Consider the determinant `"Delta"=|a_1+b_1x^2a_1x^2+b_1c_1a_2+b_2x^2a_2x^2+b_2c_2a_3+b_3x^2a_3x^2+b_3c_3|0, w h e r ea_i ,b_i , c_i R(1,2,3)a n dx Rdot` Statement 1: The value of `x` satisfying `"Delta"=0` are `x=1,-1.` Statement 2: If `|a_1b_1c_1a_2b_2c_2a_3b_3c_3|=0,t h e n"Delta"=0.`

If w is a complex cube root of unity, then value of =|a_1+b_1w a_1w^2+b_1c_1+b_1 w a_2+b_2w a_2w^2+b_2c_2+b_2 w a_3+b_3w a_3w^2+b_3c_3+b_3 w | is a. 0 b. -1 c. 2 d. none of these

consider the fourth -degree polynomial equation |{:(a_(1)+b_(1)x,,a_(1)x^(2)+b_(1),,c_(1)),(a_(2)+b_(2)x^(2),,a_(2)x^(2)+b_(2),,c_(2)),(a_(3)+b_(3)x^(2),,a_(3)x^(2)+b_(3),,c_(3)):}|=0 Without expanding the determinant find all the roots of the equation.

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

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat ka n d vec c=c_1 hat i+c_2 hat j+c_3 hat k be three nonzero vectors such that vec c is a unit vector perpendicular to both vec aa n d vec bdot If the angle between vec aa n d vec b is pi//6 , then the value of |a_1b_1c_1a_2b_2c_2a_3b_3c_3| is a. 0 b. 1 c. 1/4(a_1 2+a2 2+a3 2)(b1 2+b2 2+b3 2) d. 3/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2)

If a+b+c=0 then check the nature of roots of the equation 4a x^2+3b x+2c=0w h e r ea ,b ,c in Rdot

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

Show that a real value of x will satisfy the equation (1-i x)//(1+i x)=a-i b if a^2+b^2=1,w h e r ea ,b real.

If the equation of the locus of a point equidistant from the points (a_1, b_1) and (a_2, b_2) is (a_1-a_2)x+(b_1-b_2)y+c=0 , then the value of c is

Without expanding the determinant, prove that {:[( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) ]:} ={:[( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) ]:}