JEE Advance - Mathematics (2011 - Paper 2 Offline - No. 2)
Let $$\omega = {e^{{{i\pi } \over 3}}}$$, and a, b, c, x, y, z be non-zero complex numbers such that
$$a + b + c = x$$
$$a + b\omega + c{\omega ^2} = y$$
$$a + b{\omega ^2} + c\omega = z$$
$$a + b + c = x$$
$$a + b\omega + c{\omega ^2} = y$$
$$a + b{\omega ^2} + c\omega = z$$
Then the value of $${{{{\left| x \right|}^2} + {{\left| y \right|}^2} + {{\left| z \right|}^2}} \over {{{\left| a \right|}^2} + {{\left| b \right|}^2} + {{\left| c \right|}^2}}}$$ is
Answer
3
Explanation
The expression may not attain integral value for all a, b, c.
If we consider a = b = c, then
x = 3a
y = a(1 + $$\omega$$ + $$\omega$$2) = a(1 + i$$\sqrt3$$)
z = a(1 + $$\omega$$2 + $$\omega$$) = a(1 + i$$\sqrt3$$)
Therefore, $$|x{|^2} + |y{|^2} + |z{|^2} = 9|a{|^2} + 4|a{|^2} + 4|a{|^2} = 17|a{|^2}$$
Hence, $${{|x{|^2} + |y{|^2} + |z{|^2}} \over {|a{|^2} + |b{|^2} + |c{|^2}}} = {{17} \over {13}}$$
Note : However, if $$\omega = {e^{i(2\pi /3)}}$$, then the value of the expression is 3.
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