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(D) $1$. When a conductor is earthed,its potential becomes zero. Therefore,the potential at sphere $B$ is zero. This makes statement $(B)$ correct.$2$

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$B$ and $D$

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(D) $1$. When a conductor is earthed,its potential becomes zero. Therefore,the potential at sphere $B$ is zero. This makes statement $(B)$ correct.$2$. Due to the presence of the negatively charged sphere $A$ near sphere $B$,induction occurs. The negative charge on $A$ will attract positive charges to the inner surface of $B$ and repel negative charges to the earth. Thus,sphere $B$ will have a non-zero induced charge.$3$. Because sphere $B$ is placed very close to $A$,the electric field produced by the induced charges on $B$ will affect the distribution of charge on $A$. The charge on $A$ will be attracted towards the side facing $B$,causing the charge on $A$ to be non-uniformly distributed. This makes statement $(D)$ correct.$4$. Therefore,the correct statements are $(B)$ and $(D)$.

Two spheres $A$ and $B$ are kept very near to each other. $A$ is negatively charged and $B$ is earthed. The true statements are:
$(A)$ Charge on $B$ is zero
$(B)$ Potential at $B$ is zero
$(C)$ Charge is uniformly distributed on $A$
$(D)$ Charge is non-uniformly distributed on $A$

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Two spheres $A$ and $B$ are kept very near to each other. $A$ is negatively charged and $B$ is earthed. The true statements are:$(A)$ Charge on $B$ is zero$(B)$ Potential at $B$ is zero$(C)$ Charge is uniformly distributed on $A$$(D)$ Charge is non-uniformly distributed on $A$