The mass density of a spherical body is given by $\rho ( r ) = \frac { k } { r }$ for $r \leq R$ and $\rho ( r ) = 0$ for $r > R$, where $r$ is the distance from the center. The correct graph that describes qualitatively the acceleration, $a$ of a test particle as a function of $r$ is: (options given as graphs (1), (2), (3), (4))

$N$ moles of diatomic gas in a cylinder is at a temperature $T$. Heat is supplied to the cylinder such that the temperature remains constant but $n$ moles of the diatomic gas get converted into monoatomic gas. The change in the total kinetic energy of the gas is
(1) 0
(2) $\frac { 5 } { 2 } n R T$
(3) $\frac { 3 } { 2 } n R T$
(4) $\frac { 1 } { 2 } n R T$

A block of mass 0.1 kg is connected to an elastic spring of spring constant $640 \mathrm {~N} \mathrm {~m} ^ { - 1 }$ and oscillates in a damping medium of damping constant $10 ^ { - 2 } \mathrm {~kg} \mathrm {~s} ^ { - 1 }$. The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to-
(1) 2 s
(2) 5 s
(3) 3 s
(4) 7 s

In an experiment to determine the period of a simple pendulum of length 1 m, it is attached to different spherical bobs of radii $r _ { 1 }$ and $r _ { 2 }$. The two spherical bobs have uniform mass distribution. If the relative difference in the periods, is found to be $5 \times 10 ^ { - 4 } \mathrm {~s}$, the difference in radii, $\left| r _ { 1 } - r _ { 2 } \right|$ is best-given by
(1) 0.01 cm
(2) 0.05 cm
(3) 0.5 cm
(4) 1 cm

A standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by, $y ( x , t ) = 0.5 \sin \left( \frac { 5 \pi } { 4 } x \right) \cos ( 200 \pi t )$. What is the speed of the travelling wave moving in the positive $x$ direction? ($x$ and $t$ are in meter and second, respectively)
(1) $120 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $90 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $160 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $180 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

The following statement $(p \to q) \to [ ( \sim p \to q ) \to q ]$ is:
(1) a fallacy
(2) a tautology
(3) equivalent to $\sim p \to q$
(4) equivalent to $p \to \sim q$

As shown in the figure, forces of $10 ^ { 5 } \mathrm {~N}$ each are applied in opposite directions, on the upper and lower faces of a cube of side 10 cm , shifting the upper face parallel to itself by 0.5 cm . If the side of another cube of the same material is, 20 cm then under similar conditions as above, the displacement will be: [Figure]
(1) 1.00 cm
(2) 0.25 cm
(3) 0.37 cm
(4) 0.75 cm

When an air bubble of radius $r$ rises from the bottom to the surface of a lake, its radius becomes $\frac { 5r } { 4 }$. Taking the atmospheric pressure to be equal to 10 m height of water column, the depth of the lake would approximately be (ignore the surface tension and the effect of temperature):
(1) 10.5 m
(2) 8.7 m
(3) 11.2 m
(4) 9.5 m

A body takes 10 minutes to cool from $60 ^ { \circ } \mathrm { C }$ to $50 ^ { \circ } \mathrm { C }$. The temperature of surroundings is constant at $25 ^ { \circ } \mathrm { C }$. Then, the temperature of the body after next 10 minutes will be approximately
(1) $43 ^ { \circ } \mathrm { C }$
(2) $47 ^ { \circ } \mathrm { C }$
(3) $41 ^ { \circ } \mathrm { C }$
(4) $45 ^ { \circ } \mathrm { C }$

Two Carnot engines A and B are operated in series. Engine A receives heat from a reservoir at 600 K and rejects heat to a reservoir at temperature T . Engine B receives heat rejected by engine A and in turn rejects it to a reservoir at 100 K . If the efficiencies of the two engines A and B are represented by $\eta _ { A }$ and $\eta _ { B }$ respectively, then what is the value of $\frac { \eta _ { \mathrm { A } } } { \eta _ { \mathrm { B } } }$
(1) $\frac { 12 } { 7 }$
(2) $\frac { 12 } { 5 }$
(3) $\frac { 5 } { 12 }$
(4) $\frac { 7 } { 12 }$

