The correct answer is ক) came 

Complete sentence: He came home the day before yesterday.

Explanation:

• Words like yesterday, ago, last week/month/year, day before yesterday indicate a specific time in the past.

• When the action happened at a definite past time, the past form of the verb must be used, even if in Bengali the sentence seems present-like.

Examples:

• He came home the day before yesterday. 

• She bought the book last month. 

• Father came home yesterday. 

• I saw him long ago. 

Incorrect forms explained:

• খ) has come  → Present perfect is used for indefinite past, not for a specified past time.

• গ) had been come  → Wrong structure; “come” is intransitive and doesn’t form passive.

• ঘ) come  → Base form is incorrect; past tense came is required.

Source: A Passage To The English Language, S. M. Zakir Hussain

• The incorrect synonym pair is — খ) Intrepid: Unheroic

• Intrepid (অকুতোভয়; নিঃশঙ্ক; অসমসাহসিক) এবং Unheroic (অবীরোচিত; ভীরুতার পরিচায়ক) পরস্পর বিপরীত অর্থের, সমার্থক নয়।

• Other options:

• ক) Extirpate (সম্পূর্ণ ধ্বংস করা) ↔ Destroy (ধ্বংস/নষ্ট/বিধ্বস্ত/বরবাদ করা)

• গ) Amenable ((ব্যক্তি) সংবেদনশীল; বাধ্য; (আইন সম্বন্ধীয়) (ব্যক্তি) দায়ী) ↔ Responsible ((ব্যক্তি) আইনত অথবা নীতিগতভাবে দায়ী)

• ঘ) Temerity (হঠকারিতা; দুঃসাহস) ↔ Audacity (দুঃসাহস; স্পর্ধা; ধৃষ্টতা)

• এই শব্দগুলো পরস্পর সমার্থক (synonyms)।

Source:

1. Accessible Dictionary, Bangla Academy

2. Merriam-Webster Dictionary

3rd Conditional Rules:

• If-clause: had + past participle (V3)

• Main clause: would/could/might + have + past participle (V3)

Example from your sentence:

• If they had listened to the advice, they might have avoided the problem. 

Explanation:

• "had listened" → If-clause in past perfect

• "might have avoided" → Main clause showing hypothetical result in the past

Other options would not follow the 3rd conditional structure, so they are incorrect.