Let `a_1, a_2, a_3, ..., a_10,` be in G.P. with `a_i gt0` for `i = 1, 2, ......., 10` and `S` be the set of pairs `(r,k), r, k in NN` (the set of natural numbers) for which `|(log_e a_1^ra_2^k,log_e a_2^ra_3^k,log_e a_3^ra_4^k), (log_e a_4^ra_5^k,log_e a_5^ra_6^k,log_e a_6^ra_7^k), (log_e a_7^ra_8^k,log_e a_8^ra_9^k,log_e a_9^ra_10^k)|=0.` Then the number of elements in `S,` is

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

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

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

Let a_(1), a_(2), a_(3)……, a_(10) " be in GP with " a_(i) gt " for " I = 2,2,…..,10 and S be the 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 ltbgt Then the number of elements is S, is

If a_1,a_2,...,a_(10) are in G.P, where a_igt0 and S is a set of ordered pairs (r,k) such that [[lna_1^ra_2^k,lna_2^ra_3^k,lna_3^ra_4^k],[lna_4^ra_5^k,lna_5^ra_6^k,lna_6^ra_7^k],[lna_7^ra_8^k,lna_8^ra_9^k,lna_9^ra_(10)^k]]=0 , then number of pairs (r,k) is (a) infinitely many (b) 1 (c) 5 (d) 3

If a_1,a_2,...,a_(10) are in G.P, where a_igt0 and S is a set of ordered pairs (r,k) such that [[lna_1^ra_2^k,lna_2^ra_3^k,lna_3^ra_4^k],[lna_4^ra_5^k,lna_5^ra_6^k,lna_6^ra_7^k],[lna_7^ra_8^k,lna_8^ra_9^k,lna_9^ra_(10)^k]]=0, then number of pairs (r,k) is (a) infinitely many (b) 1 (c) 5 (d) 3

The value of 10^(log_e cot1^@)*10^(log_e cot2^@)*10^(log_e cot3^@)*...*10^(log_e cot89^@) is equal to

|log e,log e^2,log e^3],[log e^2,log e^3,log e^4],[log e^3,log e^4,log e^5]|=

The number of real solution(s) of the equation 9^(log_(3)(log_(e )x))=log_(e )x-(log_(e )x)^(2)+1 is equal to