If `|a_(1)|gt|a_(2)|+|a_(3)|,|b_(2)|gt|b_(1)|+|b_(3)|` and
`|c_(2)|gt|c_(1)|+|c_(2)|` then show that `|{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|ne0.`

Given a_(i)^(2) + b_(i)^(2) + c_(i)^(2) = 1, i = 1, 2, 3 and a_(i) a_(j) + b_(i) b_(j) + c_(i) c_(j) = 0 (i !=j, i, j =1, 2, 3) , then the value of the determinant |(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))| , is

If a_(1)^(2)+a_(2)^(2)+a_(3)^(2)=1, b_(1)^(2)+b_(2)^(2)+b_(3)^(2)=4, c_(1)^(2)+c_(2)^(2)+c_3^(2)=9, a_(1)b_(1)+a_(2)b_(2)+a_(3)b_(3)=0, a_(1)c_(1)+a_(2)c_(2)+a_(3)c_(3)=0, b_(1)c_(1)+b_(2)c_(2)+b_(3)c_(3)=0 and A[(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))] , then |A|^(4) is equal to

if a_(1)b_(1)c_(1), a_(2)b_(2)c_(2)" and " a_(3)b_(3)c_(3) are three-digit even natural numbers and Delta = |{:(c_(1),,a_(1),,b_(1)),(c_(2),,a_(2),,b_(2)),(c_(3),,a_(3),,b_(3)):}|" then " Delta is

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(2),c_(2)),(b_(3),c_(3)):}|-b_(1)|{:(a_(2),c_(2)),(a_(3),c_(3)):}|+c_(1)|{:(a_(2),b_(2)),(a_(3),b_(3)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column The value of k for which determinant |{:(2,3,-1),(-1,-2,k),(1,-4,1):}| vanishes, is "(a) -3 (b) 7/11 (c) -2 (d) 2"

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(2),c_(2)),(b_(3),c_(3)):}|-b_(1)|{:(a_(2),c_(2)),(a_(3),c_(3)):}|+c_(1)|{:(a_(2),b_(2)),(a_(3),b_(3)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column The value of the determinant |{:(2,3,4),(6,5,7),(1,-3,2):}|is: "(a) 54 (b) 40 (c) -45 (d) -40"

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(2),c_(2)),(b_(3),c_(3)):}|-b_(1)|{:(a_(2),c_(2)),(a_(3),c_(3)):}|+c_(1)|{:(a_(2),b_(2)),(a_(3),b_(3)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column. The vaue of the determinant |{:(5,1),(3,2):}|is: "(a) 4 (b) 5 (c) 6 (d) 7 "

Let overset(to)(a) =a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) , overset(to)(b) = b_(1) hat(i) +b_(2) hat(j) +b_(3) hat(k) " and " overset(to)(c) = c_(1) hat(i) +c_(2) hat(j) + c_(3) hat(k) be three non- zero vectors such that overset(to)(c ) is a unit vectors perpendicular to both the vectors overset(to)(a ) and overset(to)(b) . If the angle between overset(to)(a) " and " overset(to)(b) is (pi)/(6) then |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}|^2 is equal to

if (1+ax+bx^(2))^(4)=a_(0) +a_(1)x+a_(2)x^(2)+…..+a_(8)x^(8), where a,b,a_(0) ,a_(1)…….,a_(8) in R such that a_(0)+a_(1) +a_(2) ne 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 5.(a)/(b) " is " "____"

Let veca=a_(1)hati+a_(2)hatj+a_(3)hatk,vecb=b_(2)hatj+b_(3)hatk and vecc=c_(1)hati+c_(2)hatj+c_(3)hatk gve three non-zero vectors such that vecc is a unit vector perpendicular to both veca and vecb . If the angle between veca and vecb is pi/6 , then prove that |{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|p=1/4 (a_(1)^(2)+a_(2)^(2)+a_(3)^(2))(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))

If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =5 , then the value of Delta = |(b_(2) c_(3) - b_(3) c_(2),a_(3) c_(2) - a_(2) c_(3),a_(2) b_(3) -a_(3) b_(2)),(b_(3) c_(1) - b_(1) c_(3),a_(1) c_(3) - a_(3) c_(1),a_(3) b_(1) - a_(1) b_(3)),(b_(1) c_(2) - b_(2) c_(1),a_(2) c_(1) - a_(1) c_(2),a_(1) b_(2) - a_(2) b_(1))| is