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Let a(1),a(2),a(3), …, a(10) be in G.P. ...

Let `a_(1),a_(2),a_(3), …, a_(10)` be in G.P. with `a_(i) gt 0` for i=1, 2, …, 10 and S be te set of pairs (r, k), r, k `in` N (the set of natural numbers)
for which `|(log_(e)a_(1)^(r)a_(2)^(k),log_(e)a_(2)^(r)a_(3)^(k),log_(e)a_(3)^(r)a_(4)^(k)),(log_(e)a_(4)^(r)a_(5)^(k),log_(e)a_(5)^(r)a_(6)^(k),log_(e)a_(6)^(r)a_(7)^(k)),(log_(e)a_(7)^(r)a_(8)^(k),log_(e)a_(8)^(r)a_(9)^(k),log_(e)a_(9)^(r)a_(10)^(k))|` = 0. Then the number of elements in S is

A

4

B

2

C

10

D

infinitely

Text Solution

Verified by Experts

The correct Answer is:
D
Given, `|{:("log"_(e) a_(1)^(r)a_(2)^(k), "log"_(e) a_(2)^(r)a_(3)^(k), "log"_(e) a_(3)^(r)a_(4)^(k)), ("log"_(e) a_(4)^(r)a_(5)^(k), "log"_(e) a_(5)^(r)a_(6)^(k), "log"_(e) a_(6)^(r)a_(7)^(k)),("log"_(e) a_(7)^(r)a_(8)^(k), "log"_(e) a_(8)^(r)a_(9)^(k), "log"_(e) a_(9)^(r)a_(10)^(k)):}|`
On applying elementary operations
`C_(2) to C_(2) - C_(1) " and " C_(3) to C_(3) - C_(1)`, we get
`|{:("log"_(e) a_(1)^(r)a_(2)^(k), "log"_(e) a_(2)^(r)a_(3)^(k), -"log"_(e) a_(1)^(r)a_(2)^(k),"log"_(e) a_(3)^(r)a_(4)^(k),-"log"_(e) a_(1)^(r)a_(2)^(k)), ("log"_(e) a_(4)^(r)a_(5)^(k), "log"_(e) a_(5)^(r)a_(6)^(k),- "log"_(e) a_(4)^(r)a_(5)^(k),"log"_(e) a_(6)^(r)a_(7)^(k),-"log"_(e) a_(4)^(r)a_(5)^(k)),("log"_(e) a_(7)^(r)a_(8)^(k), "log"_(e) a_(8)^(r)a_(9)^(k),- "log"_(e) a_(7)^(r)a_(8)^(k), "log"_(e) a_(9)^(r)a_(10)^(k),-"log"_(e) a_(7)^(r)a_(8)^(k)):}| = 0`
`rArr |{:("log"_(e) a_(1)^(r)a_(2)^(k), "log"_(e) ((a_(2)^(r)a_(3)^(k))/(a_(1)^(r)a_(2)^(k))), "log"_(e) ((a_(3)^(r)a_(4)^(k))/(a_(1)^(r)a_(2)^(k)))), ("log"_(e) a_(4)^(r)a_(5)^(k), "log"_(e)((a_(5)^(r)a_(6)^(k))/(a_(4)^(r)a_(5)^(k))),"log"_(e)((a_(6)^(r)a_(7)^(k))/(a_(4)^(r)a_(5)^(k)))),("log"_(e) a_(7)^(r)a_(8)^(k), "log"_(e) ((a_(8)^(r)a_(9)^(k))/(a_(7)^(r)a_(8)^(k))), "log"_(e) ((a_(9)^(r)a_(10)^(k))/(a_(7)^(r)a_(8)^(k)))):}| = 0`
`[because "log"_(e)m-"log"_(e)n = "log"_(e)((m)/(n))]`
`[because a_(1), a_(2), a_(3)..., a_(10) " are in GP, therefore put "a_(1) = a, a_(2) = aR, a_(3) = aR^(2),....,a_(10) = aR^(9)]`
`rArr |{:("log"_(e)a^(r+k)R^(k), "log"_(e)((a^(r+k)R^(r+2k))/(a^(r+k)R^(k))), "log"_(e)((a^(r+k)R^(2r+3k))/(a^(r+k)R^(k)))),("log"_(e)a^(r+k)R^(3r+4k), "log"_(e)((a^(r+k)R^(4r+5k))/(a^(r+k)R^(3r+4k))), "log"_(e) ((a^(r+k)R^(5r+6k))/(a^(r+k)R^(3r+4k)))),("log"_(e)a^(r+k)R^(6r+7k),"log"_(e) ((a^(r+k)R^(7r+8k))/(a^(r+k)R^(6r+7k))),"log"_(e) ((a^(r+k)R^(8r+9k))/(a^(r+k)R^(6r+7k)))):}| = 0`
`rArr |{:("log"_(e)(a^(r+k)R^(k)), "log"_(e)R^(r+k), "log"_(e)R^(2r+k)),("log"_(e)a^(r+h)R^(3r+4k),"log"_(e)R^(r+k), "log"_(e)R^(2r+2k)),("log"_(e)a^(r+k)R^(6r +7k), "log"_(e)R^(r+k), "log"_(e)R^(2r+2k)):}| = 0`
`rArr |{:("log"_(e)(a^(r+k)R^(k)), "log"_(e)R^(r+k), 2"log"_(e)R^(r+k)),("log"_(e)(a^(r+h)R^(3r+4k)),"log"_(e)R^(r+k), 2"log"_(e)R^(r+k)),("log"_(e)(a^(r+k)R^(6r +7k)), "log"_(e)R^(r+k), 2"log"_(e)R^(r+k)):}| = 0`
`[because "log"m^(n) = n "log m and here log"_(e)R^(2r+2k) = "log"_(e)R^(2(r+k)) = 2"log"_(e)R^(r+h)]`
`because "Column"C_(2) "and "C_(3)` are proportional,
So, value of determinant will be zero for any value of `(r,k), r, k in N`.
`therefore `Set 'S' has infinitely many elements.
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