there was not a single topic that I could not relate to.
Because you were using the present tense just before, use "is" rather than "was." Or, explain that this was a class you took a while ago.
to the point of breaking down nature to mathematics, has appealed to me greatly.
"appeals" rather than "has appealed"
Understanding physics have given me a new perspective of the world around me.
"has," not "have"
I was never satisfied with gaining knowledge from books and locking it up in my head. I have always been keen to find the practical application of the laws and principles postulated by physics.
I feel like you can merge these into one, more effective sentence. Perhaps change the first sentence to a dependent clause: "Never satisfied with simple book-learning, I have always..."
in O-level and A-level mathematics, physics and chemistry
You don't really need a comma after "physics," but I recommend it. I like those Oxford commas =)
to choose engineering as the subject for further studies
This sounds a little awkward. Maybe say "as my major," or "for further close study."
I have always been a dreamer and a patriotic person.
I feel like you use this structure a lot: "I have always been" or "I have never been." Maybe just say "I am"?
I feel helpless when I see a eighty year old man, barely able to walk crossing a stream
"an" instead of "a" and "across" instead of "crossing"
I wonder when the countryside of my country will be illuminated with electric supplies.
"country" is a little redundant here. Maybe substitute "hills" or "plains" for "countryside"?
I wonder when the school children of my countryside of my age will encounter the most fast and efficient form of communication: Internet
"fastest and most efficient." Also, there's another "countryside" you could change. Or, you can omit it altogether because we know you're talking about your country here.
I believe knowledge of science is the biggest tool for development and engineers are the key persons to make use of that tool.
Since you're using a lot of independent clauses before then, I would change this to a dependent clause followed by an independent clause. Or (since it's already a fantastic sentence), you could add a sentence before that that isn't simply an independent clause.
choosing a specific major for the future.
This is better than you think =) It answers the question, it's straightforward, and it explains your ideas very well. I hope you get in!