The number density of molecules of a gas depends on their distance r from the origin as, $n ( r ) = n _ { 0 } e ^ { - \alpha r ^ { 4 } }$. Then the number of molecules is proportional to:
(1) $n _ { 0 } \alpha ^ { - 3 }$
(2) $\sqrt { n _ { 0 } } \alpha ^ { \frac { 1 } { 2 } }$
(3) $n _ { 0 } \alpha ^ { \frac { - 3 } { 4 } }$
(4) $n _ { 0 } \alpha ^ { \frac { 1 } { 4 } }$

A mixture of 2 moles of helium gas (atomic mass $= 4 \mathrm { u }$), and 1 mole of argon gas (atomic mass $= 40 \mathrm { u }$) is kept at 300 K in a container. The ratio of their rms speeds $\frac { V _ { \text {rms} } (\text{helium}) } { V _ { \text {rms} } (\text{argon}) }$, is close to:
(1) 0.45
(2) 2.24
(3) 0.32
(4) 3.16

A uniformly charged ring of radius $3a$ and total charge $q$ is placed in $x - y$ plane centred at origin. A point charge $q$ is moving towards the ring along the $z$-axis and has speed $v$ at $z = 4a$. The minimum value of $v$ such that it crosses the origin is:
(1) $\sqrt { \frac { 2 } { m } } \left( \frac { 1 } { 15 } \frac { q ^ { 2 } } { 4 \pi \epsilon _ { 0 } a } \right) ^ { 1 / 2 }$
(2) $\sqrt { \frac { 2 } { m } } \left( \frac { 4 } { 15 } \frac { q ^ { 2 } } { 4 \pi \epsilon _ { 0 } a } \right) ^ { 1 / 2 }$
(3) $\sqrt { \frac { 2 } { m } } \left( \frac { 1 } { 5 } \frac { q ^ { 2 } } { 4 \pi \epsilon _ { 0 } a } \right) ^ { 1 / 2 }$
(4) $\sqrt { \frac { 2 } { m } } \left( \frac { 2 } { 15 } \frac { q ^ { 2 } } { 4 \pi \epsilon _ { 0 } a } \right) ^ { 1 / 2 }$

A submarine experiences a pressure of $5.05 \times 10 ^ { 6 } \mathrm {~Pa}$ at a depth of $\mathrm { d } _ { 1 }$ in a sea. When it goes further to a depth of $\mathrm { d } _ { 2 }$, it experiences a pressure of $8.08 \times 10 ^ { 6 } \mathrm {~Pa}$. Then $\mathrm { d } _ { 2 } - \mathrm { d } _ { 1 }$ is approximately (density of water $= 10 ^ { 3 } \mathrm {~kg} / \mathrm { m } ^ { 3 }$ and acceleration due to gravity $= 10 \mathrm {~ms} ^ { - 2 }$ ):
(1) 600 m
(2) 500 m
(3) 300 m
(4) 400 m

Figure shows charge ($q$) versus voltage ($V$) graph for series and parallel combination of two given capacitors. The capacitances are:
(1) $60 \mu \mathrm {~F}$ and $40 \mu \mathrm {~F}$
(2) $50 \mu \mathrm {~F}$ and $30 \mu \mathrm {~F}$
(3) $20 \mu \mathrm {~F}$ and $30 \mu \mathrm {~F}$
(4) $40 \mu \mathrm {~F}$ and $10 \mu \mathrm {~F}$

