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HCU MCA 2013 Paper

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Question : 52 of 75

Marks: +1, -0

If a, b and c are all > 0, and are in a geometric progression as the successive terms ...,p, q, r, ... Then, what is the value of the determinant D below?

D = \begin{vmatrix} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \end{vmatrix}

0
pqr
1
pqr - abc

Solution:

(a) Let first term of GP is A and common ratio is R.

Then, a = AR

^{n-1}

⇒ log a = log A + (P - 1) log R etc.

⇒

\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} (p-1)\log R & p & 1 \\ (q-1)\log R & q & 1 \\ (r-1)\log R & r & 1 \end{vmatrix} \quad [\therefore C_1 \rightarrow C_1 - (\log A)C_3]

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

= \log R \begin{vmatrix} p & p & 1 \\ q & q & 1 \\ r & r & 1 \end{vmatrix} \quad [\therefore C_1 \rightarrow C_1 + C_3]

= 0

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