The value closest to the thermal velocity of a Helium atom at room temperature (300 K) in $\mathrm { ms } ^ { - 1 }$ is: $\left[ \mathrm { k } _ { \mathrm { B } } = 1.4 \times 10 ^ { - 23 } \mathrm {~J} / \mathrm { K } ; \mathrm { m } _ { \mathrm { He } } = 7 \times 10 ^ { - 27 } \mathrm {~kg} \right]$
(1) $1.3 \times 10 ^ { 4 }$
(2) $1.3 \times 10 ^ { 5 }$
(3) $1.3 \times 10 ^ { 2 }$
(4) $1.3 \times 10 ^ { 3 }$

5 beats/ second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m . The frequency of the fork will be:
(1) 195 Hz
(2) 251 Hz
(3) 150 Hz
(4) 300 Hz

A parallel plate capacitor with area $200 \mathrm {~cm} ^ { 2 }$ and separation between the plates 1.5 cm , is connected across a battery of emf V . If the force of attraction between the plates is $25 \times 10 ^ { - 6 } \mathrm {~N}$, the value of $V$ is approximately: $$\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \frac { \mathrm { C } ^ { 2 } } { \mathrm {~N} . \mathrm { m } } \right) ^ { 2 } \right)$$ (1) 150 V
(2) 100 V
(3) 250 V
(4) 300 V

A capacitor $\mathrm { C } _ { 1 }$ is charged up to a voltage $\mathrm { V } = 60 \mathrm {~V}$ by connecting it to battery B through switch (1). Now $\mathrm { C } _ { 1 }$ is disconnected from battery and connected to a circuit consisting of two uncharged capacitors $\mathrm { C } _ { 2 } = 3.0 \mu \mathrm {~F}$ and $\mathrm { C } _ { 3 } = 6.0 \mu \mathrm {~F}$ through a switch (2) as shown in the figure. The sum of final charges on $\mathrm { C } _ { 2 }$ and $\mathrm { C } _ { 3 }$ is: [Figure]
(1) $36 \mu \mathrm { C }$
(2) $20 \mu \mathrm { C }$
(3) $54 \mu \mathrm { C }$
(4) $40 \mu \mathrm { C }$

A constant voltage is applied between two ends of a metallic wire. If the length is halved and the radius of the wire is doubled, the rate of heat developed in the wire will be:
(1) Increased 8 times
(2) Doubled
(3) Halved
(4) Unchanged

A current of 1 A is flowing on the sides of an equilateral triangle of side $4.5 \times 10 ^ { - 2 } \mathrm {~m}$. The magnetic field at the centre of the triangle will be:
(1) $4 \times 10 ^ { - 5 } \mathrm {~Wb} / \mathrm { m } ^ { 2 }$
(2) Zero
(3) $2 \times 10 ^ { - 5 } \mathrm {~Wb} / \mathrm { m } ^ { 2 }$
(4) $8 \times 10 ^ { - 5 } \mathrm {~Wb} / \mathrm { m } ^ { 2 }$

A copper rod of mass m slides under gravity on two smooth parallel rails, with separation $l$ and set at an angle of $\theta$ with the horizontal. At the bottom, rails are joined by a resistance $R$. There is a uniform magnetic field $B$ normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is: [Figure]
(1) $\frac { \mathrm { mgR } \cos \theta } { \mathrm { B } ^ { 2 } l ^ { 2 } }$
(2) $\frac { \mathrm { mgR } \sin \theta } { \mathrm { B } ^ { 2 } l ^ { 2 } }$
(3) $\frac { \mathrm { mgR } \tan \theta } { \mathrm { B } ^ { 2 } l ^ { 2 } }$
(4) $\frac { \mathrm { mgR } \cot \theta } { \mathrm { B } ^ { 2 } l ^ { 2 } }$

