AIEEE 2003 Math Solved Paper

Given quadratic equation,
Z2+aZ+b=0
and two roots are Z1 and Z2.
∴Z1+Z2=−a and Z1Z2=b
Question says,
There are three complex numbers:

1. Origin (0)

2. Z1

3. Z2

and they form an equilateral triangle. So They are the vertices of the triangle.
[ Important Point : If Z1,Z2 and Z3 are the vertices of an equilateral triangle then -
Z12+Z22+Z32=Z1Z2+Z2Z3+Z3Z1]
In this question,
Z1=0,Z2=Z1‌ and ‌Z3=Z2
By putting those values in the equation we get,
02+Z12+Z22=0+Z1Z2+0
⇒Z12+Z22=Z1Z2
⇒Z12+Z22=b[. as Z1Z2=b]
⇒(Z1+Z2)2−2Z1Z2=b
⇒(Z1+Z2)2−2b=b
⇒(Z1+Z2)2=3b
⇒(−a)2=3b
⇒a2=3b
So Option (D) is correct.
[ Note : This question is asked to check if you know the following formula -
"If Z1,Z2 and Z3 are the vertices of an equilateral triangle then -
Z12+Z22+Z32=Z1Z2+Z2Z3+Z3Z1‌′′]