The determinant is equal to zero

MATHEMATICS · Single correct · CBSE · Mathematics · Determinants · Determinants

Problem

The determinant

\begin{vmatrix} a & b & ac + b \\ b & c & bc + c \\ ac + b & bc + c & 0 \end{vmatrix}

is equal to zero, if (1)

a, b

and

c

are in AP (2)

a, b, c

are in GP (3)

a, b

and

c

are in HP (4)

(a-c)

is a factor of

ac^2 + 2bc + c