Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: Moment of inertia of a circular disc of mass $M$ and radius $R$ about $X, Y$ axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be $I_{\mathrm{x}}, I_{\mathrm{y}}$ and $I_{\mathrm{z}}$, respectively. The respective radii of gyration about all the three axes will be the same. Reason R: A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below:
(1) Both A and R are correct but R is not the correct explanation of A.
(2) A is not correct but R is correct.
(3) A is correct but R is not correct.
(4) Both A and R are correct and R is the correct explanation of A.

Match List-I with List-II:
List-I: (a) MI of the rod (length $L$, Mass $M$, about an axis $\perp$ to the rod passing through the midpoint) (b) MI of the rod (length $L$, Mass $2M$, about an axis $\perp$ to the rod passing through one of its end) (c) MI of the rod (length $2L$, Mass $M$, about an axis $\perp$ to the rod passing through its midpoint) (d) MI of the rod (Length $2L$, Mass $2M$, about an axis $\perp$ to the rod passing through one of its end)
List-II: (i) $\frac { 8 M L ^ { 2 } } { 3 }$ (ii) $\frac { M L ^ { 2 } } { 12 }$ (iii) $\frac { 2 M L ^ { 2 } } { 3 }$
Choose the correct answer from the options given below:
(1) (a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)
(2) (a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)
(3) (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)
(4) (a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)

A mass $M$ hangs on a massless rod of length $l$ which rotates at a constant angular frequency. The mass $M$ moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular momentum of $M$ about point $A$ is $L _ { A }$ which lies in the positive $z$ direction and the angular momentum of $M$ about $B$ is $L _ { B }$. The correct statement for this system is:
(1) $L _ { A }$ and $L _ { B }$ are both constant in magnitude and direction
(2) $L _ { B }$ is constant in direction with varying magnitude
(3) $L _ { B }$ is constant, both in magnitude and direction
(4) $L _ { A }$ is constant, both in magnitude and direction

The minimum and maximum distances of a planet revolving around the Sun are $x_{1}$ and $x_{2}$. If the minimum speed of the planet on its trajectory is $v_{0}$, then its maximum speed will be:
(1) $\frac{v_{0} x_{1}^{2}}{x_{2}^{2}}$
(2) $\frac{v_{0} x_{2}^{2}}{x_{1}^{2}}$
(3) $\frac{v_{0} x_{1}}{x_{2}}$
(4) $\frac{v_{0} x_{2}}{x_{1}}$

The figure shows two solid discs with radius $R$ and $r$ respectively. If mass per unit area is the same for both, what is the ratio of MI of bigger disc around axis $AB$ (which is $\perp$ to the plane of the disc and passing through its centre) to MI of smaller disc around one of its diameters lying on its plane? Given $M$ is the mass of the larger disc. (MI stands for moment of inertia)
(1) $R ^ { 2 } : r ^ { 2 }$
(2) $2 r ^ { 4 } : R ^ { 4 }$
(3) $2 R ^ { 2 } : r ^ { 2 }$
(4) $2 R ^ { 4 } : r ^ { 4 }$

Moment of inertia of a square plate of side $l$ about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:
(1) $\frac{Ml^{2}}{6}$
(2) $\frac{2}{3}Ml^{2}$
(3) $Ml^{2}$
(4) $\frac{Ml^{2}}{12}$

The solid cylinder of length 80 cm and mass $M$ has a radius of 20 cm. Calculate the density of the material used if the moment of inertia of the cylinder about an axis $CD$ parallel to $AB$ as shown in figure is $2.7\text{ kg m}^2$.
(1) $1.49 \times 10^2\text{ kg m}^{-3}$
(2) $7.5 \times 10^1\text{ kg m}^{-3}$
(3) $14.9\text{ kg m}^{-3}$
(4) $7.5 \times 10^2\text{ kg m}^{-3}$

In Millikan's oil drop experiment, what is viscous force acting on an uncharged drop of radius $2.0 \times 10^{-5}\mathrm{~m}$ and density $1.2 \times 10^{3}\mathrm{~kg~m}^{-3}$? Take viscosity of liquid $= 1.8 \times 10^{-5}\mathrm{~N~s~m}^{-2}$. (Neglect buoyancy due to air).
(1) $5.8 \times 10^{-10}\mathrm{~N}$
(2) $3.9 \times 10^{-10}\mathrm{~N}$
(3) $1.8 \times 10^{-10}\mathrm{~N}$
(4) $3.8 \times 10^{-11}\mathrm{~N}$

