Question:
If \(a,b,c\) are in arithmetic progression, then find the value of
\[
\left|
\begin{array}{ccc}
x+2 & x+3 & x+2a\\
x+3 & x+4 & x+2b\\
x+4 & x+5 & x+2c\\
\end{array}
\right|.
\]
If \(a,b,c\) are in arithmetic progression, then find the value of
\[
\left|
\begin{array}{ccc}
x+2 & x+3 & x+2a\\
x+3 & x+4 & x+2b\\
x+4 & x+5 & x+2c\\
\end{array}
\right|.
\]
Show Hint
Whenever entries in rows or columns follow an arithmetic progression, check for linear dependence. A determinant with linearly dependent rows or columns is always zero.
Updated On: Jun 6, 2026
- \(2x\)
- \(x\)
- \(1\)
- \(0\)
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
If three elements \(a,b,c\) are in arithmetic progression, then: \[ 2b=a+c \] A determinant becomes zero if its rows or columns become linearly dependent.
Step 1: Given determinant. \[ \Delta= \left| \begin{array}{ccc} x+2 & x+3 & x+2a\\ x+3 & x+4 & x+2b\\ x+4 & x+5 & x+2c \end{array} \right| \]
Step 2: Use the A.P. condition. \[ a,b,c \text{ are in A.P.} \] Therefore, \[ a+c=2b \]
Step 3: Observe the pattern in rows.
The first two columns increase regularly from row to row. The third column also becomes linearly related because \(a,b,c\) are in arithmetic progression.
Step 4: Linear dependence.
Since the columns are linearly dependent, the determinant must be zero. \[ \Delta=0 \] \[ \therefore \text{Correct Answer is (D)} \]
If three elements \(a,b,c\) are in arithmetic progression, then: \[ 2b=a+c \] A determinant becomes zero if its rows or columns become linearly dependent.
Step 1: Given determinant. \[ \Delta= \left| \begin{array}{ccc} x+2 & x+3 & x+2a\\ x+3 & x+4 & x+2b\\ x+4 & x+5 & x+2c \end{array} \right| \]
Step 2: Use the A.P. condition. \[ a,b,c \text{ are in A.P.} \] Therefore, \[ a+c=2b \]
Step 3: Observe the pattern in rows.
The first two columns increase regularly from row to row. The third column also becomes linearly related because \(a,b,c\) are in arithmetic progression.
Step 4: Linear dependence.
Since the columns are linearly dependent, the determinant must be zero. \[ \Delta=0 \] \[ \therefore \text{Correct Answer is (D)} \]
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