138- If determinant $D = \begin{vmatrix} 1 & 1 & 1 \\ bc & ac & ab \\ ac & ab & bc \end{vmatrix}$, then what is the value of $\begin{vmatrix} a+b & b & ab \\ b+c & c & bc \\ a+c & a & ac \end{vmatrix}$?
(1) $-D$ (2) $D$ (3) $(a+b+c)D$ (4) $abcD$

138. If one unit is added to all entries of the second column of matrix $A = \begin{bmatrix} 2 & 3 & 4 \\ 5 & a & 7 \\ 3 & b & 6 \end{bmatrix}$, what number is added to the value of the original determinant of the matrix?
(1) $-3$ (2) $-2$ (3) $3$ (4) $6$
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Let $\alpha \beta \neq 0$ and $A = \left[ \begin{array} { r r r } \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ - \beta & \alpha & 2 \alpha \end{array} \right]$. If $B = \left[ \begin{array} { r r r } 3 \alpha & - 9 & 3 \alpha \\ - \alpha & 7 & - 2 \alpha \\ - 2 \alpha & 5 & - 2 \beta \end{array} \right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname { det } ( A B )$ is equal to :
(1) 64
(2) 216
(3) 343
(4) 125

Q69. If $\alpha \neq \mathrm { a } , \beta \neq \mathrm { b } , \gamma \neq \mathrm { c }$ and $\left| \begin{array} { c c c } \alpha & \mathrm { b } & \mathrm { c } \\ \mathrm { a } & \beta & \mathrm { c } \\ \mathrm { a } & \mathrm { b } & \gamma \end{array} \right| = 0$, then $\frac { \mathrm { a } } { \alpha - \mathrm { a } } + \frac { \mathrm { b } } { \beta - \mathrm { b } } + \frac { \gamma } { \gamma - \mathrm { c } }$ is equal to:
(1) 3
(2) 0
(3) 1
(4) 2