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Question

If a1,a2,a3,,an, are in GP, then the value of the determinant ∣ ∣loganlogan+1logan+2logan+3logan+4logan+5logan+6logan+7logan+8∣ ∣, is

A
0
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B
1
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C
2
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D
2
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Solution

The correct option is A 0
Since, a1,a2, are in GP.
Then, an=a1rn1
logan=loga1+(n1)logr
an+1=a1rn
logan+1=loga1+nlogr
an+2=a1rn+1
logan+2=loga1+(n+1)logr
......................
......................
an+8=a1rn+7
logan+8=loga1+(n+7)logr
Now, ∣ ∣loganlogan+1logan+2logan+3logan+4logan+5logan+6logan+7logan+8∣ ∣
=∣ ∣ ∣loga1+(n1)logrloga1+nlogrloga1+(n+1)logrloga1+(n+2)logrloga1+(n+3)logrloga1+(n+4)logrloga1+(n+5)logrloga1+(n+6)logrloga1+(n+7)logr∣ ∣ ∣
Applying R2R2R1 and R3R3R2
=∣ ∣ ∣loga1+(n1)logrloga1+(n)logrloga1+(n+1)logr3logr3logr3logr3logr3logr3logr∣ ∣ ∣
=0 ( two rows identical)

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