Option B) is the correct one. In order for the infinitive to act as an adjective, it has to modify a noun. In this case the infinitive “To note down” is modifying the noun “diary”.
Option C) and D) are incorrect as the infinitives “to cook” and “to play” are modifying the verbs “wanted” and “likes”.
Option A) is also incorrect because the infinitive “to convince” is modifying the adjective “hard”.
