If the internal energy of an ideal gas decreases by the same amount as the work done by the system,the process is

The logical statement $[\sim(\sim p \vee q) \vee (p \wedge r)] \wedge (\sim q \wedge r)$ is equivalent to

The spring balance $A$ reads $2 \ kg$ with a block $m$ suspended from it. $A$ balance $B$ reads $5 \ kg$ when a beaker with liquid is placed on the pan of the balance. The two balances are now arranged such that the hanging mass is inside the liquid in the beaker as shown in the figure. In this situation:

Two dice are rolled simultaneously. The probability that the sum of the two numbers on the dice is a prime number is:

Two particles having position vectors $r_1 = (3 \hat{i} + 5 \hat{j}) \ m$ and $r_2 = (-5 \hat{i} - 3 \hat{j}) \ m$ are moving with velocities $v_1 = (4 \hat{i} + 3 \hat{j}) \ m/s$ and $v_2 = (a \hat{i} + 7 \hat{j}) \ m/s$. If they collide after $2 \ s$,then the value of $a$ is:

$\int (1+x-x^{-1}) e^{x+x^{-1}} dx$ is equal to :