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NDA Math Determinants

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Question : 15 of 68

Marks: +1, -0

If

a_{1}, a_{2}, a_{3}

a_{9}

are in GP, then what is the value of the following determinant?

\begin{vmatrix} \ln a_{1} & \ln a_{2} & \ln a_{3} \\ \ln a_{4} & \ln a_{5} & \ln a_{6} \\ \ln a_{7} & \ln a_{8} & \ln a_{9} \end{vmatrix}

0
1
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4

Solution: 👈: Video Solution

Concept:

Let us consider sequence

a_{1}, a_{2}, a_{3} \dots a_{n}

is a G.P.

So, common ratio

= r = \frac{a_{p+1}}{a_p}

,

Where

p \in

Positive Integer Logarithmic formula:

\log

a

-\log b = \frac{\log a}{\log b}

Calculation:

Let

r

be the common ratio of the given GP

Now,

\begin{vmatrix} \ln a_{1} & \ln a_{2} & \ln a_{3} \\ \ln a_{4} & \ln a_{5} & \ln a_{6} \\ \ln a_{7} & \ln a_{8} & \ln a_{9} \end{vmatrix}

Apply,

C_{2} \rightarrow C_{2} - C_{1}

and

C_{3} \rightarrow C_{3} - C_{1}

= \begin{vmatrix} \ln a_{1} & \ln a_{2}-\ln a_{1} & \ln a_{3}-\ln a_{2} \\ \ln a_{4} & \ln a_{5}-\ln a_{4} & \ln a_{6}-\ln a_{5} \\ \ln a_{7} & \ln a_{8}-\ln a_{7} & \ln a_{9}-\ln a_{8} \end{vmatrix}

\begin{vmatrix} \ln a_{1} & \ln r & \ln r \\ \ln a_{4} & \ln r & \ln r \\ \ln a_{7} & \ln r & \ln r \end{vmatrix}

[

Since

\;\frac{a_{p}+1}{a_{p}} = r

and

\log a_{p+1} - \log a_{p} = \log \frac{a_{p+1}}{a_{p}} = \log r

]

= 0 \;\; [

Since

C_{3} = C_{2} ]

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