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General
The resistance of platinum wire at $0^{\circ}\text{C}$ is $2 \text{ } \Omega$ and $6.8 \text{ } \Omega$ at $80^{\circ}\text{C}$. The temperature coefficient of resistance of the wire isFor the following logic circuit, the truth table isA ball is projected with a velocity, $10 \text{ ms}^{-1}$, at an angle of $60^\circ$ with the vertical direction. Its speed at the highest point of its trajectory will be:The truth table for the given logic circuit isA series LCR circuit with inductance $10\text{ H}$, capacitance $10\text{ }\mu\text{F}$, resistance $50\text{ }\Omega$ is connected to an ac source of voltage, $V = 200\sin(100\pi t)\text{ volt}$. If the resonant frequency of the LCR circuit is $f_0$ and the frequency of the ac source is $f$, thenThe solids which have the negative temperature coefficient of resistance are:Two particles of mass 5 kg and 10 kg respectively are attached to the two ends of a rigid rod of length 1 m with negligible mass. The centre of mass of the system from the 5 kg particle is nearly at a distance of :The potential energy of a long spring when stretched by $2 \text{ cm}$ is $U$. If the spring is stretched by $8 \text{ cm}$, potential energy stored in it will be
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MOTION IN A STRAIGHT LINE
A ball is thrown vertically upwards. Which of the following plots represents the speed-time graph of the ball during its height if the air resistance is not ignored?A body starts from rest. What is the ratio of the distance travelled by the body during the 4th and 3rd second?The velocity of a bullet is reduced from $200 \text{ m/s}$ to $100 \text{ m/s}$ while travelling through a wooden block of thickness $10 \text{ cm}$. The retardation, assuming it to be uniform, will be:A body A starts from rest with an acceleration $a_1$. After $2$ seconds, another body B starts from rest with an acceleration $a_2$. If they travel equal distances in the $5^{\text{th}}$ second, after the start of A, then the ratio $a_1 : a_2$ is equal to:The displacement of a particle, moving in a straight line, is given by $s = 2t^2 + 2t + 4$ where $s$ is in metres and $t$ in seconds. The acceleration of the particle is:A car moving with a velocity of $10 \text{ m/s}$ can be stopped by the application of a constant force $F$ in a distance of $20 \text{ m}$. If the velocity of the car is $30 \text{ m/s}$, it can be stopped by this force in:Which of the following velocity-time graphs shows a realistic situation for a body in motion?Time taken by an object falling from rest to cover the height of $h_1$ and $h_2$ is respectively $t_1$ and $t_2$. Then the ratio of $t_1$ to $t_2$ is:
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SEMICONDUCTOR ELECTRONICS
In the following figure, the diodes which are forward biased, are (I) (II) (III) (IV). Choose the correct option from the given ones:An n-p-n transistor is connected in the common base configuration in a given amplifier. A load resistance of $800\, \Omega$ is connected in the collector circuit and the voltage drop across it is $0.8\text{ V}$. If the current amplification factor is $0.96$ and the input resistance of the circuit is $192\, \Omega$, the voltage gain and the power gain of the amplifier will respectively be:In a common-emitter transistor amplifier, the audio signal voltage across the collector is $3\text{ V}$. The resistance of the collector is $3\text{ k}\Omega$. If the current gain is $100$ and the base resistance is $2\text{ k}\Omega$, the voltage and power gain of the amplifier are:If in a p-n junction, a square input signal of $10 \text{ V}$ is applied as shown, then the output across $R_L$ will be:For transistor action:
(a) the base, emitter and collector regions should have similar size and doping concentrations.
(b) the base region must be very thin and lightly doped.
(c) the emitter-base junction is forward biased and the base-collector junction is reverse biased.
(d) both the emitter-base junction as well as the base-collector junction are forward biased.
