Question

The number of triplets $\left(a , b , c\right)$ of positive integers satisfying the equation $\begin{vmatrix} a^{3}+1 & a^{2}b & a^{2}c \\ ab^{2} & b^{3}+1 & b^{2}c \\ ac^{2} & bc^{2} & c^{3}+1 \end{vmatrix}=30$ is equal to

• A. $3$
• B. $6$
• C. $9$
• D. $12$

Answer 1

Answer: (A)

$\frac{a b c}{a b c}\begin{vmatrix} a^{3}+1 & a^{2}b & a^{2}c \\ ab^{2} & b^{3}+1 & b^{2}c \\ ac^{2} & bc^{2} & c^{3}+1 \end{vmatrix}=30$
$\Rightarrow abc\begin{vmatrix} a^{2}+\frac{1}{a} & a^{2} & a^{2} \\ b^{2} & b^{2}+\frac{1}{b} & b^{2} \\ c^{2} & c^{2} & c^{2}+\frac{1}{c} \end{vmatrix}=30$
$\Rightarrow \begin{vmatrix} a^{3}+1 & a^{3} & a^{3} \\ b^{3} & b^{3}+1 & b^{3} \\ c^{3} & c^{3} & c^{3}+1 \end{vmatrix}=30$
Apply $R_{1}\leftrightarrow R_{1}+R_{2}+R_{3}$
$\Rightarrow \left(a^{3} + b^{3} + c^{3} + 1\right)\begin{vmatrix} 1 & 1 & 1 \\ b^{3} & b^{3}+1 & b^{3} \\ c^{3} & c^{3} & c^{3}+1 \end{vmatrix}=30$
Apply $C_{2}\leftrightarrow C_{2}-C_{1},C_{3}\leftrightarrow C_{3}-C_{1}$
$\Rightarrow \left(a^{3} + b^{3} + c^{3} + 1\right)\begin{vmatrix} 1 & 0 & 0 \\ b^{3} & 1 & 0 \\ c^{3} & 0 & 1 \end{vmatrix}=30$
$\Rightarrow a^{3}+b^{3}+c^{3}+1=30\Rightarrow a^{3}+b^{3}+c^{3}=29$
$\left(a , b , c\right)\equiv \left(1,1 , 3\right)$ or $\left(1,3 , 1\right)$ or $\left(3,1 , 1\right)$