The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of $9.0 \times 10 ^ { 3 } \mathrm {~km}$. Find the mass of Mars. \{Given $\frac { 4 \pi ^ { 2 } } { G } = 6 \times 10 ^ { 11 } \mathrm {~N} ^ { - 1 } \mathrm {~m} ^ { - 2 } \mathrm {~kg} ^ { 2 }$ \}
(1) $5.96 \times 10 ^ { 19 } \mathrm {~kg}$
(2) $3.25 \times 10 ^ { 21 } \mathrm {~kg}$
(3) $7.02 \times 10 ^ { 25 } \mathrm {~kg}$
(4) $6.00 \times 10 ^ { 23 } \mathrm {~kg}$

Two blocks of masses 3 kg and 5 kg are connected by a metal wire going over a smooth pulley. The breaking stress of the metal is $\frac{24}{\pi} \times 10^2\text{ N m}^{-2}$. What is the minimum radius of the wire? (take $g = 10\text{ m s}^{-2}$)
(1) 1250 cm
(2) 1.25 cm
(3) 125 cm
(4) 12.5 cm

An ideal gas is expanding such that $PT^{3} =$ constant. The coefficient of volume expansion of the gas is:
(1) $\frac{2}{T}$
(2) $\frac{3}{T}$
(3) $\frac{1}{T}$
(4) $\frac{4}{T}$

The length of metallic wire is $l _ { 1 }$ when tension in it is $T _ { 1 }$. It is $l _ { 2 }$ when the tension is $T _ { 2 }$. The original length of the wire will be :
(1) $\frac { T _ { 2 } l _ { 1 } + T _ { 1 } l _ { 2 } } { T _ { 1 } + T _ { 2 } }$
(2) $\frac { l _ { 1 } + l _ { 2 } } { 2 }$
(3) $\frac { T _ { 2 } l _ { 1 } - T _ { 1 } l _ { 2 } } { T _ { 2 } - T _ { 1 } }$
(4) $\frac { \mathrm { T } _ { 1 } l _ { 1 } - \mathrm { T } _ { 2 } l _ { 2 } } { \mathrm {~T} _ { 2 } - \mathrm { T } _ { 1 } }$

A raindrop with radius $\mathrm { R } = 0.2 \mathrm {~mm}$ falls from a cloud at a height $\mathrm { h } = 2000 \mathrm {~m}$ above the ground. Assume that the drop is spherical throughout its fall and the force of buoyance may be neglected, then the terminal speed attained by the raindrop is: [Density of water $f _ { \mathrm { w } } = 1000 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$ and Density of air $f _ { \mathrm { a } } = 1.2 \mathrm {~kg} \mathrm {~m} ^ { - 3 } , \mathrm {~g} = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ Coefficient of viscosity of air $= 1.8 \times 10 ^ { - 5 } \mathrm {~N} \mathrm {~s} \mathrm {~m} ^ { - 2 }$ ]
(1) $250.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $43.56 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $4.94 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $14.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

The temperature of equal masses of three different liquids $x$, $y$ and $z$ are $10^\circ\text{C}$, $20^\circ\text{C}$ and $30^\circ\text{C}$ respectively. The temperature of mixture when $x$ is mixed with $y$ is $16^\circ\text{C}$ and that when $y$ is mixed with $z$ is $26^\circ\text{C}$. The temperature of mixture when $x$ and $z$ are mixed will be:
(1) $25.62^\circ\text{C}$
(2) $20.28^\circ\text{C}$
(3) $28.32^\circ\text{C}$
(4) $23.84^\circ\text{C}$

A balloon carries a total load of 185 kg at normal pressure and temperature of $27^{\circ}\mathrm{C}$. What load will the balloon carry on rising to a height at which the barometric pressure is 45 cm of Hg and the temperature is $-7^{\circ}\mathrm{C}$. Assuming the volume constant?
(1) 214.15 kg
(2) 123.54 kg
(3) 219.07 kg
(4) 181.46 kg

A refrigerator consumes an average 35 W power to operate between temperature $-10^\circ\text{C}$ to $25^\circ\text{C}$. If there is no loss of energy then how much average heat per second does it transfer?
(1) $350\text{ J s}^{-1}$
(2) $298\text{ J s}^{-1}$
(3) $263\text{ J s}^{-1}$
(4) $35\text{ J s}^{-1}$

