Paragraph: Let be the complex number representing the point M ( -1 2 , 3 2 ) and a, b, c, , , be non-zero complex numbers such that aligned & a+b+c= & a+b +c ^2= & a+b ^2+c = . aligned Question: Number of distinct complex numbers z satisfying the equation: (z+1) | array cc z+ ^2 & 1 1 & z+ array |+ | array cc 1 & z+ & ^2 array |+ ^2 | array cc & z+ ^2 ^2 & 1 array |=0 is equal to:

Step-by-step solution

(i) aligned & Given, a+b+c= , a+b +c ^2= , a+b ^2+c = & | |^2+| |^2+| |^2= + + [ = ^2 ] & aligned Now, aligned & =(a+b+c)( a + b + c ) [ ^2= ] & =|a|^2+|b|^2+|c|^2+a b +a c +b a +b c +c a +c b & = (a+b +c ^2 ) ( a + b ^2+ c ) aligned aligned & =|a|^2+|b|^2+|c|^2+a b ^2+a c +b a +b c ^2+c a ^2+c b & = (a+b ^2+c ) ( a + b + c ^2 ) & =|a|^2+|b|^2+|c|^2+a b +a c ^2+b a ^2+b c +c a +c b ^2 aligned So, array rlc + + & =3 (|a|^2+|b|^2+|c|^2 )+(a b +a c +b a +b c +c a +c b ) (1+ + ^2 ) & =3 (|a|^2+|b|^2+|c|^2 ) & [ 1+ + ^2