Force between balls will be attractive if both balls have opposite charge and Force between ball will be repulsive if both balls have same signs.
Now Given that
Ball A attracts Ball B, So A and B will have opposite signs
Ball A repels Ball C, So A and C will have same signs
Ball D attracts Ball B, So D and B will have opposite signs
Ball D has no effect on E, which means Both D and E will have zero charge on them. (Since ball B attracts ball D is also given, it will be possible because positive or negative charge of B will induce some opposite charge on ball D)
Negative charged rod attracts A, which means charge on A is +ve
Now Since A is +ve, and A attracts B, So It means B is -ve
Now Since A is +ve, and A repels C, So It means C is +ve
So correct Sequence is A = +ve, B = -ve, C = +ve, D = 0, E = 0
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