Find the coefficient of x in the determinant
`|{:((1+x)^(a_(1)b_(1)),(1+x)^(a_(1)b_(2)),(1+x)^(a_(1)b_(3))),((1+x)^(a_(2)b_(1)),(1+x)^(a_(2)b_(2)),(1+x)^(a_(2)b_(3))),((1+x)^(a_(3)b_(1)),(1+x)^(a_(3)b_(2)),(1+x)^(a_(3)b_(3))):}|`

If A=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|andB=|{:(c_(1),c_(2),c_(3)),(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)):}| then

8,A_(1),A_(2),A_(3),24.

If (a_(1)+ib_(1))(a_(2)+ib_(2))......(a_(n)+ib_(n))=A+iB , then : (a_(1)^(2)+b_(1)^(2))(a_(2)^(2)+b_(2)^(2))......(a_(n)^(2)+b_(n)^(2)) equals :

If ( a_(2)a_(3))/( a_(1) a_(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 :

Prove that |{:(a_(1)alpha_(1)+b_(1)beta_(1),a_(1)alpha_(2)+b_(1)beta_(2),a_(1)alpha_(3)+b_(1)beta_(3)),(a_(2)alpha_(1)+b_(2)beta_(1),a_(2)alpha_(2)+b_(2)beta_(2),a_(2)alpha_(3)+b_(2)beta_(3)),(a_(3)alpha_(1)+b_(3)beta_(1),a_(3)alpha_(2)+b_(3)beta_(2),a_(3)alpha_(3)+b_(3)beta_(3)):}| =0.

If the equation of the locus of a point equidistant from the points (a_(1), b_(1)) and (a_(2), b_(2)) is : (a_(1)-a_(2))x+(b_(1)-b_(2))y+c=0 , then c =

In the pair of linear equations a_(1)x +b_(1)y +c_(1)=0 and a_(2)x +b_(2)y +c_(2)=0 if (a_(1))/(a_(2)) ne (b_(1))/(b_(2)) then the

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common ratio r, then (a_(1)a_(2))/(a_(1)^(2)-a_(2)^(2))+(a_(2)a_(3))/(a_(2)^(2)-a_(3)^(2))+(a_(3)a_(4))/(a_(3)^(2)-a_(4)^(2))+"....."+(a_(n-1)a_(n))/(a_(n-1)^(2)-a_(n)^(2)) is equal to

If (1+x-2 x^(2))^(6)=1+a_(1) x+a_(z) x^(2)+l .. .+a_(12) x^(12) then a_(2)+a_(4)+..+a_(12) =

If the expansion in powers of x of the function (1)/((1-ax)(1-bx)) is a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)+ .. . . then a_(n) is: