if `(a_2a_3)/(a_1a_4)`=`(a_2+a_3)/(a_1+a_4)`=`3((a_2-a_3)/(a_1-a_4))`then `a_1,a_2,a_3,a_4`are in:

If a_1,a_2,a_3,………….a_12 are in A.P. and /_\_1 =|(a_1a_5, a_1,a_2),(a_2a_6,a_2,a_3),(a_3a_7,a_3,a_4)|, /_\_2 =|(a_2a_10, a_2,a_3),(a_3a_11,a_3,a_4),(a_4a_12,a_4,a_5)| then /_\_1:/_\_2= (A) 1:2 (B) 2:1 (C) 1:1 (D) none of these

If x,a_1,a_2,a_3,…..a_n epsilon R and (x-a_1+a_2)^2+(x-a_2+a_3)^2+…….+(x-a_(n-1)+a_n)^2le0 , then a_1,a_2,a_3………a_n are in (A) AP (B) GP (C) HP (D) none of these

If a_0, a_1, a_2, a_3 are all the positive, then 4a_0x^3+3a_1x^2+2a_2x+a_3=0 has least one root in (-1,0) if (a) a_0+a_2=a_1+a_3 and 4a_0+2a_2>3a_1+a_3 (b) 4a_0+2a_2<3a_1+a_3 (c) 4a_0+2a_2=3a_1+a_0 and 4a_0+a_2lta_1+a_3 (d) none of these

If a_1, a_2, ,a_n are in H.P., then (a_1)/(a_2+a_3++a_n),(a_2)/(a_1+a_3++a_n), ,(a_n)/(a_1+a_2++a_(n-1)) are in a. A.P b. G.P. c. H.P. d. none of these

If a_iepsilon R-[0},i=1,2,3,4 and xepsilon R and (sum_(i=1)^ a_i^2) x^2-2x(sum_(i=)^3 a_ia_(i+1)+)+sum_(i=2)^4 a_i^2ge0 then a_1,a_2,a_3,a_4 are in (A) A.P. (B) G.P. (C) H.P. (D) none of these

Evaluate: /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]|

Evaluate: /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]|

Suppose a_1, a_, are real numbers, with a_1!=0. If a_1, a_2,a_3, are in A.P., then (a) A=[(a_1,a_2,a_3),(a_4,a_5,a_6),(a_5,a_6,a_7)] is singular (where i=sqrt(-1)) (b)The system of equations a_1x+a_2y+a_3z=0,a_4x+a_5y+a_6z=0,a_7x+a_8y+a_9=0 has infinite number of solutions. (c) B=[(a_1,i a_2),(ia_2,a_1)] is non-singular (d)none of these

If a_1,a_2,a_3,……a_n are in A.P. [1/(a_1a_n)+1/(a_2a_(n-1))+1/(a_3a_(n-2))+..+1/(a_na_1)]

If a_1,a_2, a_3, a_4 be the coefficient of four consecutive terms in the expansion of (1+x)^n , then prove that: (a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot