A diák a vizsgára tanul.

In Hungarian, a/az is the definite article, so it means the, not the English indefinite article a.

Hungarian has two forms:

• a before a consonant sound: a diák, a vizsga
• az before a vowel sound: az iskola

So in this sentence, both instances of a mean the.

Yes, you can also say vizsgára tanul, and that is very common.

The difference is roughly this:

• a vizsgára tanul = the speaker has a specific exam in mind
• vizsgára tanul = more general, often just studying for an exam

So the article makes the exam sound more definite or already known from the context.

The word vizsgára is built from:

• vizsga = exam
• -ra = a case ending

This ending is called the sublative case. In concrete situations, it often means something like onto or to a surface, but with abstract nouns it can also express a goal or purpose.

So in a vizsgára tanul, the exam is the goal of the studying. That is why Hungarian uses vizsgára where English uses for the exam.

Because of vowel harmony.

Hungarian suffixes often have a back-vowel form and a front-vowel form:

• -ra
• -re

The noun vizsga contains the back vowel a, so it takes the back-vowel suffix: vizsgára.

A noun with front vowels would usually take -re instead.

Because the exam is not the direct object here.

With tanul, the direct thing being studied can take the accusative:

• magyart tanul = studies Hungarian
• matematikát tanul = studies mathematics

But an exam is usually not the content being studied. It is the target or purpose of the studying. That is why Hungarian says vizsgára tanul, not vizsgát tanul.

So:

• valamit tanul = study something
• valamire tanul = study for something

tanul is:

• present tense
• 3rd person singular
• indefinite conjugation

It agrees with a diák = the student.

It is indefinite conjugation because there is no definite direct object here. A vizsgára is not a direct object; it is a case-marked phrase expressing purpose/goal.

It can mean either, depending on context.

Hungarian does not have a separate mandatory progressive form like English is studying. The simple present tanul can express:

• a general/habitual meaning: studies
• an action going on now: is studying

So the exact English translation depends on the situation, not on a special verb form in Hungarian.

No, Hungarian word order is relatively flexible, but changing it usually changes the focus or emphasis.

This version is fairly neutral:

• A diák a vizsgára tanul.

It presents a diák as the topic, then says something about the student.

Other orders are possible, for example:

• A vizsgára tanul a diák. = emphasis on for the exam
• Tanul a diák. = emphasis on the fact that the student is studying

So Hungarian word order is often about information structure, not just grammar.

Yes. Hungarian often drops the subject if it is clear from context.

So A vizsgára tanul. is a perfectly normal sentence and means:

• He/she is studying for the exam
• or more generally is studying for the exam, depending on context

The verb form tanul already shows that the subject is 3rd person singular, though it does not show gender.

Because Hungarian uses vizsgára tanul idiomatically.

Different case endings are not interchangeable. Even if more than one ending can sometimes look possible from an English point of view, Hungarian often has a preferred fixed pattern with a given verb or expression.

With tanul in the sense of preparing for an exam, the natural form is:

• vizsgára tanul

So this is something learners should treat as a normal Hungarian pattern rather than translate word-for-word from English.