Paragraph: Let a , b and c be three real numbers satisfying [ array lll a & b & c array ] [ array lll 1 & 9 & 7 8 & 2 & 7 7 & 3 & 7 array ]= [ array lll 0 & 0 & 0 array ] Question: Let b=6 , with a and c satisfying Eq. (E). If and are the roots of the quadratic equation a x^2+b x+c=0 , then _ n=0 ^ ( 1 + 1 )^n is

Step-by-step solution

aligned & Given, [a b c]_ 1 3 [ array lll 1 & 9 & 7 8 & 2 & 7 7 & 3 & 7 array ]_ 3 3 = [ array lll 0 & 0 & 0 array ] & [ array c a+8 b+7 c 9 a+2 b+3 c 7 a+7 b+7 c array ]= [ array l 0 0 0 array ] & a+8 b+7 c=0 & 9 a+2 b+3 c=0 & a+b+c=0 & aligned On multiplying Eq. (iii) by 2 , then subtract from Eq. (ii), we get 7 a+c=0 Again multiplying Eq. (iii) by 3 , then subtract from Eq. (ii), we get array ll & 6 a-b=0 & b=6 a and c=-7 a array If b=6, a=1 and c=-7 array ll & a x^2+b x+c=0 & x^2+6 x-7=0 & (x+7)(x-1)=0 & x=1,-7