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(D) Let $a$ be the first term and $R$ be the common ratio of the geometric progression $(GP)$.Then, the $n$-th term is given by $T_n = a R^{n-1}$.Give

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$0$

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(D) Let $a$ be the first term and $R$ be the common ratio of the geometric progression $(GP)$.Then, the $n$-th term is given by $T_n = a R^{n-1}$.Given $x = T_p = a R^{p-1}$, $y = T_q = a R^{q-1}$, and $z = T_r = a R^{r-1}$.Taking the logarithm on both sides:$\log x = \log a + (p-1) \log R$$\log y = \log a + (q-1) \log R$$\log z = \log a + (r-1) \log R$Now, substitute these into the determinant:$\Delta = \left|\begin{array}{lll} \log a + (p-1) \log R & p & 1 \ \log a + (q-1) \log R & q & 1 \ \log a + (r-1) \log R & r & 1 \end{array}ight|$Using the property of determinants, we can split this into two determinants:$\Delta = \left|\begin{array}{lll} \log a & p & 1 \ \log a & q & 1 \ \log a & r & 1 \end{array}ight| + \left|\begin{array}{lll} (p-1) \log R & p & 1 \ (q-1) \log R & q & 1 \ (r-1) \log R & r & 1 \end{array}ight|$In the first determinant, the first column is $\log a$ times the third column, so it is $0$.In the second determinant, perform the operation $C_2 ightarrow C_2 - C_3$:$\Delta = 0 + \log R \left|\begin{array}{lll} p-1 & p-1 & 1 \ q-1 & q-1 & 1 \ r-1 & r-1 & 1 \end{array}ight|$Since the first and second columns are identical, the value of the determinant is $0$.Thus, $\Delta = 0 + 0 = 0$.

If $x, y, z$ are all positive and are the $p$-th, $q$-th, and $r$-th terms of a geometric progression respectively, then the value of the determinant $\left|\begin{array}{lll} \log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1 \end{array}\right|$ equals:

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