In "Hamlet" why does the queen tell lies to the king about Hamlet's madness?
Hamlet has just murdered Polonius. Murder is a pretty serious charge, and Polonius was a rather important man in the kingdom. His death will not be taken lightly--they can't just try to hide it, or pretend that it didn't happen. If Hamlet was sane, rational, and in his right mind, the repercussions of that murder would probably be pretty severe. Gertrude realizes this. She realizes that Hamlet is going to be in serious trouble for what has happened. So, to downplay Hamlet's role in the murder, she tells the king, who has the most authority to decide what will happen, that Hamlet is mad. If Hamlet was not in control of his actions because of his madness, then the repercussions will be less severe. So, she tells the king that Hamlet is
"Mad as the sea and wind when both contend
Which is the mightier...
O'er whom his very madness, like some ore
Among a mineral of metals base,
Shows itself pure. He weeps for what is done." (IV.i.7-29)
In this passage she relays the fact that she feels Hamlet is mad, out of his mind, and so not responsible for his death. Then, she softens Hamlet's role further by claiming that despite his madness, Hamlet still "weeps" for the fact that Polonius is dead. So, he is mad, but still upset over the death of the old man. She is acting to protect Hamlet, and it's not like she herself doesn't actually believe in Hamlet's madness--he has just attacked her in her room and talked to imaginary (in her mind, imaginary) ghosts wandering about the castle. So, she might just believe what she says about Hamlet's state of mind, but, for sure, she did exaggerate it to the king in order to protect her son from harsh treatment as a result of his actions. And, it works. The king decides that they must "ship him" away so that he isn't around when everything is found out. That will keep Hamlet safe from the law and people's anger for a time.
I hope that those thoughts help; good luck!