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 two lines a_1x + b_1y + c_1 = 0 and a_2x + b_2y + c_2 = 0 cut the co-ordinates axes in concyclic points. Prove that a_1a_2 = b_1b_2

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 a+c = 2b and 2/c= 1/b+1/d then prove that a:b = c:d.

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.

Find the condition so that the two equations a_1 x^2+b_1 x+c_1=0 and a_2 x^2+b_2 x+c_2=0 will have a common root.

If the lines x=a_(1)y+b_(1),z=c_(1)y+d_(1) and x=a_(2)y+b_(2),z=c_(2)y+d_(2) are perpendicular, prove that, 1+a_(1)a_(2)+c_(1)c_(2)=0 .

Find the condition on a , b ,c ,d such that equations 2a x^3+bx^2+c x+d=0 and 2ax^2+3b x+4c=0 have a common root.

Statement 1: If b c+q r=c a+r p=a b+p q=-1, then |ap a p bq b q cr c r|=0(a b c ,p q r!=0)dot Statement 2: if system of equations a_1x+b_1y+c_1=0,a_2x+b_2y+c_2=0,a_3x+b_3y+c_3=0 has non -trivial solutions |a_1b_1c_1 a_2b_2c_2 a_3b_3c_3|=0

Show that the condition for the lines x= a_1z+b_1,y = c_1z+d_1 and x = a_2z+b_2, y = c_2z+ d_2 be perpendicular is a_1a_2+c_1c_2=0

Which of the following is not the root of the equation |[x,-6,-1],[ 2,-3x,x-3],[-3, 2x,x+2]|=0? a. 2 b. 0 c. 1 d. -3