If `x ,y \ a n d \ z` are not all zero and connected by the equations `a_1x+b_1y+c_1z=0,a_2x+b_2y+c_2z=0`, and `(p_1+lambdaq_1)x+(p_2+lambdaq_2)+(p_3+lambdaq_3)z=0` , show that `lambda=-|[a_1,b_1,c_1],[a_2,b_2,c_2],[p_1,p_2,p_3]|-:|[a_1,b_1,c_1],[a_2,b_2,c_2],[q_1,q_2,q_3]|`

if x,y and z are not all zero and connected by the equations a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0 and (p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0 show that lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|

If a_1x^3 + b_1x² + c_1x + d_1 = 0 and a_2x^3 + b_2x^2+ c_2x + d_2 = 0 have a pair of repeated common roots, then prove that |[3a_1,2b_1,c_1],[3a_2,2b_2,c_2],[a_2b_1-a_1b_2,c_1a_2-c_2a_1,d_1a_2-d_2a_1]|=0

If the origin lies in the acute angle between the lines a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0 , then show that (a_1 a_2 + b_1 b_2) c_1 c_2 lt0 .

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

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

|(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

Prove that the area of the parallelogram formed by the lines a_1x+b_1y+c_1=0,a_1x+b_1y+d_1=0,a_2x+b_2y+c_2=0, a_2x+b_2y+d_2=0, is |((d_1-c_1)(d_2-c_2))/(a_1b_2-a_2b_1)| sq. units.

If the lines a_1x+b_1y+1=0,\ a_2x+b_2y+1=0\ a n d\ a_3x+b_3y+1=0 are concurrent, show that the points (a_1, b_1),\ (a_2, b_2)a n d\ (a_3, b_3) are collinear.