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The position of a particle moving on $ x $-axis is given by $ x(t) = A \sin t + B \cos^{2} t + C t^{2} + D $, where $ t $ is time. The dimension of $ \frac{ABC}{D} $ is:

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system.

The measured quantity has dimension of $[M^a L^b T^c]$. If $b = 3$, the value of $c$ is ________.

Given a charge $ q $, current $ I $ and permeability of vacuum $ \mu_{0} $. Which of the following quantity has the dimension of momentum?

Match List-I with List-II

List-I	List-II
(A) Magnetic induction	(I) $ \text{Ampere meter}^{2} $
(B) Magnetic intensity	(II) Weber
(C) Magnetic flux	(III) Gauss
(D) Magnetic moment	(IV) $ \text{Ampere meter} $

Choose the correct answer from the options given below:

The energy of a system is given as $ E(t) = \alpha^{3} e^{-\beta t} $, where $ t $ is the time and $ \beta = 0.3 \, \text{s}^{-1} $. The errors in the measurement of $ \alpha $ and $ t $ are $ 1.2\% $ and $ 1.6\% $, respectively. At $ t = 5 \, \text{s} $, maximum percentage error in the energy is:

In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic dipole moment has dimension of $ M^{P} L^{Q} T^{R} A^{S} $. The value of $ P $ and $ Q $ are:

Match the List-I with List-II.

Choose the correct answer from the options given below:

Which one of the following is the correct dimensional formula for the capacitance in $ F $? $ M $, $ L $, $ T $ and $ C $ stand for unit of mass, length, time and charge,

For the determination of refractive index of glass slab, a travelling microscope is used whose main scale contains $ 300 $ equal divisions equals to $ 15 \,\text{cm} $. The vernier scale attached to the microscope has $ 25 $ divisions equals to $ 24 $ divisions of main scale.
The least count ($ LC $) of the travelling microscope is (in $ \text{cm} $):

10.

Match List - I with List - II.

Choose the correct answer from the options given below:

11.

Match List-I with List-II

List-I	List-II
(A) Young’s Modulus	(I) $ M L^{-1} T^{-1} $
(B) Torque	(II) $ M L^{-1} T^{-2} $
(C) Coefficient of Viscosity	(III) $ M^{-1} L^{3} T^{-2} $
(D) Gravitational Constant	(IV) $ M L^{2} T^{-2} $

Choose the correct answer from the options given below:

12.

If $ \mu_{0} $ and $ \epsilon_{0} $ are the permeability and permittivity of free space, respectively, then the dimension of $ \left( \frac{1}{\mu_{0}\epsilon_{0}} \right) $ is:

13.

The expression given below shows the variation of velocity $ (v) $ with time $ (t) $, $ v = A t^2 + \frac{B t}{C + t} $. The dimension of $ ABC $ is:

14.

If $ B $ is magnetic field and $ \mu_0 $ is permeability of free space, then the dimensions of $ (B / \mu_0) $ is:

15.

The pair of physical quantities not having same dimensions is:

16.

The electric flux is $ \phi = \alpha \sigma + \beta \lambda $, where $ \lambda $ and $ \sigma $ are linear and surface charge density, respectively. $ \left( \frac{\alpha}{\beta} \right) $ represents:

17.

A person measures mass of $ 3 $ different particles as $ 435.42 \, \text{g} $, $ 226.3 \, \text{g} $ and $ 0.125 \, \text{g} $. According to the rules for arithmetic operations with significant figures, the addition of the mass of $ 3 $ particles will be:

18.

Match List – I with List – II

List – 1	List – II
(A) Angular Impulse	(I) $ [ M^0 L^2 T^{-2} ] $
(B) Latent Heat	(II) $ [ M L^2 T^{-3} A^{-1} ] $
(C) Electrical resistivity	(III) $ [ M L^2 T^{-1} ] $
(D) Electromotive force	(IV) $ [ M L^3 T^{-3} A^{-2} ] $

Choose the correct answer from the options given below ;

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