A current of 5 A passes through a copper conductor (resistivity $= 1.7 \times 10 ^ { - 8 } \Omega \mathrm {~m}$) of radius of cross-section 5 mm. Find the mobility of the charges if their drift velocity is $1.1 \times 10 ^ { - 3 } \mathrm {~ms} ^ { - 1 }$.
(1) $1.5 \mathrm {~m} ^ { 2 } \mathrm {~V} ^ { - 1 } \mathrm {~s} ^ { - 1 }$
(2) $1.8 \mathrm {~m} ^ { 2 } \mathrm {~V} ^ { - 1 } \mathrm {~s} ^ { - 1 }$
(3) $1.0 \mathrm {~m} ^ { 2 } \mathrm {~V} ^ { - 1 } \mathrm {~s} ^ { - 1 }$
(4) $1.3 \mathrm {~m} ^ { 2 } \mathrm {~V} ^ { - 1 } \mathrm {~s} ^ { - 1 }$

In the given circuit, an ideal voltmeter connected across the $10 \Omega$ resistance reads 2 V. The internal resistance $r$, of each cell is: The circuit has two cells each of EMF 1.5 V and internal resistance $r\,\Omega$.
(1) $0 \Omega$
(2) $1.5 \Omega$
(3) $0.5 \Omega$
(4) $1 \Omega$

A source of sound S is moving with a velocity of $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ towards a stationary observer. The observer measures the frequency of the source as 1000 Hz . What will be the apparent frequency of the source when it is moving away from the observer after crossing him? (Take velocity of sound in air is $350 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ )
(1) 750 Hz
(2) 857 Hz
(3) 1143 Hz
(4) 807 Hz

Four equal point charges $Q$ each are placed in the $xy$ plane at $( 0,2 ) , ( 4,2 ) , ( 4 , - 2 )$ and $( 0 , - 2 )$. The work required to put a fifth charge $Q$ at the origin of the coordinate system will be:
(1) $\frac { Q ^ { 2 } } { 4 \pi \epsilon _ { 0 } }$
(2) $\frac { Q ^ { 2 } } { 2 \sqrt { 2 } \pi \epsilon _ { 0 } }$
(3) $\frac { Q ^ { 2 } } { 4 \pi \epsilon _ { 0 } } \left( 1 + \frac { 1 } { \sqrt { 5 } } \right)$
(4) $\frac { Q ^ { 2 } } { 4 \pi \epsilon _ { 0 } } \left( 1 + \frac { 1 } { \sqrt { 3 } } \right)$

Two equal resistances when connected in series to a battery, consume electric power of 60 W. If these resistances are now connected in parallel combination to the same battery, the electric power consumed will be:
(1) 60 W
(2) 240 W
(3) 120 W
(4) 30 W

An ideal battery of emf $4 V$ and resistance $R$ are connected in series in the primary circuit of a potentiometer of length $1 m$ and resistance $5 \Omega$. The value of $R$, to give a potential difference of $5 m \mathrm {~V}$ across $10 c m$ of potentiometer wire, is:
(1) $480 \Omega$
(2) $495 \Omega$
(3) $395 \Omega$
(4) $490 \Omega$

A moving coil galvanometer allows a full scale current of $10 ^ { - 4 } \mathrm {~A}$. A series resistance of $2 \times 10 ^ { 4 } \Omega$ is required to convert the galvanometer into a voltmeter of range $0 - 5 \mathrm {~V}$. Therefore, the value of shunt resistance required to convert the above galvanometer into an ammeter of range $0 - 10 \mathrm {~mA}$ is:
(1) $100 \Omega$
(2) $200 \Omega$
(3) $300 \Omega$
(4) $10 \Omega$

In a Wheatstone bridge (see fig.), Resistances $P$ and $Q$ are approximately equal. When $R = 400 \Omega$, the bridge is balanced. On interchanging P and Q, the value of R, for balance, is $405 \Omega$. The value of $Y$ is close to
(1) 401.5 ohm
(2) 404.5 ohm
(3) 403.5 ohm
(4) 402.5 ohm