Which of the following pairs of statements is correct?A p-n photodiode is fabricated from a semiconductor with a band gap of $2.5 \text{ eV}$. It can detect a signal of wavelength:Which one of the following statements is false?Pure $\text{Si}$ at $500\text{ K}$ has an equal number of electron ($n_e$) and hole ($n_h$) concentration of $1.5 \times 10^{16} \text{ m}^{-3}$. Doping by indium increases the hole concentration $n_h$ to $4.5 \times 10^{22} \text{ m}^{-3}$. The doped semiconductor is of:
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MOVING CHARGES AND MAGNETISM
A uniform electric field and a uniform magnetic field are acting along the same direction in a certain region. If an electron is projected in the region such that its velocity is pointed along the direction of fields, then the electron:A closely wound solenoid of 2000 turns and area of cross-section $1.5 \times 10^{-4} \text{ m}^2$ carries a current of $2.0 \text{ A}$. It is suspended through its centre and perpendicular to its length, allowing it to turn in a horizontal plane in a uniform magnetic field $5 \times 10^{-2} \text{ T}$ making an angle of $30^{\circ}$ with the axis of the solenoid. The torque on the solenoid will beA metallic rod of mass per unit length of 0.5 kg m⁻¹ is lying horizontally on a smooth inclined plane which makes an angle of 30° with the horizontal. The rod is not allowed to slide down by flowing a current through it when a magnetic field of induction of 0.25 T is acting on it in the vertical direction. What is the current flowing through the rod to keep it stationary?A voltmeter has a resistance of $G$ ohms and range $V$ volts. The value of resistance used in series to convert it into a voltmeter of range $nV$ volts is:The current sensitivity of a moving coil galvanometer is 5 div/mA and its voltage sensitivity (angular deflection per unit voltage applied) is 20 div/V. The resistance of the galvanometer is:A long solenoid of radius $1 \text{ mm}$ has $100$ turns per mm. If $1 \text{ A}$ current flows in the solenoid, the magnetic field strength at the centre of the solenoid is:A square loop ABCD carrying a current $i$ is placed near and coplanar with a long straight conductor XY carrying a current $I$. The net force on the loop will be:An electron moving in a circular orbit of radius $r$ makes $n$ rotations per second. The magnetic field produced at the centre has magnitude:
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UNITS AND MEASUREMENTS
In an experiment, four quantities $a$, $b$, $c$ and $d$ are measured with percentage error $1\%$, $2\%$, $3\%$ and $4\%$ respectively. Quantity $P$ is calculated as follows: $P=\frac{a^3b^2}{cd}$. Percentage error in $P$ is:Dimensional formula for volume elasticity isThe velocity $v$ of a particle at time $t$ is given by $v = at + \frac{b}{t+c}$, where $a$, $b$ and $c$ are constants. The dimensions of $a$, $b$ and $c$ are respectively:Which one has the dimensions different from the remaining three?Taking into account the significant figures, what is the value of $9.99\text{ m} - 0.0099\text{ m}$?Of the following quantities, which one has dimensions different from the remaining three?If the unit of length and force be increased four times, then the unit of energy is:Which of the following pairs of physical quantities has the same dimensions?
