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Question

If a,b,c are pth,qth and rth terms of a GP, then ⎡⎢⎣logap1logbq1logcr1⎤⎥⎦ is equal to -

A
0
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B
1
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C
logabc
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D
pqr
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Solution

The correct option is A 0
As a,b,c are pth,qth,rth term of G.P,, then
Let common ratio of G.P. is R and first term is A
a=ARp1,b=ARq1,c=ARr1

Let Δ=∣ ∣logap1logbq1logcr1∣ ∣Δ=∣ ∣logap1logbq1logcr1∣ ∣

Applying R2R2R1;R3R3R1, we get
Δ=∣ ∣logap1logblogaqp0logclogarp0∣ ∣

Expanding along C3, we get
Δ=(rp)(logbloga)(qp)(logcloga)
=(rp)(logba)(qp)(logca)
=(rp)(logARq1ARp1)(qp)(logARr1ARp1)
=(rp)(logRqp)(qp)(logR(rp))=(rp)(qp)logR(qp)(rp)logR=0

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