В анкете нужно написать имя, фамилию и отчество без ошибок.

Because after в meaning in, Russian usually uses the prepositional case for location.

• анкета = questionnaire / form
• в анкете = in the form

This is a very common pattern:

• в книге = in the book
• в школе = at school
• в анкете = in the form

So анкета changes to анкете because it is the place where something is written.

Нужно means something like it is necessary, one must, or you need to.

In this sentence, it gives a general instruction, not a personal statement. It does not literally say you; it means:

• нужно написать = it is necessary to write / you need to write

This is a very common way in Russian to express obligation in a neutral, impersonal way.

Compare:

• Мне нужно написать письмо. = I need to write a letter.
• Здесь нужно ждать. = One must wait here / You have to wait here.

Because нужно is followed by an infinitive.

The structure is:

• нужно + infinitive

So:

• нужно написать = need to write
• нужно читать = need to read
• нужно заполнить = need to fill out

Russian often uses this structure where English would use need to + verb.

This is about aspect.

• писать = imperfective, focuses on the process, repeated action, or ongoing action
• написать = perfective, focuses on completing the action

In a form/questionnaire context, the idea is usually to write the information down completely, so написать is natural.

Here it means something like:

• You need to write down / enter your name, surname, and patronymic.

If you used писать, it would sound more like focusing on the act of writing itself rather than the completed result.

Отчество is a patronymic, a name derived from a person's father's first name.

In Russian-speaking cultures, a full formal name often includes:

• имя = first name
• фамилия = surname / family name
• отчество = patronymic

For example, if someone's name is Иван Сергеевич Петров:

• Иван = first name
• Петров = surname
• Сергеевич = patronymic, meaning son of Sergey

For women, patronymics have different endings:

• Сергеевич = male
• Сергеевна = female

So this sentence is specifically about writing your full official name in the Russian style.

These words are the direct objects of написать, so they are in the accusative case.

Their accusative forms are:

• имя → имя
• фамилия → фамилию
• отчество → отчество

Why do some change and others do not?

• имя and отчество are neuter nouns, and for inanimate neuter nouns, nominative and accusative are usually the same.
• фамилия is a feminine noun ending in -я, so in the accusative singular it becomes -ю.

So:

• написать имя
• написать фамилию
• написать отчество

Russian often leaves out possessive words like your when the meaning is obvious from context.

In English, we naturally say:

• Write your name

In Russian, especially in instructions, it is very common to say simply:

• написать имя, фамилию и отчество

It is understood that these are your own details because you are filling out the form.

If needed, Russian could say:

• ваше имя, вашу фамилию и ваше отчество

But in this sentence that would sound more explicit than necessary.

Без means without, and it is followed by the genitive case.

So:

• ошибки = errors / mistakes (nominative plural)
• без ошибок = without errors / without mistakes

This is a fixed and very common pattern:

• без сахара = without sugar
• без воды = without water
• без проблем = without problems
• без ошибок = without mistakes

So the phrase means that the information should be written correctly.

In this context, без ошибок most naturally means without mistakes in the written information.

For a form, that usually suggests:

• correct spelling
• correct official form of the name
• no accidental writing errors

So it can include spelling mistakes, but also more generally any mistakes in how the name information is entered.

Russian can list these in different orders depending on context.

This sentence uses:

• имя, фамилию и отчество = first name, surname, and patronymic

That is a natural order when listing personal details in a general way.

But in official documents and databases, you also often see:

• фамилия, имя, отчество

That order is especially common in forms, records, and alphabetical lists.

So this sentence is normal; it is just not the only possible order.

Yes. Анкета can mean several closely related things, depending on context:

• questionnaire
• form
• application form
• personal information form

It often refers to a document where you enter personal details. So in this sentence, в анкете is best understood as on/in the form.

No. Russian often uses impersonal constructions where English would use you, one, or a passive structure.

Here:

• В анкете нужно написать...

literally feels like:

• In the form, it is necessary to write...

But natural English would usually be:

• On the form, you need to write...
• The form requires you to write...

So the sentence is complete even though there is no explicit subject.

Yes. Russian uses в with анкета, but English may prefer either in or on, depending on style.

• в анкете literally = in the form
• natural English often = on the form

Both are reasonable translations in practice. Russian and English do not always match exactly in preposition choice.