Question

The value of \[ \begin{vmatrix} 1 & a & b + c \\ 1 & b & c + a \\ 1 & c & a + b \end{vmatrix} \] is:

• A. $a+b+c$
• B. $(a-b)(b-c)(c-a)$
• C. 0
• D. 1

Hint:

Summing variables often reveals identical rows or columns in cyclic determinants.

Answer 1

The Correct Option is C

Step 1: Concept
Use determinant properties to simplify the calculation.
Step 2: Analysis
Apply the column operation $C_{3} \rightarrow C_{3} + C_{2}$. The third column becomes $(a+b+c, a+b+c, a+b+c)$. Taking $(a+b+c)$ common from $C_{3}$, the column becomes $(1, 1, 1)$.
Step 3: Conclusion
Now $C_{1}$ and $C_{3}$ are identical. A determinant with two identical columns is zero.
Final Answer: (C)