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Match List-I with List-II.

Chose the correct answer from the options given below:

What is the dimensional formula of \( a b^{-1} \) in the equation \( \left( P + \dfrac{a}{V^{2}} \right) ( V - b ) = R T \), where letters have their usual meaning.

Given below are two statements:
Statement (I) : Dimensions of specific heat is \( [ L^{2} T^{-2} K^{-1} ] \)
Statement (II) : Dimensions of gas constant is \( [ M L^{2} T^{-1} K^{-1} ] \)
In the light of the above statements, choose the most appropriate answer from the options given below.

Given below are two statements :
Statement (I) : Planck's constant and angular momentum have same dimensions.
Statement (II) : Linear momentum and moment of force have same dimensions.
In the light of the above statements, choose the correct answer from the options given below :

A force is represented by \( F = a x^{2} + b t^{1/2} \) where \( x \) = distance and \( t \) = time. The dimensions of \( \frac{b^{2}}{a} \) are:

Least count of a vernier caliper is \(\displaystyle \frac{1}{20N} \) cm. The value of one division on the main scale is \( 1 \, \text{mm} \). Then the number of divisions of main scale that coincide with \( N \) division of vernier scale is:

Consider two physical quantities \( A \) and \( B \) related to each other as \( E = \dfrac{B - x^2}{A t} \) where \( E, x \) and \( t \) have dimensions of energy, length and time respectively. The dimension of \( AB \) is:

The de-Broglie wavelength associated with a particle of mass \(m\) and energy \(E\) is \( \dfrac{h}{\sqrt{2 m E}} \). The dimensional formula for Planck's constant is:

Applying the principle of homogeneity of dimensions, determine which one is correct, where \( T \) is time period, \( G \) is gravitational constant, \( M \) is mass, \( r \) is radius of orbit.

10.

Match List-I with List-II

Choose the correct answer from the options given below:

11.

The dimensional formula of latent heat is:

12.

Match List-I with List-II.

List-I (Number)	List-II (Significant figure)
(A) \( 1001 \)	(I) \( 3 \)
(B) \( 010.1 \)	(II) \( 4 \)
(C) \( 100.100 \)	(III) \( 5 \)
(D) \( 0.0010010 \)	(IV) \( 6 \)

Choose the correct answer from the options given below:

13.

The angle between vector \( \vec{Q} \) and the resultant of \( (2\vec{Q} + 2\vec{P}) \) and \( (2\vec{Q} - 2\vec{P}) \) is:

14.

The equation of state of real gas is given by \(\displaystyle \left( p + \frac{a}{V^{2}} \right)(V-b)=RT \), where \( P \), \( V \) and \( T \) are pressure, volume and temperature respectively and \( R \) is the universal gas constant. The dimensions of \(\displaystyle \frac{a}{b^{2}} \) is similar to that of:

15.

Three vectors \( \overrightarrow{OP} \), \( \overrightarrow{OQ} \) and \( \overrightarrow{OR} \) each of magnitude \( A \) are acting as shown in figure. The resultant of the three vectors is \( A\sqrt{x} \). The value of \( x \) is _______.

16.

For three vectors \( \vec{A} = (-x\hat{i} - 6\hat{j} - 2\hat{k}) \), \( \vec{B} = (-\hat{i} + 4\hat{j} + 3\hat{k}) \) and \( \vec{C} = (-8\hat{i} - \hat{j} + 3\hat{k}) \), if \( \vec{A}\cdot(\vec{B}\times\vec{C}) = 0 \), then value of \( x \) is ________.

17.

There are \( 100 \) divisions on the circular scale of a screw gauge of pitch \( 1 \, \text{mm} \). With no measuring quantity in between the jaws, the zero of the circular scale lies \( 5 \) divisions below the reference line. The diameter of a wire is then measured using this screw gauge. It is found that \( 4 \) linear scale divisions are clearly visible while \( 60 \) divisions on circular scale coincide with the reference line. The diameter of the wire is:

18.

The dimensional formula of angular impulse is:

19.

The equation of stationary wave is: \( y = 2a \sin\!\left( \dfrac{2\pi n t}{\lambda} \right)\cos\!\left( \dfrac{2\pi x}{\lambda} \right) \). Which of the following is NOT correct:

20.

If mass is written as \( m = k c^{P} G^{-1/2} h^{1/2} \) then the value of \( P \) will be: (Constants have their usual meaning with \( k \) a dimensionless constant).

21.

If \( G \) be the gravitational constant and \( u \) be the energy density then which of the following quantity have the dimensions as that of the \( \sqrt{uG} \):

22.

The measured value of the length of a simple pendulum is \( 20 \, \text{cm} \) with \( 2 \, \text{mm} \) accuracy. The time for \( 50 \) oscillations was measured to be \( 40 \, \text{seconds} \) with \( 1 \, \text{seconds} \) resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is \( N\% \). The value of \( N \) is:

23.

In an expression \( a \times 10^{b} \).

24.

If \( \varepsilon_0 \) is the permittivity of free space and \( E \) is the electric field, then \( \varepsilon_0 \) \(E^{2} \) has the dimensions:

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