mongodb nested object array query

With your current documents structure you need to use the $where operator which does a JavaScript evaluation.

db.collection.find(function(){
    for (var key in this.questions) {     
        if (this.questions[key]["o2"] == "Bad")         
            return true; 
    }
})

But I suggest you change your document structure because generally you don't want to use $where. To do that you need to use the Bulk() Api.

var bulk = db.collection.initializeUnorderedBulkOp();
var questions = [];
var subdoc = {};
var count  = 0;

db.collection.find().forEach(function(doc) { 
    for (var key in doc.questions) {     
        subdoc["o2"] = doc.questions[key]["o2"];    
        subdoc["o1"] = doc.questions[key]["o1"];    
        subdoc["_namespace"] = doc.questions[key]["_namespace"]; 
        subdoc["question"] = doc.questions[key]["question"];   
        subdoc["id"] = key;     
        questions.push(subdoc); 
        subdoc = {};
    } 

    bulk.find({ "_id": doc._id }).updateOne({
        "$set": { "questions": questions }});
    count++; 
    if (count % 1000 == 0) {  
        // Execute per 1000 operations and re-init.   
        bulk.execute();    
        bulk = db.collection.initializeUnorderedBulkOp(); 
    } 
})

// Clean up queues
if ( count % 1000 != 0 ) { 
    bulk.execute(); 
}

Then you documents look like this:

> db.collection.find().pretty()
{
        "_id" : ObjectId("55bf22a73d64bd3c495adf3d"),
        "_data_type" : "questionnaire",
        "title" : "Food quality",
        "_namespace" : "questionnaire",
        "description" : "tell us your opinion",
        "created_time" : ISODate("2015-08-03T08:13:27.874Z"),
        "questions" : [
                {
                        "o2" : "Bad",
                        "o1" : "Good",
                        "_namespace" : "questionnaire::question",
                        "question" : "How do you rate our food quality",
                        "id" : "sub_id_1"
                },
                {
                        "o2" : "Bad",
                        "o1" : "Good",
                        "_namespace" : "questionnaire::question",
                        "question" : "How do you rate the colouring of our dishes ",
                        "id" : "sub_id_2"
                },
                {
                        "o2" : "I don't care!",
                        "o1" : "It is greate",
                        "_namespace" : "questionnaire::question",
                        "question" : "How do you like our couches",
                        "id" : "sub_id_3"
                }
        ],
        "updated_time" : ISODate("2015-08-03T08:20:37.982Z")
}

You query become simpler and can use indexes if any.

db.collection.find({ "questions.o2": "Bad" })