The length of the latus rectum of the curve represented by $x=3(\cos t+\sin t)$ and $y=4(\cos t-\sin t)$ is:
$\frac{32 \sqrt{2}}{3}$
$9 \sqrt{2}$
$\frac{9}{\sqrt{2}}$
$\frac{9}{2}$
If the function $f(x)=x^4-31 x^2+\boldsymbol{a} x+5$ has a turning point at $x=1$, then the value of ' $\boldsymbol{a}$ ' is $\_\_\_\_$ and the function attains a $\_\_\_\_$ at $x=1$
$a=50$, local minima
$a=58$, local maxima
$a=58$, local minima
$a=-50$, local maxima
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?
$$ \frac{13}{313} $$
$$ \frac{8}{21} $$
$$ \frac{4}{9} $$
$$ \frac{8}{13} $$
$$ \int \frac{e^{\log \left(1+\frac{1}{x^2}\right)}}{x^2+\frac{1}{x^2}} d x= $$
$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x^2+1}{x \sqrt{2}}\right)+C $$
$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x^2-1}{\sqrt{2} x}\right)+C $$
$$ -\frac{1}{\sqrt{2}} \tan ^{-1}\left(x-\frac{1}{x}\right)+C $$
$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(x-\frac{1}{x}\right)+C $$
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