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 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) $a=L,b=LT,c=T^2$

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 $l =\frac{v_o}{\alpha }(1-e^{\alpha t})$ , 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

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) $\frac{0.1}{5.3}\times 100$

The pressure on a square plate is measured by measuring the force on the plate and the length of the sides of the plate. If the maximum error in the measurement of force and length are respectively 4% and 2%. The maximum error in the measurement of pressure is 

(1) 1%

(2) 2%

(3) 6%

(4) 8%

While measuring the acceleration due to gravity by a simple pendulum, a student makes a positive error of 1% in the length of the pendulum and a negative error of 3% in the value of time period. His percentage error in the measurement of g by the relation $g=4{\pi }^2(l/T^2)$ will be

(1) 2%

(2) 4%

(3) 7%

(4) 10%