If `x_(i) = a_(i)b_(i)c_(i)= I = 1,2,3` are three - digit positive integers such that each `x_(i)` is a multiple of 19, then for some interger n, prove that `|(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))|` is divisible by 19.

If a_(r) =(cos 2 r pi + i sin 2 r pi)^(1/9) , then the value of |(a_(1),a_(2),a_(3)),(a_(4),a_(5),a_(6)),(a_(7),a_(8),a_(9))| a)1 b)-1 c)0 d)None of these

Write the first three terms of the sequence defined by a_(1)= 2 ,a_(n+1) = (2a_(n)+3)/(a_(n)-:2)

If a , a_(1) , a_(2) ,a_(3) , cdots a_(2n), b are in A.P. and a, g_(1) , g_(2) , g_(3) , cdots , g_(2n), b are in G.P. and h is the H.M of a and b then prove that (a_(1)+a_(2n))/(g_(1)g_(2n))+(a_(2)+a_(2n-1))/(g_(2)g_(2n-1))+cdots+ (a_(n)+a_(n+1))/(g_(n)g_(n+1))=(2n)/(h)

Let a_(1), a_(2), cdots and b_(1), b_(2) cdots be arithmetic progression such that a_(1) = 25, b_(1) = 75 and a_(100) + b_(100) = 100 , then the sum of first hundred terms of the progression a_(1) + b_(1), a_(2) + b_(2), cdots is

If a_(1), a_(2), …, a_(50) are in GP, then " "(a_(1)-a_(3)+a_(5)-…+a_(49))/(a_(2)-a_(4)+a_(6)-…+a_(50)) is equal to :

If a_(1), a_(2), a_(3), ………., a_(n) are in AP with common difference 5 and if a_(i)a_(j) ne -1 for i, j = 1, 2, …….., n, then tan^(-1)((5)/(1+a_(1)a_(2))) + tan^(-1)((5)/(1+a_(2)a_(3))) +……..+ tan^(-1)((5)/(1+a_(n-1)a_(n))) is equal to a) tan^(-1)((5)/(1+a(n)a_(n-1))) b) tan^(-1)((5)/(1+a(n)a_(n))) c) tan^(-1)((5n-5)/(1+a(n)a_(n+1))) d) tan^(-1)((5n-5)/(1+a_(1)a_(n)))

If A=[a_(ij)]_(2xx2) where a_(ij)=i+j , then A is equal to a) [(1,1),(2,2)] b) [(1,2),(1,2)] c) [(1,2),(3,4)] d) [(2,3),(3,4)]