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Rotational motion
From a circular ring of mass $M$ and radius $R$, an arc corresponding to a $90^\circ$ sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is $K$ times $MR^2$. The value of $K$ will be:The moment of inertia of a uniform circular disc of radius $R$ and mass $M$ about an axis touching the disc at its edge and normal to the disc is:A solid cylinder of mass $2 \text{ kg}$ and radius $4 \text{ cm}$ is rotating about its axis at the rate of $3 \text{ rpm}$. The torque required to stop after $2\pi$ revolutions is:Three identical spherical shells, each of mass $m$ and radius $r$ are placed as shown in the figure. Consider an axis $XX'$, which is touching two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the $XX'$ axis is:A solid cylinder of mass $3 \text{ kg}$ is rolling on a horizontal surface with a velocity of $4 \text{ ms}^{-1}$. It collides with a horizontal spring of force constant $200 \text{ Nm}^{-1}$. The maximum compression produced in the spring will be:The moment of inertia of a thin rod about an axis passing through its mid-point and perpendicular to the rod is $2400\text{ g cm}^2$. The length of the $400\text{ g}$ rod is nearly:A solid sphere is in rolling motion. In rolling motion, a body possesses translational kinetic energy ($K_t$) as well as rotational kinetic energy ($K_r$) simultaneously. The ratio $K_t : (K_t + K_r)$ for the sphere will be:An energy of $484 \text{ J}$ is spent in increasing the speed of a flywheel from $60 \text{ rpm}$ to $360 \text{ rpm}$. The moment of inertia of the flywheel is:
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LAWS OF MOTION
A block A of mass $m_1$ rests on a horizontal table. A light string connected to it passes over a frictionless pulley at the edge of the table and from its other end, another block B of mass $m_2$ is suspended. The coefficient of kinetic friction between block A and the table is $\mu_k$. When block A is sliding on the table, the tension in the string is:A block of mass 4 kg is suspended through two light spring balances A and B connected in series. Then A and B will read respectively:A lift of mass $1000 \text{ kg}$ is moving with an acceleration of $1 \text{ m/s}^2$ in the upward direction. Tension developed in the string, which is connected to the lift, is:The distance covered by a body of mass $5 \text{ g}$ having linear momentum $0.3 \text{ kg m/s}$ in $5 \text{ s}$ is:A particle of mass $m$ is projected with velocity $v$ making an angle of $45^\circ$ with the horizontal. When the particle lands on the level ground, the magnitude of the change in its momentum will be:It is easier to draw up a wooden block along a smooth inclined plane than to haul it vertically, principally because:When two surfaces are coated with a lubricant, then they:A block of mass $m$ is in contact with the cart $C$ as shown in the figure. The coefficient of static friction between the block and the cart is $\mu$. The acceleration $a$ of the cart that will prevent the block from falling satisfies:
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NUCLEI
In the nuclear reaction $X(n, \alpha){}_{3}^{7}\mathrm{Li}$, the term $X$ will be:The half-life of a radioactive sample undergoing $\alpha$-decay is $1.4 \times 10^{17}\text{ s}$. If the number of nuclei in the sample is $2.0 \times 10^{21}$, the activity of the sample is nearly equal to:If the nuclear radius of $^{27}\text{Al}$ is 3.6 Fermi, the approximate nuclear radius of $^{64}\text{Cu}$ in Fermi is:The half-life of a radioactive substance is 30 minutes. The time (in minute) taken between 40% decay and 85% decay of the same radioactive substance is:The fraction of the original number of radioactive atoms that disintegrates (decays) during the average lifetime of a radioactive substance will be:A nucleus of mass number 189 splits into two nuclei having mass numbers 125 and 64. The ratio of the radius of two daughter nuclei respectively is:In the nuclear emission stated above: ${}_{82}^{290}X \xrightarrow{\alpha} Y \xrightarrow{e^+} Z \xrightarrow{\beta^-} P \xrightarrow{e^-} Q$, the mass number and atomic number of the product Q respectively, are:The energy equivalent of 0.5 g of a substance is:
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RAY OPTICS AND OPTICAL INSTRUMENTS
