Learn Hardcore Russian: Старая папка лежит на письменном столе рядом со скотчем. - The old folder lies on the desk next to the Scotch tape.

Because adjectives in Russian must agree with their nouns in gender, number and case. The noun папка is feminine, singular, and in the nominative case, so its modifying adjective takes the feminine nominative singular ending -ая.

What case is письменном столе in, and why?

It’s in the Prepositional case. The preposition на when indicating location (“on something”) requires the Prepositional. A masculine noun like стол takes -е in the Prepositional singular (столе), and its adjective письменный changes to письменном.

Why do we use на with письменном столе instead of в?

In Russian you use в (“in”) for enclosed spaces or containers. A desk is treated as a surface you put things on, so “on the desk” is на столе, not в столе.

Why is it рядом со скотчем and not just рядом скотчем?

The adverb рядом (“next to”) must be followed by с + the Instrumental case to link it to the object. Without с (or со) the phrase would be ungrammatical.

What case is скотчем and why does it end in -ем?

Скотчем is in the Instrumental case because of the preposition с/со. Masculine nouns ending in a consonant form the Instrumental singular with -ом or -ем; скотч becomes скотчем.

Why do we say со скотчем instead of с скотчем?

Russian often changes с to со before words that begin with certain consonant clusters (here ск-) to make pronunciation smoother. Both mean “with,” but со is preferred in this context.

Could we replace лежит with another verb like стоит or находится?

Yes.

лежит literally means “lies” (flat on a surface).
стоит means “stands” (upright).
находится means “is located” (more formal, neutral).
You choose based on how the object physically sits or the style you want.