IIT JEE , AIEEE Forum - Mathematics , Algebra

hit_ur_heart

there isnt much to do in this , proceed this way..................

sin(x1)z^3 + sin(x2)z^2 + sin(x3)z + sin(x4) = 3
taking modulus on both side, we get
|sin(x1)z^3 + sin(x2)z^2 + sin(x3)z + sin(x4)| = |3|
but
|sin(x1)z^3 + sin(x2)z^2 + sin(x3)z + sin(x4)| <= |sin(x1)z^3| + |sin(x2)z^2| + |sin(x3)z| + |sin(x4)| .

but as we know that |sin(theta)| <= 1, for any value of theta.
=> 3 <= |z^3| + |z^2| +|z| + 1
=> |z|^3 + |z|^2 + |z| - 2 >= 0
now |z| is non-negative real number ,

Consider the function f(x) = x^3 + x^2 + x - 2
as this function will be strictly increasing ( verify it) , there will be only one root , let it be "r".

=> (|z| -r) ( some quadratic function of |z| with no real root ) >= 0
=> |z| >= r
Since maix value of |z| given in option is 2/3 , but f( 2/3) < 0
=> r > 2/3
=> |z| > 2/3
If u have any query regarding any step , u can post back

Ask & Discuss Questions with Community & Experts