At the centre of a fixed large circular coil of radius R , a much smaller circular coil of radius $r$ is placed. The two coils are concentric and are in the same plane. The larger coil carries a current I. The smaller coil is set to rotate with a constant angular velocity $\omega$ about an axis along their common diameter. Calculate the emf induced in the smaller coil after a time $t$ of its start of rotation.
(1) $\frac { \mu _ { 0 } I } { 2 R } \omega r ^ { 2 } \sin \omega t$
(2) $\frac { \mu _ { 0 } I } { 4 R } \omega \pi r ^ { 2 } \sin \omega t$
(3) $\frac { \mu _ { 0 } I } { 2 R } \omega \pi r ^ { 2 } \sin \omega t$
(4) $\frac { \mu _ { 0 } \mathrm { I } } { 4 \mathrm { R } } \omega \mathrm { r } ^ { 2 } \sin \omega \mathrm { t }$

A plane polarized monochromatic EM wave is travelling in a vacuum along $z$ direction such that at $\mathrm { t } = \mathrm { t } _ { 1 }$ it is found that the electric field is zero at a spatial point $z _ { 1 }$. The next zero that occurs in its neighbourhood is at $z _ { 2 }$. The frequency of the electromagnetic wave is:
(1) $\frac { 3 \times 10 ^ { 8 } } { \left| z _ { 2 } - z _ { 1 } \right| }$
(2) $\frac { 6 \times 10 ^ { 8 } } { \left| z _ { 2 } - z _ { 1 } \right| }$
(3) $\frac { 1.5 \times 10 ^ { 8 } } { \left| z _ { 2 } - z _ { 1 } \right| }$
(4) $\frac { 1 } { t _ { 1 } + \frac { \left| z _ { 2 } - z _ { 1 } \right| } { 3 \times 10 ^ { 8 } } }$

A convergent doublet of separated lenses, corrected for spherical aberration, has resultant focal length of 10 cm . The separation between the two lenses is 2 cm . The focal lengths of the component lenses are:
(1) $18 \mathrm {~cm} , 20 \mathrm {~cm}$
(2) $10 \mathrm {~cm} , 12 \mathrm {~cm}$
(3) $12 \mathrm {~cm} , 14 \mathrm {~cm}$
(4) $16 \mathrm {~cm} , 18 \mathrm {~cm}$

If the de Broglie wavelengths associated with a proton and an $\alpha$-particle are equal, then the ratio of velocities of the proton and the $\alpha$-particle will be:
(1) $1 : 4$
(2) $1 : 2$
(3) $4 : 1$
(4) $2 : 1$

An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of $8 : 27$ . The ratio of the radii of the nuclei (assumed to be spherical) is:
(1) $8 : 27$
(2) $2 : 3$
(3) $3 : 2$
(4) $4 : 9$

The carrier frequency of a transmitter is provided by a tank circuit of a coil of inductance $49 \mu \mathrm { H }$ and a capacitance of 2.5 nF . It is modulated by an audio signal of 12 kHz . The frequency range occupied by the side bands is:
(1) $18 \mathrm { kHz } - 30 \mathrm { kHz }$
(2) $63 \mathrm { kHz } - 75 \mathrm { kHz }$
(3) $442 \mathrm { kHz } - 466 \mathrm { kHz }$
(4) $13482 \mathrm { kHz } - 13494 \mathrm { kHz }$

The de-Broglie's wavelength of electron present in first Bohr orbit of '$H$' atom is:
(1) $4 \times 0.529 \AA$
(2) $2 \pi \times 0.529 \AA$
(3) $\frac { 0.529 } { 2 \pi } \AA$
(4) $0.529 \AA$

Two 5 molal solutions are prepared by dissolving a non-electrolyte, non-volatile solute separately in the solvents X and Y . The molecular weights of the solvents are $\mathrm { M } _ { \mathrm { X } }$ and $\mathrm { M } _ { \mathrm { Y } }$, respectively where $\mathrm { M } _ { X } = \frac { 3 } { 4 } \mathrm { M } _ { \mathrm { Y } }$. The relative lowering of vapour pressure of the solution in X is "m" times that of the solution in Y . Given that the number of moles of solute is very small in comparison to that of solvent, the value of "m" is:
(1) $\frac { 3 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 4 } { 3 }$