MCQ
$\mathop {\lim }\limits_{m \to \infty } \,{\left( {\cos \frac{x}{m}} \right)^m} = $
- A$0$
- B$e$
- C$1/e$
- ✓$1$
$ = \mathop {\lim }\limits_{m \to \infty } {\left[ {1 - \left( { - \cos \frac{x}{m} + 1} \right)} \right]^m}$
$ = \mathop {\lim }\limits_{m \to \infty } {\left[ {1 - 2{{\sin }^2}\frac{x}{{2m}}} \right]^m}$
$ = {e^{\mathop {\lim }\limits_{m \to \infty } - \left( {2{{\sin }^2}\frac{x}{{2m}}} \right)\,m}}$
$ = {e^{\mathop {\lim }\limits_{m \to \infty } - 2{{\left( {\frac{{\sin \frac{x}{{2m}}}}{{x/2m}}} \right)}^2}\left( {\frac{{{x^2}}}{{4{m^2}}}} \right)\,m}}$
$ = {e^{ - 2\mathop {\lim }\limits_{m \to \infty } \frac{{{x^2}}}{{4m}}}} = {e^0} = 1$.
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$1.$ The coordinates of $\mathrm{A}$ and $\mathrm{B}$ are
$(A)$ $(3,0)$ and $(0,2)$
$(B)$ $\left(-\frac{8}{5}, \frac{2 \sqrt{161}}{15}\right)$ and $\left(-\frac{9}{5}, \frac{8}{5}\right)$
$(C)$ $\left(-\frac{8}{5}, \frac{2 \sqrt{161}}{15}\right)$ and $(0,2)$
$(D)$ $(3,0)$ and $\left(-\frac{9}{5}, \frac{8}{5}\right)$
$2.$ The orthocentre of the triangle $\mathrm{PAB}$ is
$(A)$ $\left(5, \frac{8}{7}\right)$ $(B)$ $\left(\frac{7}{5}, \frac{25}{8}\right)$
$(C)$ $\left(\frac{11}{5}, \frac{8}{5}\right)$ $(D)$ $\left(\frac{8}{25}, \frac{7}{5}\right)$
$3.$ The equation of the locus of the point whose distances from the point $\mathrm{P}$ and the line $\mathrm{AB}$ are equal, is
$(A)$ $9 x^2+y^2-6 x y-54 x-62 y+241=0$
$(B)$ $x^2+9 y^2+6 x y-54 x+62 y-241=0$
$(C)$ $9 x^2+9 y^2-6 x y-54 x-62 y-241=0$
$(D)$ $x^2+y^2-2 x y+27 x+31 y-120=0$
Give the answer question $1,2$ and $3.$