In what ways does Rikki exemplify the curiosity of a mongoose in his first days in bungalow in "Rikki-tikki-tavi"?    

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When Rikki-tikki-tavi washes up in the bungalow’s garden, he is immediately curious rather than frightened.  We are told that a real mongoose is consumed with curiosity.  Rikki-tikki definitely demonstrates this.  As soon as he recovers from almost drowning, he begins playing around.

The motto of all the mongoose family is "Run and find out"; and Rikki-tikki was a true mongoose. He looked at the cotton-wool, decided that it was not good to eat, ran all around the table, sat up and put his fur in order, scratched himself, and jumped on the small boy's shoulder.

Rikki is not afraid of people, and likes to climb on them and sit in their laps.  His biggest concern is whether or not things are edible.  He also wants to inspect the house as soon as possible.  Almost immediately he is looking around rather than sleeping.

Teddy’s father knows that having a mongoose around can be a very good thing when you are in India.  The garden may have snakes, and the snakes may come into the house.  A mongoose is the best defense against snakes.

"I don't like that," said Teddy's mother; "he may bite the child." "He'll do no such thing," said the father. "Teddy's safer with that little beast than if he had a bloodhound to watch him. If a snake came into the nursery now --- "

Rikki-tikki proves to be a very brave mongoose.  In his first encounter with the cobras, they are almost more afraid of him than he is of them.  Rikki-tikki refuses to be backed down, and cannot be tricked.  Rikki is their enemy from then on, and he makes it his goal to remove all snakes from the garden.  It is not easy, but he succeeds in killing Nag, Karait, and Nagaina and the garden is safe for people and animals.