Paragraph: Match the condition of column-I with corresponding number of real roots of f(x)=0 is column-II and number of points of non-derivability of y=f(|x|) in column-III, where f(x)=a x^2+b x+c . ( array l|l|l Column-I & Column-II & Column-III (I) a^2+b^2+c^2-a b-b c-c a 0 & (i) 0 & (P) 0 (II) a^2+b^2+c^2+a b+b c+c a 0 & (ii) 1 & (Q) 1 (III) 3 (a^2+b^2+c^2+1 ) 2(a+b+c+a b+b c+c a) & (iii) 2 & (R) 3 (IV) a^2+b^2+c^

Step-by-step solution

I) array ll & a^2+b^2+c^2-a b-b c-c a 0 & (a-b)^2+(b-c)^2+(c-a)^2 0 a=b=c & f(x)=a x^2+a x+a & D < 0 & no real root & |f(|x|)| is always derivable except one point. & (I) (i) (Q) array (II) array ll & a^2+b^2+c^2+a b+b c+c a 0 (a+b)^2+(b+c)^2+(c+a)^2 0 & a+b=0, b+c=0, c+a=0 & a=b=c=0 & f(x)=a x^2+b x+c is an identify & f(x)=0 has infinite roots & f(x) is always derivable. & (II) (iv) (P) array (III) aligned & (a-1)^2+(b-1)^2+(c-1)^2+(a-b)^2+(b-c)^2+(c-a)^2 0 & a=b=c=1 aligned f(x)=0 has no real root and |f(|x|)| is