Is maate is correct word for mother in sanskrit?

Answer 1

No. The correct word in Sanskrit for mother is mAtR^i [मातृ] the last vowel R^i to be pronounced similar to the uvular r from German French but with the tip of the tongue vibrating instead of the uvula.

This mAtR^i is the mUlashabda or what is in tchnical grammatical terms called as the prAtipadika in Paninian Grammar system. This in the Nominative case singular takes the form of mAtA [माता]and is as such in a sentence, like mama mAta suvarchalA to mean "My mother is Suvarchalaa"

In the addressing mode this takes the form of mAtaH = [मातः] to mean O! Mother! as in mayi kR^ipAM kuru mAtaH! O! Mother bless me!

mAte = maate on th other hand is the address form of the word mAtA in Hindi, or may be other vernaculars!