MCQ
$\int\limits_0^\infty {} [2 e^{-x}]\, dx$ where $[x]$ denotes the greatest integer function is
- A$0$
- ✓$ln\, 2$
- C$e^2$
- D$2/e$
alternatively, put $e^{-x} = t ; - x = ln\, t ; dx = \frac{1}{t}\,dt$
$\int\limits_0^1 {[2t]\frac{1}{t}dt} $ $-\int\limits_0^1 {[2t]\frac{{dt}}{t}} $ ;
$\int\limits_0^{\frac{1}{2}} {0\,dt} $ $+\int\limits_{\frac{1}{2}}^{\frac{1}{2}} {\frac{{dt}}{t}} $
$=\left. {\ln \,t} \right]_{\frac{1}{2}}^{\,1}$
$= [1] - ln\, \frac{1}{2} = ln\, 2$
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$S_n(x)=\sum_{k=1}^n \cot ^{-1}\left(\frac{1+k(k+1) x^2}{x}\right)$
where for any $x \in R , \cot ^{-1} x \in(0, \pi)$ and $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then which of the following
statements is (are) $TRUE$?
$(A)$ $S _{10}( x )=\frac{\pi}{2}-\tan ^{-1}\left(\frac{1+11 x ^2}{10 x }\right)$, for all $x >0$
$(B)$ $\lim _{n \rightarrow \infty} \cot \left(S_n(x)\right)=x$, for all $x>0$
$(C)$ The equation $S_3(x)=\frac{\pi}{4}$ has a root in $(0, \infty)$
$(D)$ $\tan \left( S _{ n }( x )\right) \leq \frac{1}{2}$, for all $n \geq 1$ and $x >0$
$2 x+4 y+2 a z=b$
$x+2 y+3 z=4$
$2 x-5 y+2 z=8$
which of the following is NOT correct?