If \( a, b, c \) are non-zero real numbers such that \( \int_{0}^{1}\left(1+\cos ^{8} x\right)\l...

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If \( a, b, c \) are non-zero real numbers such that \( \int_{0}^{1}\left(1+\cos ^{8} x\right)\left(a x^{2}+b x+c\right) d x \) \( =\int_{0}^{2}\left(1+\cos ^{8} x\right)\left(a x^{2}+b x+c\right) d x=0 \), then the equation \( a x^{2}+b x+c=0 \) will have
(a) one root between 0 and 1 and another between 1 and 2
(b) both the roots between 0 and 1
(c) both the roots between 1 and 2
(d) None of the above
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