If `a_1x+b_1y+c_1z=0, a_2x+b_2y+c_2z=0, a_3x+b_3y+c_3z=0` and `|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0` , then the given system has, 1) more than two solutions 2) one trivial and one non trivial solutions 3) no solution 4) only trivial solution (0, 0, 0)

|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0 then the system of equations a_1x+b_1y+c_1z=0, a_2x+b_2y+c_2z=0, a_3x+b_3y+c_3z=0 has

Consider the system of equations a_(1) x + b_(1) y + c_(1) z = 0 a_(2) x + b_(2) y + c_(2) z = 0 a_(3) x + b_(3) y + c_(3) z = 0 If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =0 , then the system has

If Delta =|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)| then the value of |(2a_1+3b_1+4c_1,b_1,c_1),(2a2+3b_2+4c_2,b_2,c_2),(2a_3+3b_3+4c_3,b_3,c_3)| is equal to

Consider the following system of equations a_1x+b_1y+c_1z=d_1, a_2x+b_2y+c_2z=d_2, a_3x+b_3y+c_3z=d_3 Let /_\= |(a_1,b_1,c_1), (a_2,b_2,c_2), (a_3,b_3,c_3)|, /_\_1= |(d_1,b_1,c_1), (d_2,b_2,c_2), (d_3,b_3,c_3)|, ,/_\_2=|(a_1,d_1,c_1), (a_2,d_2,c_2), (a_3,d_3,c_3)|,, /_\_3=|(a_1,b_1,cd_1), (a_2,b_2,d_2), (a_3,b_3,d_3)| , The given system of equations will have i. unique solution if /_\!=0 ii. infinitely many solutions if /_\=/_\_1=/_\_3=0 . iii. no solution if /_\=0 and any of /_\_1, /_\_2, /_\_3 is none zero. On the basis of above informatioin answer thefollowing questions for the following system of linear equations. . 2x+ay+6z=8, x+2y+bz=5, x+y+3z=4 The given system of equatioin has unique solution if (A) a=2,b=2 (B) a!=2,b=3 (C) a!=2, b!=3 (D) a=2,b!=3

Consider the following system of equations a_1x+b_1y+c_1z=d_1, a_2x+b_2y+c_2z=d_2, a_3x+b_3y+c_3z=d_3 Let /_\= |(a_1,b_1,c_1), (a_2,b_2,c_2), (a_3,b_3,c_3)|, /_\_1= |(d_1,b_1,c_1), (d_2,b_2,c_2), (d_3,b_3,c_3)|, ,/_\_2=|(a_1,d_1,c_1), (a_2,d_2,c_2), (a_3,d_3,c_3)|,, /_\_3=|(a_1,b_1,cd_1), (a_2,b_2,d_2), (a_3,b_3,d_3)| , The given system of equations will have i. unique solution if /_\!=0 ii. infinitely many solutions if /_\=/_\_1=/_\_3=0 . iii. no solution if /_\=0 and any of /_\_1, /_\_2, /_\_3 is none zero. On the basis of above informatioin answer thefollowing questions for the following system of linear equations. x+y+z=6, x+2y+3z=14, 2x+5y+lamda=mu The given system of equations has no solution if (A) lamda=8, mu=10 (B) lamda!=8, muepsilon R (C) lamda=8, mu!=10 (D) lamda!=8, mu!=10

Consider the following system of equations a_1x+b_1y+c_1z=d_1, a_2x+b_2y+c_2z=d_2, a_3x+b_3y+c_3z=d_3 Let /_\= |(a_1,b_1,c_1), (a_2,b_2,c_2), (a_3,b_3,c_3)|, /_\_1= |(d_1,b_1,c_1), (d_2,b_2,c_2), (d_3,b_3,c_3)|, ,/_\_2=|(a_1,d_1,c_1), (a_2,d_2,c_2), (a_3,d_3,c_3)|,, /_\_3=|(a_1,b_1,cd_1), (a_2,b_2,d_2), (a_3,b_3,d_3)| , The given system of equations will have i. unique solution if /_\!=0 ii. infinitely many solutions if /_\=/_\_1=/_\_3=0 . iii. no solution if /_\=0 and any of /_\_1, /_\_2, /_\_3 is none zero. On the basis of above informatioin answer thefollowing questions for the following system of linear equations. . x+y+z=6, x+2y+3z=14, 2x+5y+lamda=mu The given system of equations has infinite solution if (A) lamda=8, mu=36 (B) lamda!=8, muepsilon R (C) lamda=8, mu!=36 (D) lamda!=8, mu!=36

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=

Consider a system of linear equation in three variables x,y,z a_1x+b_1y+ c_1z = d_1 , a_2x+ b_2y+c_2z=d_2 , a_3x + b_3y + c_3z=d_3 The systems can be expressed by matrix equation [(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)][(x),(y),(z)]=[(d_1),(d_2),(d_3)] if A is non-singular matrix then the solution of above system can be found by X = A^(-1)B , the solution in this case is unique. if A is a singular matrix i.e. then the system will have no solution (i.e. it is inconsistent) if Where Adj A is the adjoint of the matrix A, which is obtained by taking transpose of the matrix obtained by replacing each element of matrix A with corresponding cofactors. Now consider the following matrix. A=[(a,1,0),(1,b,d),(1,b,c)], B=[(a,1,1),(0,d,c),(f,g,h)], U=[(f),(g),(h)], V=[(a^2),(0),(0)], X=[(x),(y),(z)] The system AX=U has infinitely many solutions if :

If the lines a_1x+b_1y+1=0,a_2x+b_2y+1=0 and a_3x+b_3y+1=0 are concurrent, show that the point (a_1, b_1),(a_1, b_2) and (a_3, b_3) are collinear.

If u=a_1x+b_1y+c_1=0,v=a_2x+b_2y+c_2=0, and (a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2), then the curve u+k v=0 is (a)the same straight line u (b)different straight line (c)not a straight line (d)none of these