The equation of the plane passing through the point $(1,1,1)$ and through the line of intersection of $x+2 y-z+1=0$ and $3 x-y-4 z+3=0$ is

A manufacturing company produces two items, A and B. Each toy should be processed by two machines, I and II. Machine I can be operated for maximum 10 hours 40 minutes. It takes 20 minutes for an item of A and 15 minutes for B. Machine II can be operated for a total time at 8 hours 20 minutes. It takes 5 minutes for an item A and 8 minutes for B . The profit per item of $A$ is $Rs 25$ and per item of $B$ is ₹ 18 . The formulation of an L.P.P. to maximize the profit (where $x$ is number of items A and $y$ is the number of item $B$ ) is

$$ \begin{array}{r}If\,\,\,\, y=\tan ^{-1}\left(\frac{1}{1+x+x^2}\right)+\tan ^{-1}\left(\frac{1}{x^2+3 x+3}\right) +\tan ^{-1}\left(\frac{1}{x^2+5 x+7}\right) \end{array} $$

then the value of $y^{\prime}(0)$ is

If $0 \leq x \leq \pi$ and $81^{\sin ^2 x}+81^{\cos ^2 x}=30$ Then $x$ takes the value