12th Class
Mathematics
Definite Integrals
Question Bank
Critical Thinking
question_answer
Let a, b, c be non-zero real numbers such that \[\int_{0}^{1}{(1+{{\cos }^{8}}x)(a{{x}^{2}}+bx+c)\,dx}=\int_{0}^{2}{(1+{{\cos }^{8}}x)(a{{x}^{2}}+bx+c)\,dx}\]Then the quadratic equation \[a{{x}^{2}}+bx+c=0\] has [IIT 1981; CEE 1993]
A)No root in (0, 2)
B)At least one root in (0, 2)
C)A double root in (0, 2)
D)None of these
Correct Answer:
B
Solution :
We have \[\int_{0}^{2}{f(x)dx=\int_{0}^{1}{f(x)dx+\int_{1}^{2}{f(x)dx}}}\],
where \[f(x)=(a{{x}^{2}}+bx+c)(1+{{\cos }^{8}}x)\]
If \[f(x)>0(<0)\,x\in (1,\,2)\] then \[\int_{1}^{2}{f(x)dx>0(<0)}\].
Thus \[f(x)=(1+{{\cos }^{8}}x)(a{{x}^{2}}+bx+c)\] must be positive for some value of x in [1, 2] and must be negative for some value of x in [1, 2].
As \[(1+{{\cos }^{8}}x)\ge 1\] it follows that if \[g(x)=a{{x}^{2}}+bx+c,\]then there exist some \[\alpha ,\beta \in (1,\,2)\] such that \[g(\alpha )>0\]and \[g(\beta )<0\]. Since g is continuous on R, therefore there exist some c between \[\alpha \]and \[\beta \]such that \[g(c)=0\]. Thus \[a{{x}^{2}}+bx+c\]=0 has at least one root in (1, 2) and hence in (0, 2).