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If |{:(a,b,a),(b,c,1),(c,a,1):}|=2010 an...

If `|{:(a,b,a),(b,c,1),(c,a,1):}|=2010` and if `|{:(c-a,c-b,ab),(a-b,a-c,bc),(b-c,b-a,ca):}|-|{:(c-a,c-b,c^(2)),(a-b,a-c,a^(2)),(b-c,b-a,b^(2)):}|=p` , then the number of positive divisors of `p` is

A

`36`

B

`49`

C

`64`

D

`81`

Text Solution

Verified by Experts

The correct Answer is:
D
`(d)` `|{:(c-a,c-b,ab-c^(2)),(a-b,a-c,bc-a^(2)),(b-c,b-a,ca-b^(2)):}|=|{:(a,b,1),(b,c,1),(c,a,1):}|^(2)=(2010)^(2)`
`=(2xx3xx6xx67)^(2)=2^(2)3^(2)5^(2)(67)^(2)`
No. of divisors of `P=(2+1)(2+1)(2+1)(2+1)=81`
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