$10\, g$ of ice at $-20^{\circ}C$ is dropped into a calorimeter containing $10\, g$ of water at $10^{\circ}C$. The specific heat of water is twice that of ice. When equilibrium is reached,the calorimeter will contain:

$A$ vibratory motion is represented by $x = 2A \cos \omega t + A \cos (\omega t + \frac{\pi}{2}) + A \cos (\omega t + \pi) + \frac{A}{2} \cos (\omega t + \frac{3\pi}{2})$. The resultant amplitude of the motion is:

The $PV$ diagram shows four different possible reversible processes performed on a monatomic ideal gas. Process $A$ is isobaric (constant pressure). Process $B$ is isothermal (constant temperature). Process $C$ is adiabatic. Process $D$ is isochoric (constant volume). For which process(es) does the temperature of the gas decrease?

The curve satisfying the differential equation,$ydx - (x + 3y^2) dy = 0$ and passing through the point $(1, 1)$,also passes through the point

$\lim _{x \rightarrow 8} \frac{\sqrt{1+\sqrt{1+x}}-2}{x-8}$ is equal to

If $p \rightarrow (q \vee r)$ is false,then the truth values of $p, q, r$ are respectively