A horizontal ray of light is incident on a right-angled prism with a prism angle of $6^{\circ}$. If the refractive index of the material of the prism is 1.5, then the angle of emergence will be:A concave lens with a focal length of $-25 \text{ cm}$ is sandwiched between two convex lenses, each with a focal length of $40 \text{ cm}$. The power (in diopters) of the combined lens system would be:The refracting angle of a prism is $A$, and the refractive index of the material of the prism is $\cot(A/2)$. The angle of minimum deviation is:A small telescope has an objective of focal length $140\text{ cm}$ and an eyepiece of focal length $5.0\text{ cm}$. The magnifying power of the telescope for viewing a distant object is:An astronomical telescope has an objective and eyepiece of focal lengths $40\text{ cm}$ and $4\text{ cm}$ respectively. To view an object $200\text{ cm}$ away from the objective, the lenses must be separated by a distance of:A plano-convex lens fits exactly into a plano-concave lens. Their plane surfaces are parallel to each other. If lenses are made of different materials of refractive indices $\mu_1$ and $\mu_2$ and $R$ is the radius of curvature of the curved surface of the lenses, then the focal length of the combination is:A double convex lens has a focal length of 25 cm. The radius of curvature of one of the surfaces is double of the other. What would be the radii if the refractive index of the material of the lens is 1.5?The power of a biconvex lens is 10 dioptre and the radius of curvature of each surface is 10 cm. The refractive index of the material of the lens is:
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ELECTRIC CHARGES AND FIELDS
Two identical charged spheres suspended from a common point by two massless strings of lengths l are initially at a distance d (d << l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then v varies as a function of the distance x between the spheres, as:Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude $17.0 \times 10^{-22} \, \text{C/m}^2$. The electric field between the plates is:Two point charges $q_A = 3 \, \mu\text{C}$ and $q_B = -3 \, \mu\text{C}$ are located $20 \, \text{cm}$ apart in a vacuum. The electric field at the midpoint $O$ of the line $AB$ joining the two charges is:The electric field in a certain region is acting radially outward and is given by E = Ar. A charge contained in a sphere of radius 'a' centered at the origin of the field will be given by:The electric field due to a uniformly charged solid sphere of radius R as a function of the distance from its centre is represented graphically by -Four-point +ve charges of the same magnitude (Q) are placed at four corners of a rigid square frame as shown in the figure. The plane of the frame is perpendicular to Z-axis. If a –ve point charge is placed at a distance z away from the above frame (z<<L) thenShown below is a distribution of charges. The flux of electric field due to these charges through the surface S is:Two point charges A and B, having charges +Q and −Q respectively, are placed at a certain distance apart and the force acting between them is F. If 25% charge of A is transferred to B, then the force between the charges becomes:
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WORK, ENERGY AND POWER
A $100 \text{ g}$ iron ball having velocity $10 \text{ m/s}$ collides with a wall at an angle $30^\circ$ and rebounds with the same angle. If the period of contact between the ball and wall is $0.1 \text{ second}$, then the force experienced by the wall is:An object of mass $500 \text{ g}$ initially at rest is acted upon by a variable force whose $x$-component varies with $x$ in the manner shown. The velocities of the object at the points $x=8 \text{ m}$ and $x=12 \text{ m}$ would have the respective values of nearly:When a body moves with a constant speed along a circleForce $F$ on a particle moving in a straight line varies with distance $d$ as shown in the figure. The work done on the particle during its displacement of $12$ m is:A particle moves with a velocity $(5\hat i-3\hat j+6\hat k)\text{ ms}^{-1}$ horizontally under the action of a constant force $(10\hat i+10\hat j+20\hat k)\text{ N}$. The instantaneous power supplied to the particle is:In a circus stuntman rides a motorbike in a circular track of radius $R$ in the vertical plane. The minimum speed at highest point of track will be:The minimum work done in pulling up a block of wood weighing $2 \text{ kN}$ for a length of $10 \text{ m}$ on a smooth plane inclined at an angle of $15^\circ$ with the horizontal is (given $\sin 15^\circ = 0.2588$):A man fires a bullet of mass $200 \text{ g}$ at a speed of $5 \text{ m/s}$. The gun is of one kg mass. By what velocity the gun rebounds backwards?
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ALTERNATING CURRENT
In an alternating current circuit consisting of elements in series, the current increases on increasing the frequency of the supply. Which of the following elements are likely to constitute the circuit?
