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. $\begin{vmatrix}1 & a b & \frac{1}{a}+\frac{1}{b} \\ 1 & b c & \frac{1}{b}+\frac{1}{c} \\ 1 & c a & \frac{1}{c}+\frac{1}{a}\end{vmatrix}$
• 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) $\begin{vmatrix}1 & a b & \frac{1}{a}+\frac{1}{b} \\ 1 & b c & \frac{1}{b}+\frac{1}{c} \\ 1 & c a & \frac{1}{c}+\frac{1}{a}\end{vmatrix}$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) $\begin{vmatrix}\log _x x y z & \log _x y & \log _x z \\ \log _y x y z & 1 & \log _y z \\ \log _z x y z & \log _z y & 1\end{vmatrix}$