Dimensional Analysis

The force $F$ is expressed in terms of distance $x$ and time $t$ as $F=a\sqrt{x}+b{t}^2$. The dimensions of $\mathrm{a}/\mathrm{b}$ is

Which of the following is not dimensionless?

The number of particles crossing the unit area perpendicular to the $z$-axis per unit time is given by $N=-D\left(\frac{n_2-n_1}{z_2-z_1}\right)$, where $n_1$ and $n_2$ are the numbers of particles per unit volume at $z_1$ and $z_2$, respectively, along $z$-axis. What is the dimensional formula for the diffusion constant $D$?

Choose the incorrect statement:

If $E=$ energy, $G=$ gravitational constant, $I=$ impulse and $M=$ mass, the dimension $\frac{G{I}^{2}M}{E^2}$ is same as that of,

If $A=B+\frac{C}{D+E},$ the dimensions of $B$ and $C$ are $\left[{\mathrm{M}}^0{\mathrm{LT}}^{-1}\right]$ and $\left[{\mathrm{M}}^0{\mathrm{LT}}^0\right],$ respectively. Find the dimensions of $A\mathit{,}\mathit{ }D$ and $\mathrm{E}$.

The potential energy $U$ of a particle varies with distance $x$ from a fixed origin as $U=\frac{A\sqrt{x}}{x^2+B}$, where$A$ and$B$ are dimensional
constants. The dimensional formula for $AB$ is,
 

The dimensions of capacitance in $M\mathit{,}\mathit{ }L\mathit{,}\mathit{ }T$ and $C$ (Coulomb) is given as,
 

Two quantities $A$ and $B$ are related by the relation $\frac{A}{B}=m,$ where $m$ is linear mass density and$A$ is force. The dimensions of $B$ will be same as that of,
 

When dimensions of a given physical quantity are given, the physical quantity is unique.
 

$\text{ Match the list-I with list-II }$

$\begin{array}{|llll|}\hline & & & \\ \text{ P }& \begin{array}{l}\text{ Boltzmann }\\ \text{ constant }\end{array}& \text{ (I) }& \left[{\mathrm{ML}}^0{\mathrm{T}}^0\right]\\ \text{ Q }& \begin{array}{l}\text{ Coefficient of }\\ \text{ viscosity }\end{array}& \text{ (II) }& \left[{\mathrm{ML}}^{-1}{\mathrm{T}}^{-1}\right]\\ \text{ R }& \text{ Water equivalent }& \text{ (III) }& \left[{\mathrm{MLT}}^{-3}{\mathrm{K}}^{-1}\right]\\ \text{ S }& \begin{array}{l}\text{ Coefficient of }\\ \text{ thermal }\\ \text{ conductivity }\end{array}& \text{ (IV) }& \left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{K}}^{-1}\right]\\ \hline\end{array}$

Match the following two columns.
$\begin{array}{|llll|}\hline & \text{ Column I }& \mathrm{Column} \mathrm{II}& \\ \text{ (a) }& \begin{array}{l}\text{ Electrical }\\ \text{ resistance }\end{array}& \text{ (p) }& \left[{\mathrm{M}}^1{\mathrm{L}}^3{\mathrm{T}}^{-3}{\mathrm{A}}^{-2}\right]\\ \text{ (b) }& \text{ Electrical potential }& \text{ (q) }& \left[{\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{A}}^{-2}\right]\\ \text{ (c) }& \text{ Specific resistance }& \text{ (r) }& \left[{\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{A}}^{-1}\right]\\ \text{ (d) }& \begin{array}{l}\text{ Specific }\\ \text{ conductance }\end{array}& \text{ (s) }& \text{ None of these }\\ \hline\end{array}$

 

In the relation $P=\frac{\alpha }{\beta }{\mathrm{e}}^{\frac{-\alpha Z}{k\theta }},P$ is pressure, $Z$ is the distance, $\mathrm{k}$ is Boltzmann's constant and $\theta$ is the temperature. The dimensional formula of $\beta$ will be 
 

A physical quantity of the dimensions of length that can be formed out of$c,$$G$ and $\frac{{\mathrm{e}}^2}{4\pi {ϵ}_0}$ [$c$ is the velocity of light, $G$ is the universal constant of gravitation and $e$ is a charge]
 

A spherical liquid drop is placed on a horizontal plane. A small disturbance causes the volume of the drop to oscillate. The time period of oscillation $\left(T\right)$ of the liquid drop depends on radius $\left(r\right)$ of the drop, density $\left(\rho \right)$ and surface tension $\left(S\right)$ of the liquid. Which among the following will be a possible expression for $T$ (where, $k$is a dimensionless constant)?

Planck's constant $\left(h\right)$, speed of light in vacuum $\left(c\right)$ and Newton's gravitational constant $\left(G\right)$ are three fundamental constants. Which of the following combinations of these has the dimension of length? 

If $X=3Y{Z}^2$ then the dimension of $Y$in $\mathrm{MKS}$ system, if  $X$ and $Z$ are the dimension of capacity and magnetic field, respectively is
 

Force is given by the expression $\mathrm{F}=\mathrm{Acos}\left(\mathrm{B}x\right)+\mathrm{Ccos}\left(\mathrm{Dt}\right)$ where $\mathrm{x}$ is displacement and $t$ is time. The dimension of$\left[\frac{D}{B}\right]$ is same as that of 
 

If $E\text{,}M\text{,}J$ and $G$, respectively denote energy, mass, angular momentum and gravitational constant, then$\frac{E{J}^2}{M^5{G}^2}$ has the dimensions of 

The velocity $v$ Of a particle at time $t$ Is given by, $v=at+\frac{b}{t+c},$ where $a, b$ and $c$ Are constants. The dimensions of $a, b$ and $c$ Are respectively