A mass of 5 kg is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length 4 m has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed?
(1) $4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
(2) $8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
(3) $5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
(4) $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

A cylindrical container of volume $4.0 \times 10^{-3}\text{ m}^3$ contains one mole of hydrogen and two moles of carbon dioxide. Assume the temperature of the mixture is 400 K. The pressure of the mixture of gases is:
[Take gas constant as $8.3\text{ J mol}^{-1}\text{K}^{-1}$]
(1) $24.9 \times 10^3\text{ Pa}$
(2) $249 \times 10^1\text{ Pa}$
(3) $24.9 \times 10^5\text{ Pa}$
(4) 24.9 Pa

The two thin coaxial rings, each of radius $a$ and having charges $+Q$ and $-Q$ respectively are separated by a distance of $s$. The potential difference between the centres of the two rings is:
(1) $\frac{Q}{2\pi\varepsilon_0}\left[\frac{1}{a} - \frac{1}{\sqrt{s^2 + a^2}}\right]$
(2) $\frac{Q}{4\pi\varepsilon_0}\left[\frac{1}{a} - \frac{1}{\sqrt{s^2 + a^2}}\right]$
(3) $\frac{Q}{4\pi\varepsilon_0}\left[\frac{1}{a} + \frac{1}{\sqrt{s^2 + a^2}}\right]$
(4) $\frac{Q}{2\pi\varepsilon_0}\left[\frac{1}{a} + \frac{1}{\sqrt{s^2 + a^2}}\right]$

A parallel-plate capacitor with plate area $A$ has separation $d$ between the plates. Two dielectric slabs of dielectric constant $K_1$ and $K_2$ of same area $\frac{A}{2}$ and thickness $\frac{d}{2}$ are inserted in the space between the plates. The capacitance of the capacitor will be given by:
(1) $\frac{\varepsilon_0 A}{d}\left(\frac{1}{2} + \frac{K_1 K_2}{K_1 + K_2}\right)$
(2) $\frac{\varepsilon_0 A}{d}\left(\frac{1}{2} + \frac{2(K_1 + K_2)}{K_1 K_2}\right)$
(3) $\frac{\varepsilon_0 A}{d}\left(\frac{1}{2} + \frac{K_1 + K_2}{K_1 K_2}\right)$
(4) $\frac{\varepsilon_0 A}{d}\left(\frac{1}{2} + \frac{K_1 K_2}{2(K_1 + K_2)}\right)$

Five identical cells each of internal resistance $1\Omega$ and emf 5 V are connected in series and in parallel with an external resistance $R$. For what value of $R$, current in series and parallel combination will remain the same?
(1) $1\Omega$
(2) $5\Omega$
(3) $25\Omega$
(4) $10\Omega$

Two resistors $R _ { 1 } = ( 4 \pm 0.8 ) \Omega$ and $R _ { 2 } = ( 4 \pm 0.4 ) \Omega$ are connected in parallel. The equivalent resistance of their parallel combination will be:
(1) $( 4 \pm 0.4 ) \Omega$
(2) $( 2 \pm 0.4 ) \Omega$
(3) $( 4 \pm 0.3 ) \Omega$
(4) $( 2 \pm 0.3 ) \Omega$

Consider the combination of two capacitors $C _ { 1 }$ and $C _ { 2 }$, with $C _ { 2 } > C _ { 1 }$, when connected in parallel, the equivalent capacitance is 10 times the equivalent capacitance of the same connected in series. Calculate the ratio of capacitors, $\frac { C _ { 2 } } { C _ { 1 } }$.
(1) $4 + \sqrt { 15 }$
(2) $2 + \sqrt { 15 }$
(3) 9
(4) $\frac { 15 } { 4 }$

An electric bulb of 500 W at 100 V is used in a circuit having a 200 V supply. Calculate the resistance $R$ to be connected in series with the bulb so that the power delivered by the bulb is 500 W.
(1) $30\,\Omega$
(2) $5\,\Omega$
(3) $20\,\Omega$
(4) $10\,\Omega$

Two ions of masses 4 amu and 16 amu have charges $+2e$ and $+3e$ respectively. These ions pass through the region of the constant perpendicular magnetic field. The kinetic energy of both ions is the same. Then:
(1) lighter ion will be deflected less than heavier ion
(2) lighter ion will be deflected more than heavier ion
(3) both ions will be deflected equally
(4) no ion will be deflected