The correct answer is option 4, which states that the fundamental cause of the first Karnataka war between the British and the French was the war of Austrian succession, which led to Britain and France becoming each other`s enemy in every part of the world.

Option 1 suggests that the participation of the English and the French in the native politics of Karnataka was the cause of the war. While it is true that their participation played a role, it was not the fundamental cause of the war.

Option 2 states that the ambition of Dupleix, a French official, to drive out the English from the South was the cause of the war. While Dupleix did play a significant role in the conflict, his ambition alone was not the sole cause.

Option 3 mentions the trade rivalry between the English and the French. While trade rivalry did exist between the two nations, it was not the primary cause of the war.

In summary, the correct answer is option 4, as it accurately identifies the war of Austrian succession as the fundamental cause of the first Karnataka war between the British and the French.