In a competition, $A, B$, and $C$ are participating. The probability that $A$ wins is twice that of $B$, and the probability that $B$ wins is twice that of $C$. What is the probability that $A$ loses?

The most abundant prokaryotes helpful to humans in making curd from milk and in production of antibiotics are categorised as

$A \rightarrow D$ is an endothermic reaction occurring in $3$ steps (elementary).
$(i)$ $A \rightarrow B$ $\Delta H_{i} = +ve$
$(ii)$ $B \rightarrow C$ $\Delta H_{ii} = -ve$
$(iii)$ $C \rightarrow D$ $\Delta H_{iii} = -ve$
Which of the following graphs between potential energy ($y$-axis) vs reaction coordinate ($x$-axis) correctly represents the reaction profile of $A \rightarrow D$?

$x=\cos ^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right), y=\sin ^{-1}\left(\frac{t}{\sqrt{1+t^2}}\right) \Rightarrow \frac{d y}{d x}$ is equal to

The diameters of a circle are along $2x+y-7=0$ and $x+3y-11=0$. Then,the equation of this circle,which also passes through $(5,7)$,is

The uncertainties in the velocities of two particles $A$ and $B$ are $0.05 \ m \ s^{-1}$ and $0.02 \ m \ s^{-1}$ respectively. The mass of $B$ is five times that of mass $A$. What is the ratio of uncertainties $\left(\frac{\Delta x_A}{\Delta x_B}\right)$ in their positions?