MCQ
$4\, {\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}} = $
- A$\frac{\pi }{2}$
- B$\frac{\pi }{3}$
- ✓$\frac{\pi }{4}$
- DNone of these
$ = 2\, {\tan ^{ - 1}}\left[ {\frac{{\frac{2}{5}}}{{1 - \frac{1}{{25}}}}} \right] - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}}$
$ = 2\, {\tan ^{ - 1}}\left( {\frac{5}{{12}}} \right) - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}}$
$ = {\tan ^{ - 1}}\left[ {\frac{{\frac{5}{6}}}{{1 - \frac{{25}}{{144}}}}} \right] - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}}$
$ = {\tan ^{ - 1}}\left( {\frac{{120}}{{119}}} \right) - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}}$
$ = {\tan ^{ - 1}}\left( {\frac{{120}}{{119}}} \right) + {\tan ^{ - 1}}\left[ {\frac{{\frac{1}{{99}} - \frac{1}{{70}}}}{{1 + \frac{1}{{99}}.\frac{1}{{70}}}}} \right]$
$ = {\tan ^{ - 1}}\left( {\frac{{120}}{{119}}} \right) + {\tan ^{ - 1}}\left( {\frac{{ - 29}}{{6931}}} \right)$
$ = {\tan ^{ - 1}}\frac{{120}}{{119}} - {\tan ^{ - 1}}\frac{{29}}{{6931}} = {\tan ^{ - 1}}\frac{{120}}{{119}} - {\tan ^{ - 1}}\frac{1}{{239}}$
$ = {\tan ^{ - 1}}\left[ {\frac{{\frac{{120}}{{119}} - \frac{1}{{239}}}}{{1 + \frac{{120}}{{119}} \times \frac{1}{{239}}}}} \right] $
$= {\tan ^{ - 1}}(1) = \frac{\pi }{4}$.
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$P(x) = 2x^3 + x^2 + 3x - 2? $
$(i)$ It has exactly one positive real root.
$(ii)$ It has either one or three negative roots.
$(iii)$It has a root between $0$ and $1.$
$(iv)$ It must have exactly two real roots.
$(v)$ It has a negative root between $- 2$ and $-1.$
$(vi)$ It has no complex roots.
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