z 1 ∣ = ∣ Z 2 ∣ = ∣ Z 3 ∣ = 1 ∑ cyclic 1 , 2 , 3 Z 1 2 Z 2 Z 3 = − 1 \displaystyle{{ Z }_{ 1 },{ Z }_{ 2 },{ Z }_{ 3 }\in C\quad \\ \left| { Z }_{ 1 } \right| =\left| { Z }_{ 2 } \right| =\left| { Z }_{ 3 } \right| =1\\ \sum _{ \text{cyclic} }^{ 1,2,3 }{ \cfrac { { { Z }_{ 1 } }^{ 2 } }{ { { Z }_{ 2 } }{ { Z }_{ 3 } } } } =-1}

Let a = ∣ Z 1 + Z 2 + Z 3 ∣ \displaystyle{a=\left| { Z }_{ 1 }+{ Z }_{ 2 }+{ Z }_{ 3 } \right| }

Let an Set A A contains all possible values of a a . Then find the value of ∑ a ∈ A a \displaystyle{\sum _{ a\in A }^{ }{ a } }

This is not original.


The answer is 3.

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Karan Shekhawat by Karan Shekhawat , 25, India

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1 solution

Nishu Sharma
Apr 29, 2015

It is simple Problem , But Problem setter Makes it harder in looks , but it is So simple .

I'am giving only Hint's (which are sufficient for a maths student)

Try to use following identities :

1 1 - ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 ( a . b + b . c + c . a ) (a+b+c)^2=a^2+b^2+c^2+2(a.b+b.c+c.a)

2 2 - Z Z ˉ = ∣ Z ∣ 2 Z\bar { Z } ={ \left| Z \right| }^{ 2 }

You will Calculate easily .. just give a try !

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you please solve it totally

Ram Sita - 3 years, 8 months ago

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