Test Index

VITEEE 2009 Paper

Question : 88 of 120

Marks: +1, -0

p^{\text{th}}, q^{\text{th}} \text{ and } r^{\text{th}}

\begin{vmatrix} \log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1 \end{vmatrix}

equals

Solution:

Let a and R be the first term and common ratio of a Gp respectively.

So,

T_{p}=aR^{p-1}=x

T_{q}=aR^{q-1}=y

and

T_{r}=aR^{r-1}=z

⇒ log x = log a + (p - 1) log R

log y = log a + (q - 1) log R

and log z = log a + (r - 1) log R

∴

\begin{vmatrix} \log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1 \end{vmatrix} = \begin{vmatrix} \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{vmatrix}

\begin{vmatrix} \log a & p & 1 \\ \log a & q & 1 \\ \log a & r & 1 \end{vmatrix}

\begin{vmatrix} (p-1)\log R & p & 1 \\ (q-1)\log R & q & 1 \\ (r-1)\log R & r & 1 \end{vmatrix}

= log a

\begin{vmatrix} 1 & p & 1 \\ 1 & q & 1 \\ 1 & r & 1 \end{vmatrix} + \log R \begin{vmatrix} p-1 & p-1 & 1 \\ q-1 & q-1 & 1 \\ r-1 & r-1 & 1 \end{vmatrix}_{(C_2 \rightarrow C_2 - C_3)}

= 0 + 0 = 0 (∵ two columns are identical)

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