E, m, l and G denote energy, mass, angular momentum and gravitational constant respectively, then the dimension of $\frac{E{l}^2}{m^5{G}^2}$ are

1. Angle

2. Length

3. Mass

4. Time

From the equation \(\tan \theta=\dfrac{rg}{v^2}\), one can obtain the angle of banking \(θ\) for a cyclist taking a curve (the symbols have their usual meanings). Then say, it is:

1. Both dimensionally and numerically correct

2. Neither numerically nor dimensionally correct

3. Dimensionally correct only

4. Numerically correct only

A dimensionally consistent relation for the volume V of a liquid of coefficient of viscosity η flowing per second through a tube of radius r and length l and having a pressure difference p across its end, is

1. $V=\frac{\pi p{r}^4}{8\eta l}$

2. $V=\frac{\pi \eta l}{8p{r}^4}$

3. $V=\frac{8p\eta l}{\pi r^4}$

4. $V=\frac{\pi p\eta }{8l{r}^4}$

The velocity \(v\) (in \(\text{cm/sec}\)) of a particle is given in terms of time \(t\) (in \(\text{sec}\)) by the relation \(v=at+\dfrac{b}{t+c};\) the dimensions of \(a,~b\) and \(c\) are:

1. \(a = L^{2} , \textrm{ } b = T , \textrm{ } c = L T^{2}\)

2. \(a = L T^{2} , \textrm{ } b = L T , \textrm{ } c = L\)

3. \(a = L T^{- 2} , b = L , \textrm{ } c = T\)

4. \(a = L , \textrm{ } b = L T , \textrm{ } c = T^{2}\)

From the dimensional consideration, which of the following equation is correct ?

1. $T=2\pi \sqrt{\frac{R^3}{GM}}$

2. $T=2\pi \sqrt{\frac{GM}{R^3}}$

3. $T=2\pi \sqrt{\frac{GM}{R^2}}$

4. $T=2\pi \sqrt{\frac{R^2}{GM}}$

The position of a particle at time \(t\) is given by the relation \(l=\dfrac{v_0}{\alpha}\left(1-e^{\alpha t}\right)\) , where \(v_0\) is a constant and \(α > 0\). The dimensions of \(v_0\) and \(α\) are respectively:
1. \(\left[M^0L^1T^{-1}\right]\ \text{and}\ \left[T^{-1}\right]\)
2. \(\left[M^0L^1T^0\right] \text{and} \left[T^{-1}\right]\)
3. \(\left[M^0L^1T^{-1}\right] \text{and} \left[LT^{-2}\right]\)
4. \(\left[M^0L^1T^{-1}\right] \text{and} \left[T\right]\) 

The dimensions of $\frac{a}{b}$ in the equation $P=\frac{a-t^2}{bx}$, where P is pressure, x is distance and t is time, are 

1. MT^–2

2. M²LT^–3

3. ML³T^–1

4. LT^–3

Dimensions of $\frac{1}{{\mu }_0{\epsilon }_0}$, where symbols have their usual meaning, are 

1. [LT^–1]

2. [L^–1T]

3. [L^–2T²]

4. [L²T^–2]

The dimensions of $e^2/4\pi {\epsilon }_{0}hc$, where $e,{\epsilon }_0,h$ and c are electronic charge, electric permittivity, Planck’s constant and velocity of light in vacuum respectively 

1. $\left[M^0{L}^0{T}^0\right]$

2. $\left[M^1{L}^0{T}^0\right]$

3. $\left[M^0{L}^1{T}^0\right]$

4. $\left[M^0{L}^0{T}^1\right]$

If radius of the sphere is (5.3 ± 0.1)cm. Then percentage error in its volume will be 

1. $3+6.01\times \frac{100}{5.3}$

2. $\frac{1}{3}\times 0.01\times \frac{100}{5.3}$

3. $\left(\frac{{\displaystyle 3\times 0.1}}{{\displaystyle 5.3}}\right)\times 100$

4. $\frac{0.1}{5.3}\times 100$