MCQ
If $\phi \,(x) = {\log _5}\,{\log _3}\,x;$ then $\phi '\,(e)$ is equal to
- A$ e\ log\ 5$
- B$- e\ log\ 5$
- ✓$\frac{1}{{e\,\ln \,5}}$
- D$\frac{log 5}{{e}}$
$\phi(\mathrm{x})=\log _{\mathrm{s}}\left(\frac{\ln \mathrm{x}}{\ln 3}\right)$
$\phi ({\rm{x}}) = {\log _{\rm{s}}}{\rm{lnx}} - {\log _{\rm{s}}}\ln 3$
$\phi(x)=\frac{\ln \ln x}{\ln 5}-\log _{5} \ln 3$
$\phi^{\prime}(x)=\frac{1}{\ln 5} \times \frac{1}{\ln x} \times \frac{1}{x}$
${\phi ^\prime }(e) = \frac{1}{{e\ln S}}$
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Be three lines such that $\mathrm{L}_1$ is perpendicular to $\mathrm{L}_2$ and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then the point which lies on $\mathrm{L}_3$ is
Match the conditions / expressions in Column $I$ with statements in Column $II$ and indicate your answers by darkening the appropriate bubbles in $4 \times 4$ matrix given in the $ORS$.
| Column $I$ | Column $II$ |
| $(A)$ If $-1 < x < 1$, then $f$ ( $x$ ) satisfies | $(p)$ $ 0 < $ f (x) $ < 1$ |
| $(B)$ If $1 < x < 2$, then $f(x)$ satisfies | $(q)$ $\mathrm{f}(\mathrm{x}) < 0$ |
| $(C)$ If $3 < x < 5$, then $f(x)$ satisfies | $(r)$ $ \mathrm{f}(\mathrm{x}) > 0$ |
| $(D)$ If $x > 5$, then $f(x)$ satisfies | $(s)$ $ f (\mathrm{x}) < 1$ |
$(S_1)$ there exists $\mathrm{x}_{1}, \mathrm{x}_{2} \in(2,4), \mathrm{x}_{1}<\mathrm{x}_{2}$, such that $f^{\prime}\left(x_{1}\right)=-1$ and $f^{\prime}\left(x_{2}\right)=0$
$(S_2)$ there exists $\mathrm{x}_{3}, \mathrm{x}_{4} \in(2,4), \mathrm{x}_{3}<\mathrm{x}_{4}$, such that $f$ is decreasing in $\left(2, x_{4}\right)$, increasing in $\left(x_{4}, 4\right)$ and $2 f^{\prime}\left(x_{3}\right)=\sqrt{3} f\left(x_{4}\right)$.
Then