The energy $E$ of a system is a function of time $t$ and is given by $E(t) = \alpha t - \beta t^3$,where $\alpha$ and $\beta$ are constants. The dimensions of $\alpha$ and $\beta$ are

Force $(F)$ and density $(d)$ are related as $F = \frac{\alpha}{\beta + \sqrt{d}}$. The dimensions of $\alpha$ are:

The efficiency of an engine is given by $\eta = \frac{\alpha \beta}{\sin \theta} \cdot \log_{e} \frac{\beta x}{kT}$, where $\alpha$ and $\beta$ are constants. If $T$ is the absolute temperature, $k$ is the Boltzmann constant, $\theta$ is angular displacement, and $x$ is distance, then the incorrect statement is:

$A$ modified gravitational potential is given by $V = -\frac{GM}{r} + \frac{A}{r^2}$. If the constant $A$ is expressed in terms of gravitational constant $G$, mass $M$, and velocity of light $c$, then from dimensional analysis, $A$ is:

Given that the displacement of an oscillating particle is given by $y = A \sin(Bx + Ct + D)$. The dimensional formula for $ABCD$ is: (Here $x$ is position,$t$ is time)

If $E$ and $E_0$ represent the energies, and $t$ and $t_0$ represent the times, which of the following relations is dimensionally correct?