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Questions & Answers about Она должна спать.
Why does the word for "must" end in "-а" in this sentence?
Because the Russian word for "must" (должен) is actually a short-form adjective, not a verb! Adjectives must agree with the gender of the noun they describe. Since Она (she) is feminine, we use the feminine form должна.
Is должна conjugated like the verbs хотеть and мочь?
Why doesn't the verb спать change its ending to match "she"?
When you use a modal word expressing obligation, ability, or desire, the main action verb stays in its dictionary form (the infinitive). Just like with хотеть (to want) or мочь (can), the second verb doesn't change. So it stays спать (to sleep) rather than changing to спит (she sleeps).
Could I use надо or нужно instead to say "She needs to sleep"?
You could, but the grammar would change entirely! Должна is used with a regular subject (Она должна). Words like надо or нужно require the person who needs to do something to be in the Dative case. You would have to say Ей надо спать (literally "To her it is necessary to sleep"). Both are common, but должна often feels a bit stronger, like a strict obligation.