If $x_1, x_2, x_3, \ldots, x_{13}$ are in A.P then the value of $$\begin{vmatrix} e^{x_1} & e^{x_4} & e^{x_7} \\ e^{x_4} & e^{x_7} & e^{x_{10}} \\ e^{x_7} & e^{x_{10}} & e^{x_{13}} \end{vmatrix}$$ is ____
Step-by-Step Solution
Key Concept: Since x₁, x₂, ..., x₁₃ are in A.P. with common difference d, the elements form a pattern where eˣ¹, eˣ⁴, eˣ⁷, eˣ¹⁰, eˣ¹³ are successive terms of a geometric progression (eˣ¹·eᵌᵈ, eˣ¹·e²ᵈ, etc.). After factoring out exponential terms from rows, the resulting matrix has all identical rows, making the determinant zero.
Factor out $e^{A}$ from $R_1$, $e^{4}$ from $R_2$, and $e^{3t}$ from $R_3$ to obtain $\Delta = e^{A+4+3t} \Delta_1$ where $\Delta_1$ is a determinant with entries involving powers of $e^d$ where $d$ is the common difference. Since all rows become identical when simplified, $\Delta_1 = 0$, therefore $\Delta = 0$.
Correct Answer: 0