Two electric bulbs, rated at $( 25 \mathrm {~W} , 220 \mathrm {~V} )$ and $( 100 \mathrm {~W} , 220 \mathrm {~V} )$, are connected in series across a 220 V voltage source. If the 25 W and 100 W bulbs draw powers $P _ { 1 }$ and $P _ { 2 }$ respectively, then:
(1) $P _ { 1 } = 4 \mathrm {~W} , P _ { 2 } = 16 \mathrm {~W}$
(2) $P _ { 1 } = 9 \mathrm {~W} , P _ { 2 } = 16 \mathrm {~W}$
(3) $P _ { 1 } = 16 \mathrm {~W} , P _ { 2 } = 9 \mathrm {~W}$
(4) $P _ { 1 } = 16 \mathrm {~W} , P _ { 2 } = 4 \mathrm {~W}$

In an experiment, electrons are accelerated, from rest, by applying a voltage of 500 V. Calculate the radius of the path if a magnetic field 100 mT is then applied. [Charge of the electron $= 1.6 \times 10 ^ { - 19 } \mathrm { C }$, Mass of the electron $= 9.1 \times 10 ^ { - 31 } \mathrm {~kg}$]
(1) $7.5 \times 10 ^ { - 3 } \mathrm {~m}$
(2) $7.5 \times 10 ^ { - 2 } \mathrm {~m}$
(3) 7.5 m
(4) $7.5 \times 10 ^ { - 4 } \mathrm {~m}$

A proton and an $\alpha$-particle (with their masses in the ratio of $1 : 4$ and charges in the ratio of $1 : 2$) are accelerated from rest through a potential difference $V$. If a uniform magnetic field $(B)$ is set up perpendicular to their velocities, the ratio of the radii $r _ { p } : r _ { \alpha }$ of the circular paths described by them will be:
(1) $1 : 3$
(2) $1 : \sqrt { 2 }$
(3) $1 : 2$
(4) $1 : \sqrt { 3 }$

The resistive network shown below is connected to a D.C. source of 16 V. The power consumed by the network is 4 Watt. The value of R is:
(1) $16\,\Omega$
(2) $8\,\Omega$
(3) $6\,\Omega$
(4) $1\,\Omega$

Two wires $A$ \& $B$ are carrying currents $I _ { 1 }$ and $I _ { 2 }$ as shown in the figure. The separation between them is $d$. A third wire C carrying a current I is to be kept parallel to them at a distance $x$ from A such that the net force acting on it is zero. The possible values of $x$ are:
(1) $x = \pm \frac { \mathrm { I } _ { 1 } \mathrm {~d} } { \left( \mathrm { I } _ { 1 } - \mathrm { I } _ { 2 } \right) }$
(2) $x = \left( \frac { \mathrm { I } _ { 1 } } { \mathrm { I } _ { 1 } + \mathrm { I } _ { 2 } } \right) \mathrm { d }$ and $x = \frac { \mathrm { I } _ { 2 } } { \left( \mathrm { I } _ { 1 } - \mathrm { I } _ { 2 } \right) } \mathrm { d }$
(3) $x = \left( \frac { \mathrm { I } _ { 2 } } { \mathrm { I } _ { 1 } + \mathrm { I } _ { 2 } } \right) \mathrm { d }$ and $x = \left( \frac { \mathrm { I } _ { 2 } } { \mathrm { I } _ { 1 } - \mathrm { I } _ { 2 } } \right) \mathrm { d }$
(4) $x = \left( \frac { \mathrm { I } _ { 1 } } { \mathrm { I } _ { 1 } - \mathrm { I } _ { 2 } } \right) \mathrm { d }$ and $x = \frac { \mathrm { I } _ { 2 } } { \left( \mathrm { I } _ { 1 } + \mathrm { I } _ { 2 } \right) } \mathrm { d }$

