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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Given below are two statements:
Statement I:
$\displaystyle \int_{-a}^{a} f(x),dx = \int_{0}^{a} [f(x)+f(-x)],dx$
Statement II:
$\displaystyle \int_{0}^{1} \sqrt{(1+x)(1+x^3)},dx \le \dfrac{15}{8}$
In the light of the above statements, choose the most appropriate answer from the options given below:
Solution
Statement I:
This is a standard property of definite integrals.
So Statement I is true.
Statement II:
Using AM ≥ GM:
$(1+x)(1+x^3) \le \left(\dfrac{(1+x)+(1+x^3)}{2}\right)^2$
So,
$\sqrt{(1+x)(1+x^3)} \le \dfrac{2 + x + x^3}{2}$
Integrating from $0$ to $1$:
$\displaystyle \int_0^1 \sqrt{(1+x)(1+x^3)},dx \le \dfrac{15}{8}$
Statement II is true.
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