Prove that the value of each the following determinants is zero: `|a_1l a_1+m b_1b_1a_2l a_2+m b_2b_2a_3l a_3+m b_3b_3|`

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

If (b_2-b_1)(b_3-b_1)+(a_2-a_1)(a_3-a_1)=0 , then prove that the circumcenter of the triangle having vertices (a_1,b_1),(a_2,b_2) and (a_3,b_3) is ((a_2+a_3)/2,(b_(2+)b_3)/2)

If A_1,B_1,C_1, are respectively, the cofactors of the elements a_1, b_1,c_1, of the determinant "Delta"=|[a_1b_1c_1][a_2b_2c_2][a_3b_3c_3]|,"Delta"!=0 , then the value of |[B_2C_2][B_3C_3]| is equal to a. a1^2Δ b. a_1Δ c. a_1^2Δ d. a1 2^2Δ

Consider the determinant Delta = |[a_1+b_1x^2,a_1x^2+b_1,c_1],[a_2+b_2x^2,a_2x^2+b_2,c_2],[a_3+b_3x^2,a_3x^2+b_3,c_3]| = 0 , \ w h e r e \ a_i ,b_i , c_i in R \ (i = 1,2,3) \ a n d \ x in R . Statement 1: The value of x satisfying Delta=0 are x=1,-1. Statement 2: If |[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]|=0,t h e n \ Delta=0.

If w is a complex cube root of unity, then value of =|[a_1+b_1w, a_1w^2+b_1,c_1+b_1 w],[ a_2+b_2w, a_2w^2+b_2,c_2+b_2 w],[ a_3+b_3w ,a_3w^2+b_3,c_3+b_3 w] | is a. 0 b. -1 c. 2 d. none of these

The total number of infections (one-one into mappings) from {a_1,a_2,a_3,a_4} to {b_1,b_2,b_3,b_4,b_5,b_6,b_7} is

Prove that the sum of the G.P. (a+b+….l) is (b l - a^(2))/(b-a) .

Let's find the L.C.M of the following algebraic expressions. 3a^2 - 5ab - 12b^2, a^5 - 27a^2 b^3, 9a^2 + 24ab + 16b^2

If x_i=a_i b_i c_i,i=1,2,3 are three-digit positive integer such that each x_1 is a multiple of 19, then for some integers 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.

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat k and vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a and vec b . If the angle between a and b is pi/6, then prove that |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2=1/4(a_1 ^2+a_2 ^2+a_3 ^2)(b_1 ^2+b_2 ^2+b_3 ^2)