MCQ
The largest term in the sequence ${a_n} = {{{n^2}} \over {{n^3} + 200}}$ is given by
- A${{529} \over {49}}$
- B${8 \over {89}}$
- ✓${{49} \over {543}}$
- DNone of these
$f(x) = \frac{{{x^2}}}{{({x^3} + 200)}}$.....(i)
$f'(x) = x\frac{{(400 - {x^3})}}{{{{({x^3} + 200)}^2}}} = 0$
When $x = {(400)^{1/3}}$ $( \because x \ne 0)$
$x = {(400)^{1/3}} - h \Rightarrow f'(x) > 0$
$x = {(400)^{1/3}} + h \Rightarrow f'(x) < 0$
$\therefore $$f(x)$ has maxima at $x = {(400)^{1/3}}$
Since $7 < {(400)^{1/3}} < 8,$ either ${a_7}$ or ${a_8}$ is the greatest term of the sequence.
$ \because {a_7} = \frac{{49}}{{543}}$ and ${a_8} = \frac{8}{{89}}$ and $\frac{{49}}{{543}} > \frac{8}{{89}}$
$\therefore $ ${a_7} = \frac{{49}}{{543}}$ is the greatest term.
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Match each entry in $List-I$ to the correct entry in $List-II$.
| $List-I$ | $List-II$ |
| ($P$) $\gamma$ equals | ($1$) $-\hat{i}-\hat{j}+\hat{k}$ |
| ($Q$) A possible choice for $\hat{n}$ is | ($2$) $\sqrt{\frac{3}{2}}$ |
| ($R$) $\overline{O R_1}$ equals | ($3$) $1$ |
| ($S$) A possible value of $\overline{O R_1} \cdot \hat{n}$ is | ($4$) $\frac{1}{\sqrt{6}} \hat{i}-\frac{2}{\sqrt{6}} \hat{j}+\frac{1}{\sqrt{6}} \hat{k}$ |
| ($5$) $\sqrt{\frac{2}{3}}$ |
The correct option is