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If you're looking for a pretty in depth look at the math and physics of music, Musimathics is a pretty good book.

I tried to read Arnold's "Mathematical problems from 5 to 15" when I was 22, there are two possibilities...

• I was actually 10 years old, judging by the amount of questions I could answer.
  
  
• As I had 22 and 22∉{5,6,7,...,15}, this forbid me to answer all of it.
  
  To understand the joke, see the last problem of this list, it's easy to find a copy of it with Google.
  
  But seriously, not being able to answer problems from past grades is not a big deal, we forget stuff all the time, just try to study it again. When you try to solve problems from past grades without success, it's probably because you don't remember the pragmatic way to answer it and when trying to answer it, you try to "invent" a solution. I think that this is a great opportunity to contemplate how superb mathematics is: These pragmatic solutions we learn at school, even from very basic mathematics took centuries to be developed in the form it is presented to you. Newton's "Standing on the shoulders of giants" couldn't be more fit.
  
  You should take a look at Dörrie's "100 great problems of elementary mathematics: Their history and solution"

For your first question, you'll likely enjoy reading this:


The Psychology of Invention in the Mathematical Field

> I can solve though but the thought why i am doing this is always alarming inside, go and ask any teacher or students as why they do these maths? They will say it's for Grades!

Eek, you have a terrible history of teachers.

>Don't know how many students give up maths just because of wrong Teacher.

For sure.

Starting with calculus/analysis, the book most undergraduate students in America start with is this one. Not every concept starts with real-life examples, but every chapter and section includes actual real-life examples.