The displacement of a particle from its mean position (in metre) is given by $y = 0.2 \sin(10\pi t + 1.5\pi) \cos(10\pi t + 1.5\pi)$. The motion of the particle is:

If $x y \neq 0, x+y \neq 0$ and $x^m y^n=(x+y)^{m+n}$,where $m, n \notin N$,then $\frac{d y}{d x}$ is equal to

$A$ vessel contains $110\,g$ of water. The heat capacity of the vessel is equivalent to $10\,g$ of water. The initial temperature of the water in the vessel is $10\,^{\circ}C$. If $220\,g$ of hot water at $70\,^{\circ}C$ is poured into the vessel,the final temperature,neglecting radiation loss,will be ........ $^{\circ}C$.

An electrochemical cell is fueled by the combustion of butane at $1$ bar and $298 K$ . Its cell potential is $\frac{ X }{ F } \times 10^3$ volts, where $F$ is the Faraday constant. The value of $X$ is $...$ .
Use : Standard Gibbs energies of formation at $298 K$ are : $\Delta_f G_{ CO _2}^{ o }=-394 kJ mol ^{-1}; \Delta_f G_{\text {water }}^{ o }=-237 kJ mol ^{-1} ; \Delta_f G_{\text {butane }}^{ o }=-18 kJ mol ^{-1}$

The energy of an electron in the $3^{rd}$ orbit of a hydrogenic atom is $-E$. The energy of an electron in the first orbit will be

If $\frac{3x^2 + 5}{(x^2 + 1)^2} = \frac{a}{x^2 + 1} + \frac{b}{(x^2 + 1)^2}$,then $(a, b) = $