(a) Only resistor
(b) Resistor and an inductor
(c) Resistor and a capacitor
(d) Only a capacitorAn AC source given by $V = V_m \sin(\omega t)$ is connected to a pure inductor $L$ in a circuit and $I_m$ is the peak value of the AC current. The instantaneous power supplied to the inductor is:A transformer having efficiency of 90% is working on 200 V and 3 kW power supply. If the current in the secondary coil is 6A, the voltage across the secondary coil and the current in the primary coil respectively are:A standard filament lamp consumes 100 W when connected to a 200 V AC mains supply. The peak current through the bulb will be:The net impedance of the circuit (as shown in the figure) will be:A 12 V, 60 W lamp is connected to the secondary of a step-down transformer, whose primary is connected to AC mains of 220 V. Assuming the transformer to be ideal, what is the current in the primary winding?The variation of EMF with time for four types of generators is shown in the figures. Which amongst them can be called AC voltage?An inductor of inductance L, a capacitor of capacitance C and a resistor of resistance R are connected in series to an AC source of potential difference V volts. The potential difference across L, C and R is 40 V, 10 V and 40 V, respectively. The amplitude of the current flowing through the LCR series circuit is $10\sqrt{2}$ A. The impedance of the circuit will be:
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Oscillations
A simple pendulum performs simple harmonic motion about $x = 0$ with an amplitude $a$ and time period $T$. The speed of the pendulum at $x = rac{a}{2}$ will be:The restoring force of a spring, with a block attached to the free end of the spring, is represented by:The displacement-time ($x-t$) graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at $t = 2$ s is:A particle executing simple harmonic motion has a kinetic energy of $K = K_0 \cos^2(\omega t)$. The values of the maximum potential energy and the total energy are, respectively:When two displacements represented by $y_1 = a \sin(\omega t)$ and $y_2 = b \cos(\omega t)$ are superimposed, the motion is:From the given functions, identify the function which represents a periodic motion:A particle executes linear simple harmonic motion with an amplitude of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then, its time period in seconds is:Match List-I with List-II.
**List-I (Graphs)**
(a) Displacement-time graph with decreasing amplitude (Damped)
(b) Displacement-time graph with constant amplitude (Undamped)
(c) Force-displacement graph (Linear relationship, F = -kx)
(d) Energy-time graph (Constant total energy)
**List-II (Situations)**
(i) Total mechanical energy is conserved
(ii) Bob of a pendulum is oscillating under negligible air friction
(iii) Restoring force of a spring
(iv) Bob of a pendulum is oscillating along with air friction
Choose the correct answer from the options given below:
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ELECTROSTATIC POTENTIAL AND CAPACITANCE
The variation of electrostatic potential with radial distance $r$ from the centre of a positively charged metallic thin shell of radius $R$ is given by the graph:Six charges +q, -q, +q, -q, +q and -q are fixed at the corners of a hexagon of side d as shown in the figure. The work done in bringing a charge q₀ to the centre of the hexagon from infinity is: (ε₀ - permittivity of free space)A, B and C are three points in a uniform electric field. The electric potential is:The electric potential at a point in free space due to a charge $Q$ coulomb is $Q \times 10^{11}$ V. The electric field at that point is:Four point charges -Q, -q, 2q and 2Q are placed, one at each corner of the square. The relation between Q and q for which the potential at the centre of the square is zero, isIf potential in a region is expressed as $V(x,y,z) = 6xy - y + 2yz$, the electric field at point $(1, 1, 0)$ is:Two spheres of radius $a$ and $b$ respectively are charged and joined by a wire. The ratio of the electric field at the surface of the spheres is:Two metal spheres, one of radius $R$ and the other of radius $2R$ respectively have the same surface charge density $\sigma$. They are brought in contact and separated. What will be the new surface charge densities on them?
