The correct option is
A Mr. Ghosh was a successful businessman, but he was an irresponsible husband.
A simple sentence has only one subject and predicate. It has a finite verb with a completed thought. A compound sentence has at least two parts (also referred to as a clause). Each part has its own subject and predicate and makes good sense by itself. Hence each clause is independent of the other and called the main clause or independent clause. The main clauses are connected using a co-ordinating conjunction (for, and, nor, but, or, yet, so).A complex sentence has at least one main clause (which is an independent clause) and one or more dependent clauses, connected using a subordinate conjunction. The dependent clause cannot stand by itself and make good sense. It depends on the main clause.
Option A: The two clauses viz (i) "Mr. Ghosh was a successful businessman" and (ii) "He was an irresponsible husband", can stand independently and make good sense. They are independent clausesconnected using the co-ordinating conjunction-"but". Hence, this is a compound sentence and this answer is correct.
Option B: This is a simple sentence as it contain a single clause viz, "Mr. Ghosh was an irresponsible husband". It has a single subject ("Mr. Ghosh") and predicate ("was an irresponsible husband"). The words, "Despite being a successful businessman" is a phrase. It is not a clause as it does not contain a subject and a predicate. Hence, this option is incorrect.
Options C and D: Both these sentence are complex sentences as they use the subordinating conjunctions, "Even though" and "because". These connect independent and dependent clauses to form complex sentences.
In Option C, the clause, "He was an irresponsible husband". The clause, "Even though Mr. Ghosh was a successful businessman" is a dependent clause, which cannot stand on its own and make good sense.
In Option D, the clause, "Mr. Ghosh was a successful businessman". The clause, "because he was an irresponsible husband" is a dependent clause. Further, this sentence is not meaningful as being an irresponsible husband cannot be the reason for Mr. Ghosh to be a successful businessman.
Hence, both options C and D are incorrect.