Advika chooses one of three scarves every morning: Red, Blue, or Green.
The probability she chooses Red is $20 \%$.
The probability she chooses Blue is twice the probability of choosing Red.
On the remaining days she wears a Green scarf.
Once a scarf is chosen, she decides whether to wear a Hat (H) and Sunglasses (S).
These choices are independent of each other but depend on the scarf colour:
$$ \begin{array}{|l|l|l|} \hline \text { Scarf colour } & \mathbf{P ( H )} & \mathbf{P ( S )} \\ \hline \text { Red } & 0.5 & 0.8 \\ \hline \text { Blue } & 0.4 & 0.5 \\ \hline \text { Green } & 0.1 & 0.5 \\ \hline \end{array} $$
Advika is spotted outdoors wearing both a Hat and Sunglasses.
What is the probability that she is wearing the Red scarf?
$$ \frac{13}{313} $$
$$ \frac{8}{21} $$
$$ \frac{4}{9} $$
$$ \frac{8}{13} $$
$$ \int \frac{e^{\log \left(1+\frac{1}{x^2}\right)}}{x^2+\frac{1}{x^2}} d x= $$
$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x^2+1}{x \sqrt{2}}\right)+C $$
$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x^2-1}{\sqrt{2} x}\right)+C $$
$$ -\frac{1}{\sqrt{2}} \tan ^{-1}\left(x-\frac{1}{x}\right)+C $$
$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(x-\frac{1}{x}\right)+C $$
Which of the following is NOT a comer point of the feasible region determined by the constraints:
$$ \begin{aligned} & x+2 y \leq 4 \\ & x+y \geq 2 \\ & x \geq 0 \text { and } y \geq 0 \end{aligned} $$
$(0,2)$
$(4,0)$
$(0,0)$
$(2,0)$
The angle between the two lines whose direction cosines satisfy the relations $\boldsymbol{l}+\boldsymbol{m}+\boldsymbol{n}=\mathbf{0}$ and $\boldsymbol{l}^{\mathbf{2}}=\boldsymbol{m}^{\mathbf{2}}+\boldsymbol{n}^{\mathbf{2}}$ is
$\frac{\pi}{2}$
$\frac{\pi}{4}$
$\frac{\pi}{3}$
$\frac{\pi}{6}$
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