If $4$ moles of an ideal monoatomic gas at temperature $400 ~K$ is mixed with $2$ moles of another ideal monoatomic gas at temperature $700 ~K$,the temperature of the mixture is

If $f: N \rightarrow Z$ is defined by $f(n)=\begin{cases} 2 & \text{if } n=3k, k \in Z \\ 10 & \text{if } n=3k+1, k \in Z \\ 0 & \text{if } n=3k+2, k \in Z \end{cases}$,then $\{n \in N: f(n)>2\}$ is equal to

In the case of a simple pendulum executing $SHM$,at $t=0$,the bob is not at the mean position. The graph drawn between the tension $(T)$ in the string and time $(t)$ is

An inclined plane makes an angle of $30^{\circ}$ with the horizontal. $A$ solid sphere rolling down this inclined plane from rest without slipping has a linear acceleration equal to ........

The metal ion present in vitamin $B_{12}$ is

$A$ mass $M$ attached to a horizontal spring executes $S.H.M.$ of amplitude $A_1$. When the mass $M$ passes through its mean position,a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio $\left(\frac{A_1}{A_2}\right)$ is