Querying an array of arrays in MongoDB

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Interesting question, This will do the trick

 db.multiArr.find({'Keys':{$elemMatch:{$elemMatch:{$in:['carrot']}}}})

$elemMatch used to check if an element in an array matches the specified match expression.
so nested $elemMatch will go deeper into nested arrays

Test data
    

db.multiArr.insert({"ID" : "fruit1","Keys" : [["apple", "carrot", "banana"]]})
db.multiArr.insert({"ID" : "fruit2","Keys" : [["apple", "orange", "banana"]]})


db.multiArr.find({'Keys':{$elemMatch:{$elemMatch:{$in:['carrot']}}}})
{ "_id" : ObjectId("506555212aeb79b5f7374cbf"), "ID" : "fruit1", "Keys" : [ [ "apple", "carrot", "banana" ] ] }

db.multiArr.find({'Keys':{$elemMatch:{$elemMatch:{$in:['banana']}}}})

{ "_id" : ObjectId("506555212aeb79b5f7374cbf"), "ID" : "fruit1", "Keys" : [ [ "apple", "carrot", "banana" ] ] }
{ "_id" : ObjectId("5065587e2aeb79b5f7374cc0"), "ID" : "fruit2", "Keys" : [ [ "apple", "orange", "banana" ] ] }

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