| Let\[{{I}_{n}}=\int\limits_{I}^{e}{{{(\ell nx)}^{n}}dx,\,\,n\in N}\] |
| Statement-1: \[{{I}_{1}},\,\,\,{{I}_{2}},\,\,{{I}_{3}},...\]is an increasing sequence. |
| Statement-2: \[\ln \,\,\,x\] is an increasing function. |
A) Statement-1 is false, Statement-2 is true.
B) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
C) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D) Statement-1 is true, Statement-2 is false.
Correct Answer: A
Solution :
Statement-II is true, as if\[f(x)=\ln x\], then \[f'(x)=\frac{1}{x}>0\](as\[x>0\], so that\[f(x)\]is defined) Statement-I is not true as \[0<\ln x<1,\,\,\forall x\in (1,\,\,e)\]and hence \[{{(\ln x)}^{n}}\] decreases as \[n\] is increasing. So that \[{{I}_{n}}\] is a decreasing sequence.
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