Two players $A$ and $B$ alternatively toss 3 coins simultaneously. The player who gets 2 heads and 1 tail first, wins the game. If game continues until someone wins and if $A$ begins the game, the probability that B wins the game is

If $X$ is a Poisson variate satisfying the condition $3 P(X=2)=P(X=4)$, then $P(X=6)=$

Let $A=(1,2), B=(2,1), C=(-1,-1)$ be three points. If $P$ is a point such that the area of the quadrilateral $P A B C$ is twice the area of the $\triangle P A B$, then the equation of the locus of $P$ is

When the origin is shifted to the point $(h, k)$ by translating the coordinates axes, the equation $S \equiv 2 x^2-x y+y^2+2 x+3 y+1=0$ is changed to $S \equiv a x^2+2 h x y+b y^2-3=0$. Again by rotating the coordinate axes about the new origin through the angle $\theta$ in the positive direction, $S^{\prime}=0$ is changed to $A x^2+B y^2+C=0$. Then, $h+k+\tan 2 \theta=$