Let a,b,c  be non zero real numbers such that ∫01(1+cos8x)(ax2+bx+c)dx=∫02(1+cos8x)(ax2+bx+c)dx  then the quadratic equation ax2+bx+c=0  has

∫01f(x)dx=∫02f(x)dx  =∫01f(x)dx+∫12f(x)dx⇒∫12f(x)dx=0 ∴∫12(1+cos8x)(ax2+bx+c)dx=0If ax2+bx+c=0  has no root in (1,2) then the integrand is either positive or negative and the integral cannot vanish.Hence ax2+bx+c=0  has atleast one root in (1, 2)