Question

Which of the following determinant(s) vanish(es) ?

• A. $\begin{vmatrix}1 & bc & bc ( b + c ) \\ 1 & ca & ca ( c + a ) \\ 1 & ab & ab ( a + b )\end{vmatrix}$
• B. $\begin{vmatrix}1 & \text { ab } & \frac{1}{ a }+\frac{1}{ b } \\ 1 & \text { bc } & \frac{1}{ b }+\frac{1}{ c } \\ 1 & ca & \frac{1}{ c }+\frac{1}{ a }\end{vmatrix}$
• C. $\begin{vmatrix}0 & a - b & a - c \\ b - a & 0 & b - c \\ c - a & c - b & 0\end{vmatrix}$
• D. $\begin{vmatrix}\log _x x y z & \log _x y & \log _x z \\ \log _y x y z & 1 & \log _y z \\ \log _z x y z & \log _z y & 1\end{vmatrix}$

Answer 1

Answer: (A), (B), (C), (D)

Correct answer is (a) $\begin{vmatrix}1 & bc & bc ( b + c ) \\ 1 & ca & ca ( c + a ) \\ 1 & ab & ab ( a + b )\end{vmatrix}$Correct answer is (b) $\begin{vmatrix}1 & \text { ab } & \frac{1}{ a }+\frac{1}{ b } \\ 1 & \text { bc } & \frac{1}{ b }+\frac{1}{ c } \\ 1 & ca & \frac{1}{ c }+\frac{1}{ a }\end{vmatrix}$Correct answer is (c) $\begin{vmatrix}0 & a - b & a - c \\ b - a & 0 & b - c \\ c - a & c - b & 0\end{vmatrix}$Correct answer is (d) $\begin{vmatrix}\log _x x y z & \log _x y & \log _x z \\ \log _y x y z & 1 & \log _y z \\ \log _z x y z & \log _z y & 1\end{vmatrix}$