A hypothetical equation is given as $\rho = a \sqrt{P} + bT^2$, where $\rho$ is density, $P$ is pressure, and $T$ is temperature. The dimensions of $\frac{a}{b}$ are:

Consider a new system of units in which c(speed of light in vacuum), h(Planck’s constant) and G(gravitational constant) are taken as fundamental units. Which of the following would correctly represent mass in this new system?

If the unit of length and force be increased four times, then the unit of energy is

The time period ($T$) of a simple pendulum depends on its length ($l$) and acceleration due to gravity ($g$). Which of the following is dimensionally consistent with the principle of homogeneity?

The velocity of a wave is given by $v = \sqrt{\frac{T}{\mu}} + ax^2$, where $T$ is tension, $\mu$ is linear density, and $x$ is displacement. What are the dimensions of $a$?

If the speed $v$ of transverse waves on a stretched string depends on the tension $T$ in the string and the mass per unit length $μ$ of the string, which of the following is dimensionally consistent?

Let L denote the self-inductance of coil which is in series with a capacitor of capacitance C. Which of the following has the unit second?

The time period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}} + kx$, where $l$ is the length, $g$ is acceleration due to gravity, and $x$ is displacement. The dimensions of $k$ are:

Given the equation $P = \frac{a}{V} - \frac{b}{V^2} + cT$, where $P$ is pressure, $V$ is volume, and $T$ is temperature. What are the dimensions of $abc$?

The damping force of an oscillating particle is observed to be proportional to velocity. The constant of proportionality can be measured in

Length is measured in metre and time in second as usual. But a new unit of mass is so chosen that G=1. This new unit of mass is equal to