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

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

If (k-1),(2k+1),(6k+3) are in GP then k=?

If a_(1), a_(2), a_(3), …., a_(9) are in GP, then what is the value of the following determinant? |(ln a_(1), ln a_(2), ln a_(3)),(lna_(4), lna_(5), ln a_(6)), (ln a_(7), ln a_(8), ln a_(9))|

If (2k-1, k) is a solution of the equation 10 x-9y=12 , then k= (a) 1 (b) 2 (c) 3 (d) 4

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 .^36C_(r+1) xx (k^2-3) = 6 xx .^35C_r , then number of ordered pairs (r, k) are (where kinI ).

If a_1,a_2,……….,a_(n+1) are in A.P. prove that sum_(k=0)^n ^nC_k.a_(k+1)=2^(n-1)(a_1+a_(n+1))

Find the values of k for which the following pair of linear equations has infinitely many solutions : 2x - 3y = 7, ( k + 1 ) x + ( 1 - 2k) y = ( 5k - 4) .