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 roots common, 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 a_(1) x^(3) + b_(1)x^(2) + c_(1)x + d_(1) = 0 and a_(2)x^(3) + b_(2)x^(2) + c_(2)x + d_(2) = 0 a pair of repeated roots common, then prove that |{:(3a_(1)", "2b_(1) ", "c_(1)),(3a_(2)", " 2b_(2)", "c_(1)),(a_(2)","b_(1)- a_(1)b_(2)", "c_(2)a_(1)-c_(2)a_(1)", "d_(1)a_(2)-d_(2)a_(1)):}|=0

If the lines a_1x+b_1y+c_1=0 and a_2x+b_2y+c_2=0 cut the coordinae axes at concyclic points, then prove that |a_1a_2|=|b_1b_2|

The line a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 are perpendicular if:

If (a_1)/( a_2) =(b_1)/( b_2) = (C_1) /( c_2) where a_1x +b_1y +c_1=0 and a_2x +b_2y +c_2 =0, then the given pair of linear equation has …………… solutions

|[2a_1b_1, a_1b_2+a_2b_1, a_1b_3+a_3b_1] , [a_1b_2+a_2b_1, 2a_2b_2, a_2b_3+a_3b_2] , [a_1b_3+a_3b_1, a_3b_2+a_2b_3, 2a_3b_3]|=

Statement 1: If b c+q r=c a+r p=a b+p q=-1, t h e n|a p a p b q b q c r 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_2y+c^3=0 has non -trivial solutions |a_1b_1c_1a_2b_2c_2a_3b_3c_3|=0

If x ,ya n dz are not all zero and connected by the equations a_1x+b_1y+c_1z=0,a_2x+b_2y+c_2z=0,a n d(p_1+lambdaq_1)x+(p_2+lambdaq_2)+(p_3+lambdaq_3z=0) , show that lambda=-|a_1b_1c_1a_2b_2c_2p_1p_2p_3|-:|a_1b_1c_1a_2b_2c_2q_1q_2q_3|

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

If the equation x^2+b x-a=0"and" x^2-a x+b=0 have a common root, then a. a+b=0 b. a=b c. a-b=1 d. a+b=1