If $\sin \mathrm{A}+\sin \mathrm{B}=x$ and $\cos \mathrm{A}+\cos \mathrm{B}=y$, then $\sin (A+B)=$

Let mean and standard deviation of probability distribution

$$ \begin{array}{|c|c|c|c|c|} \hline \mathrm{X}=x & -3 & 0 & 1 & \alpha \\ \hline \mathrm{P}(\mathrm{X}=x) & \frac{1}{4} & \mathrm{~K} & \frac{1}{4} & \frac{1}{3} \\ \hline \end{array} $$

be $\mu$ and $\sigma$ respectively and if $\sigma-\mu=2$ then $\sigma=$

In a game a man wins $Rs \,\, 40$ if he gets 5 or 6 on a throw of a fair die and loses ₹ 20 for getting any other number on the die. If he decides to throw the die either till he gets a five or six or to a maximum of three throws, then his expected gain/loss (in rupees) is

The rate at which a substance cools in moving air, is proportional to the difference between the temperature of the substance and that of air. The temperature of air is 290 K and the substance cools from 370 K to 330 K in 10 minutes. Then the time to cool the substance upto 295 K is