I don’t understand this connection between the words nose and narc.

You're right, Maverick. There is no connection between "narc" with a "c" and "nose". Sorry. I goofed. I should have written "nark", not "narc", everywhere. [Anu posted a subnote on "narc", short for "narcotics", which I didn't use because it isn't the "Word of the Day".]

So, your comment on "narc" is right on the nose.

As for "keen nose" and "landing on the nose", it gives "nose" a badly needed nose job. :)

re "tosh?": Merriam-Webster Online supplies this defn.
"sheer nonsense : BOSH, TWADDLE"

But, you're right! You narked me with this one. I thought a "tosh" is a butt, but it's not.

A "tush" is a butt.
'tush Function: noun
Etymology: perhaps modification of Yiddish tokhes, from Hebrew tahath under, beneath
slang : BUTTOCKS

Thanks for your ? mark. I amended my limerick. :)