In free space, a particle $A$ of charge $1 \mu \mathrm { C }$ is held fixed at point $P$. Another particle $B$ of the same charge and mass $4 \mu \mathrm {~g}$ is kept at a distance of 1 mm from $P$. If $B$ is released, then its velocity at a distance of 9 mm from $P$ is: [Take $\frac { 1 } { 4 \pi \epsilon _ { 0 } } = 9 \times 10 ^ { 9 } \mathrm {~N} \mathrm {~m} ^ { 2 } \mathrm { C } ^ { - 2 }$ ]
(1) $1.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $1.5 \times 10 ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $2.0 \times 10 ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $3.0 \times 10 ^ { 4 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

There are two long co-axial solenoids of same length $l$. The inner and outer coils have radii $r _ { 1 }$ and $r _ { 2 }$ and number of turns per unit length $\mathrm { n } _ { 1 }$ and $\mathrm { n } _ { 2 }$, respectively. The ratio of mutual inductance to the self-inductance of the inner-coil is:
(1) $\frac { n _ { 1 } } { n _ { 2 } }$
(2) $\frac { n _ { 2 } } { n _ { 1 } } \cdot \frac { r _ { 1 } } { r _ { 2 } }$
(3) $\frac { n _ { 2 } } { n _ { 1 } } \cdot \frac { r _ { 2 } ^ { 2 } } { r _ { 1 } ^ { 2 } }$
(4) $\frac { n _ { 2 } } { n _ { 1 } }$

A proton, an electron, and a Helium nucleus, have the same energy. They are in circular orbits in a plane due to magnetic field perpendicular to the plane. Let $\mathrm { r } _ { \mathrm { p } } , \mathrm { r } _ { \mathrm { e } }$ and $\mathrm { r } _ { \mathrm { He } }$ be their respective radii, then,
(1) $r _ { e } > r _ { p } = r _ { H e }$
(2) $r _ { e } < r _ { p } = r _ { H e }$
(3) $\mathrm { r } _ { \mathrm { e } } < \mathrm { r } _ { \mathrm { p } } < \mathrm { r } _ { \mathrm { He } }$
(4) $r _ { e } > r _ { p } > r _ { H e }$

Space between two concentric conducting spheres of radii a and $\mathrm { b } ( \mathrm { b } > \mathrm { a } )$ is filled with a medium of resistivity $\rho$. The resistance between the two spheres will be:
(1) $\frac { \rho } { 4 \pi } \left( \frac { 1 } { a } + \frac { 1 } { b } \right)$
(2) $\frac { \rho } { 2 \pi } \left( \frac { 1 } { a } + \frac { 1 } { b } \right)$
(3) $\frac { \rho } { 4 \pi } \left( \frac { 1 } { a } - \frac { 1 } { b } \right)$
(4) $\frac { \rho } { 2 \pi } \left( \frac { 1 } { a } - \frac { 1 } { b } \right)$

A particle of mass $m$ and charge $q$ is in an electric and magnetic field given by $$\overrightarrow { \mathrm { E } } = 2 \hat { i } + 3 \hat { j } ; \overrightarrow { \mathrm { B } } = 4 \hat { j } + 6 \hat { k }$$ The charged particle is shifted from the origin to the point $\mathrm { P } ( x = 1 ; y = 1 )$ along a straight path. The magnitude of the total work done is:
(1) (0.35)q
(2) $5 q$
(3) $( 2.5 ) q$
(4) $( 0.15 ) q$

A transformer consisting of 300 turns in the primary and 150 turns in the secondary gives output power of 2.2 kW. If the current in the secondary coil is 10 A, then the input voltage and current in the primary coil are:
(1) 440 V and 20 A
(2) 220 V and 20 A
(3) 440 V and 5 A
(4) 220 V and 10 A

A square loop is carrying a steady current I and the magnitude of its magnetic dipole moment is m . If this square loop is changed to a circular loop and it carries the same current, the magnitude of the magnetic dipole moment of circular loop will be:
(1) $\frac { 4 m } { \pi }$
(2) $\frac { 3 m } { \pi }$
(3) $\frac { 2 m } { \pi }$
(4) $\frac { m } { \pi }$