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. \(10\%\)

The length, breadth, and thickness of a block are given by \(l = 12 \ \text{cm}, b = 6 \ \text{cm}\ \text{and}\ t = 2.45 \ \text{cm}.\) The volume of the block according to the idea of significant figures should be:
1. \(1\times 10^2 \, \text{cm}^3\)
2. \(2 \times 10^2 \, \text{cm}^3\)
3. \(1.764 \times 10^2 \, \text{cm}^3\)
4. None of these

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%

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 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. $\left[M^0{L}^0{T}^1\right]$

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 $\frac{a}{b}$ 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 \(l=\dfrac{v_0}{\alpha}\left(1-e^{\alpha t}\right)\) , 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. \(\left[M^0L^1T^{-1}\right] \text{and} \left[T\right]\) 

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