Bahadur Shah (First) was born in the year 1643.

Option 1: 1543 - This option is incorrect because Bahadur Shah (First) was not born in 1543.

Option 2: 1643 - This is the correct answer. Bahadur Shah (First) was indeed born in 1643.

Option 3: 1743 - This option is incorrect because Bahadur Shah (First) was born much earlier, in 1643.

Option 4: 1843 - This option is incorrect because Bahadur Shah (First) was not born in 1843.

In conclusion, Bahadur Shah (First) was born in 1643.