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

1. $a=L^2,b=T,c=L{T}^2$

2. $a=L{T}^2,b=LT,c=L$

3. $a=L{T}^{-2},b=L,c=T$

4. $a=L,b=LT,c=T^2$

The SI unit of surface tension is

1. Dyne/cm

2. Newton/cm

3. Newton/metre

4. Newton-metre

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

The position of a particle at time t is given by the relation $l<mspace\; width="1em"></mspace>=\frac{v_o}{\alpha }(1-e^{\alpha t})$ , where v₀ is a constant and α > 0. The dimensions of v₀ and α are respectively

1. $M^0{L}^1{T}^{-1}$ and T^–1

2. $M^0{L}^1{T}^0$ and T^–1

3. $M^0{L}^1{T}^{-1}$ and LT^–2

4. $M^0{L}^1{T}^{-1}$ and T

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. $V=\frac{\pi p\eta }{8l{r}^4}$

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. $T=2\pi \sqrt{\frac{R^2}{GM}}$

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]

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 $\tan \theta =\frac{rg}{v^2}$, 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