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Question

Assertion :Consider $$I_{1}=\int_{0}^{\frac{\displaystyle \pi }{4}}e^{x^{2}}dx, I_{2}=\int_{0}^{\frac{\displaystyle \pi }{4}}e^{x}dx, I_{3}=\int_{0}^{\frac{\displaystyle \pi }{4}}e^{x^{2}}\cos\:x\:dx, I_{4}=\int_{0}^{\frac{\displaystyle \pi }{4}}e^{x^{2}}\sin\:x\:dx$$,
then $$I_{2}> I_{1}> I_{3}> I_{4}.$$ Reason: For $$x(0, 1),x> x^{2}$$ and $$\sin x> \cos x.$$


A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

The correct option is C Assertion is correct but Reason is incorrect
$$\displaystyle x> x_{2}^{2}, \forall \:x\in \left ( 0, \frac{\pi }{4} \right )$$ or $$\displaystyle e^{x}> e^{x^{2}}\:x\in \left ( 0, \frac
{\pi }{4} \right )$$
$$\displaystyle \cos x= \sin x\:\forall x\in \left ( 0, \frac{\pi }{4} \right )$$
or $$\displaystyle e^{x^{2}}\cos x> e^{x^{2}}\sin x$$
or $$\displaystyle e^{x}> e^{x^{2}}> e^{x^{2}}\cos x> e^{x^{2}}\sin x\:\forall x\in \left ( 0, \frac{\pi }{4} \right )$$
or $$\displaystyle I_{2}> I_{1}> I_{3}> I_{4}$$

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