If `a_r=(cos2rpi+i sin 2rpi)^(1//9)` , then prove that `|[a_1,a_2,a_3],[a_4,a_5,a_6],[a_7,a_8,a_9]|=0`.

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]|

If alpha_r=(cos2rpi+isin2rpi)^(1/10) , then |(alpha_1, alpha_2, alpha_4),(alpha_2, alpha_3, alpha_5),(alpha_3, alpha_4, alpha_6)|= (A) alpha_5 (B) alpha_7 (C) 0 (D) none of these

If a_1, a_2, a_3,54,a_6,a_7, a_8, a_9 are in H.P., and D=|[a_1,a_2,a_3],[5, 4,a_6],[a_7,a_8,a_9]| , then the value of [D]i sw h e r e[dot] represents the greatest integer function

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 Z_r=cos((2rpi)/5)+isin((2rpi)/5),r=0,1,2,3,4,... then z_1z_2z_3z_4z_5 is equal to

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

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

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 (1+a x+b x^2)^4=a_0+a_1x+a_2x^2+....+a_8x^8,w h e r ea ,b ,a_0, a_1, ,a_8 in R such that a_0+a_1+a_2!=0 and |(a_0,a_1,a_2),(a_1,a_2,a_0),(a_2,a_0,a_1)|=0 , then the value of 5a/b is _______.