According to British Council -
A বা An এর ব্যবহার ঠিক করা হয় পরবর্তী শব্দটি কোন ধ্বনি (Sound) দিয়ে শুরু হচ্ছে তার উপর ভিত্তি করে। যেন ফলশ্রুতিতে উচ্চারণ (Pronunciation) সহজ হয়।
We use 'a' when the word that follows it begins with a consonant sound. We use 'an' when it's followed by a vowel sound. This makes pronunciation easier.

According to Merriam-Webster Dictionary -
Choosing between the indefinite articles a and an is determined by the sound of the following word. If the word begins with a consonant sound you would use a, such as "a dog" and "a balloon," as well as "a one" and "a unicorn." If the word begins with a vowel sound, use an, such as in "an honorable man," and with spoken acronyms like "an FBI agent." Words like historic, with a pronounced "H," can use either a or an.

According to Cambridge Dictionary -
We use 'a' in front of consonant sounds and 'an' in front of vowel sounds. So, for instance, someone asked whether to use 'a' or 'an' before the abbreviation 'MBA'. The answer is 'an' because the sound of the first letter is /em/, so it begins with a vowel sound, even though the letter itself is a consonant.

According to Purdue University -
The choice of the article is based upon the phonetic (sound) quality of the first letter in a word, not on the orthographic (written) representation of the letter.

We use 'a' before a consonant sound.
Example: A car, a house, a big truck, a wheel. Warning: Some words that begin with a vowel letter in writing have a consonant sound.
Example: a university, a one-year-old child.
We use 'an' before a vowel sound.
Example: An apple, an old shoe, an orchestra, an umbrella.

Warning: Some words that begin with a consonant letter in writing have a vowel sound.
Example: an hour, an MP3 player.

[বাজারের প্রচলিত অনেক বইয়ে উত্তরটি ভুল দেওয়া থাকতে পারে। উপরের রেফারেন্স সহ আলোচনার মাধ্যমে আশা করি কনফিউশন ক্লিয়ার হয়ে যাবে]