MCQ
The function $f(x) = \log (1 + x) - {{2x} \over {2 + x}}$ is increasing on
- ✓(0, $\infty $)
- B($ - \infty $, 0)
- C$( - \infty ,\infty )$
- DNone of these
$ \Rightarrow f'(x) = \frac{1}{{1 + x}} - \frac{{(2 + x).(2 - 2x)}}{{{{(2 + x)}^2}}}$
==> $f'(x) = \frac{{{x^2}}}{{(x + 1){{(x + 2)}^2}}}$
Obviously, $f'(x) > 0$ for all $x > 0$
Hence $f(x)$ is increasing on $(0,\infty )$.
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$\overrightarrow{ r }=\hat{ i }+\lambda(-\hat{ i }+2 \hat{ j }+2 \hat{ k }), \lambda \in R \text { and }$
$\overrightarrow{ r }=\mu(2 \hat{ i }-\hat{ j }+2 \hat{ k }), \mu \in R$
respectively. If $L _3$ is a line which is perpendicular to both $L _1$ and $L _2$ and cuts both of them, then which of the following options describe(s) $L _3$ ?
$(1)$ $\overrightarrow{ r }=\frac{1}{3}(2 \hat{ i }+\hat{ k })+ t (2 \hat{ i }+2 \hat{ j }-\hat{ k }), t \in R$
$(2)$ $\overrightarrow{ i }=\frac{2}{9}(2 \hat{ i }-\hat{ j }+2 \hat{ k })+ t (2 \hat{ i }+2 \hat{ j }-\hat{ k }), t \in R$
$(3)$ $\overrightarrow{ r }=t(2 \hat{ i }+2 \hat{ j }-\hat{ k }), t \in R$
$(4)$ $\overrightarrow{ r }=\frac{2}{9}(4 \hat{ i }+\hat{ j }+\hat{ k })+ t (2 \hat{ i }+2 \hat{ j }-\hat{ k }), t \in R$