If `Delta=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|` and `Delta_1=|(a_1+pb_1,b_1+qc_1,c_1+ra_1),(a_2+pb_2,b_2+qc_2,c _2+ra_2),(a_3+pb_3,b_3+qc_3,c_3+ra_3)|` then `Delta_1=`

Let Delta_1=|{:(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3):}| , Delta_2=|{:(6a_1,2a_2,2a_3),(3b_1,b_2,b_3),(12c_1, 4c_2,4c_3):}| and Delta_3=|{:(3a_1+b_1 , 3a_2+b_2 , 3a_3+b_3),(3b_1,3b_2,3b_3),(3c_1,3c_2,3c_3):}| then Delta_3-Delta_2=kDelta_1 , find k.

|{:(1,1,1),(a^2,b^2,c^2),(a^3,b^3,c^3):}|=(b-c)(c-a)(a-b)(bc+ca+ab)

If D= |{:(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3):}| and A_1,B_1,C_1 etc. are the respective cofactors of the elements a_1,b_1,c_1 etc. then D will be-

If the points ( a_1,b_1),(a_2,b_2) and (a_1+a_2,b_1+b_2) are collinear ,show that a_1b_2=a_2b_1 .

If a_1b_1c_1,a_2b_2c_2 and a_3b_3c_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]| ,t h e n Delta i s divisible by 2 but not necessarily by 4 divisible by 4 but not necessarily by 8 divisible by 8 none of these

If the determinant of the matrix [(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))] is denoted by D, then the determinant of the matrix [(a_(1)+3b_(1)-4c_(1),b_(1),4c_(1)),(a_(2)+3b_(2)-4c_(2),b_(2),4c_(2)),(a_(3)+3b_(3)-4c_(3),b_(3),4c_(3))] will be -

Find the value of lambda for which |{:(2a_1+b_1 , 2a_2+b_2 , 2a_3+b_3),(2b_1+c_1, 2b_2+c_2 , 2b_3+c_3),(2c_1+a_1,2c_2+a_2, 2c_3+a_3):}|=lambda|{:(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3):}|

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 A_1B_1C_1 , A_2B_2C_2 and A_3B_3C_3 are three digit numbers, each of which is divisible by k, then Delta = |(A_1,B_1,C_1),(A_2,B_2,C_2),(A_3,B_3,C_3)| is divisible by ____

If |x_1y_1 1x_2y_2 1x_3y_3 1|=|a_1b_1 1a_2b_2 1a_3b_3 1| then the two triangles with vertices (x_1, y_1),(x_2,y_2),(x_3,y_3) and (a_1,b_1),(a_2,b_2),(a_3,b_3) are equal to area (b) similar congruent (d) none of these