Vandermonde Determinant: |1 a $a^2$| |1 b $b^2$| = (a-b)(b-c)(c-a) |1 c $c^2$|

This appears frequently in JEE. Memorize the result AND the sign convention.

Circulant Determinant (a, b, c): |a b c| |c a b| = -($a^3$ + $b^3$ + $c^3$ - 3abc) = -(a+b+c)(a^{2+b}^{2+c}^{2-ab-bc-ca}) |b c a|

Determinant with arithmetic progression entries: If rows or columns form an AP, often the determinant is zero (because one row becomes a linear combination of others).

Differentiation of determinants: Differentiate one row at a time, keeping others fixed. Sum the resulting determinants. This appears in problems like "If f(x) = |...|, find f'(0)."