Mathematics Q.1

$lim_{ x\rightarrow 0 }\frac { { \ln { \left( { \left( \cos { \left( x \right) } \right) }^{ a } \right) } } }{ { x }^{ b } } \\ =lim_{ x\rightarrow 0 }\frac { { a\ln { \left( { \left( \cos { \left( x \right) } \right) } \right) } } }{ { x }^{ b } } \\$

Let us Now Apply L'Hôpital's Rule

$lim_{ x\rightarrow 0 }\frac { { -a\tan { (x) } } }{ { bx }^{ b-1 } } \\ =lim_{ x\rightarrow 0 }\frac { { \tan { (x) } } }{ { x } } \frac { -a }{ b } \frac { 1 }{ { x }^{ b-2 } }$

$=lim_{ x\rightarrow 0 }\frac { -a }{ b } \frac { 1 }{ { x }^{ b-2 } }$

Now for Limit to be Finite

$b-2=\left\{ 0,-1,-2,-3,-4...... \right\} \\ b=\left\{ 2,1,0,-1,-2,-3...... \right\} \\ \\$

But b can Only be $b=\left\{ 2,1 \right\}$ as it is an outcome of a Dice.

Now probability is $\\ \\ P=\frac { No\quad of\quad was\quad to\quad select\quad "a" }{ Total\quad No\quad of\quad was\quad to\quad select\quad "a" } .\frac { No\quad of\quad was\quad to\quad select\quad "b" }{ Total\quad No\quad of\quad was\quad to\quad select\quad "b" } \\ P=\frac { 6 }{ 6 } \frac { 2 }{ 6 } =\frac { 1 }{ 3 } =0.33333333333333333333$