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MOTION IN A PLANE
An object is thrown along a direction inclined at an angle of $45^\circ$ with the horizontal direction. The horizontal range of the particle is equal to:A particle moves so that its position vector is given by $\vec{r} = \cos(\omega t)\hat{x} + \sin(\omega t)\hat{y}$, where $\omega$ is a constant. Which of the following is true?For a projectile projected at angles $(45^{\circ}-\theta)$ and $(45^{\circ}+\theta)$, the horizontal ranges described by the projectile are in the ratio of:A body is moving with velocity $30 \text{ m/s}$ towards east. After $10 \text{ s}$ its velocity becomes $40 \text{ m/s}$ towards north. The average acceleration of the body is:A ball is projected with kinetic energy $E$ at an angle of $45^\circ$ to the horizontal. At the highest point during its flight, its kinetic energy will be:Two particles $A$ and $B$, move with constant velocities $\vec{v}_1$ and $\vec{v}_2$ respectively. At the initial moment, their position vectors are $\vec{r}_1$ and $\vec{r}_2$ respectively. The condition for particles $A$ and $B$ for their collision will be:A particle starting from the origin $(0,0)$ moves in a straight line in the $(x,y)$ plane. Its coordinates at a later time are $(\sqrt{3}, 3)$. The path of the particle makes an angle of __________ with the $x$-axis:If a particle moves in a circle describing equal angles in equal times, its velocity vector:
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WAVE
If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is:When a string is divided into three segments of lengths $l_1, l_2$ and $l_3$, the fundamental frequencies of these three segments are $\nu_1, \nu_2$ and $\nu_3$ respectively. The original fundamental frequency ($\nu$) of the string is:Two periodic waves of intensities $I_1$ and $I_2$ pass through a region at the same time in the same direction. The sum of the maximum and minimum intensities is:The two nearest harmonics of a tube closed at one end and open at the other end are $220 \text{ Hz}$ and $260 \text{ Hz}$. What is the fundamental frequency of the system?A transverse wave propagating along the $x$-axis is represented by: $y(x,t) = 8.0\sin(0.5\pi x - 4\pi t - \frac{\pi}{4})$, where $x$ is in meters and $t$ in seconds. The speed of the wave is:The fundamental frequency in an open organ pipe is equal to the third harmonic of a closed organ pipe. If the length of the closed organ pipe is $20 \text{ cm}$, the length of the open organ pipe is:The number of possible natural oscillations of the air column in a pipe closed at one end of a length of $85 \text{ cm}$ whose frequencies lie below $1250 \text{ Hz}$ is: (velocity of sound $340 \text{ ms}^{-1}$)A source of unknown frequency gives $4 \text{ beats/s}$ when sounded with a source of known frequency $250 \text{ Hz}$. The second harmonic of the source of unknown frequency gives five beats per second when sounded with a source of frequency $513 \text{ Hz}$. The unknown frequency is
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Thermodynamics
The volume ($V$) of a monatomic gas varies with its temperature ($T$), as shown in the graph. The ratio of work done by the gas to the heat absorbed by it when it undergoes a change from state $A$ to state $B$ will be:A carnot engine having an efficiency of $\frac{1}{10}$ as a heat engine, is used as a refrigerator. If the work done on the system is $10\text{ J}$, the amount of energy absorbed from the reservoir at lower temperature is:An engine has an efficiency of $\frac{1}{6}$. When the temperature of the sink is reduced by $62^{\circ}\text{C}$, its efficiency is doubled. The temperature of the source is:A refrigerator works between $4^{\circ}\text{C}$ and $30^{\circ}\text{C}$. It is required to remove $600 \text{ calories}$ of heat every second to keep the temperature of the refrigerated space constant. The power required will be: (Take, $1 \text{ cal} = 4.2 \text{ Joules}$)When $1 \text{ kg}$ of ice at $0^{\circ}\text{C}$ melts into water at $0^{\circ}\text{C}$, the resulting change in its entropy, taking the latent heat of ice to be $80 \text{ cal/g}$, is:One mole of an ideal diatomic gas undergoes a transition from $A$ to $B$ along a path $AB$ as shown in the figure. The change in internal energy of the gas during the transition isThe coefficient of performance of a refrigerator is $5$. If the temperature inside freezer is $-20^\circ\text{C}$, the temperature of the surroundings to which it rejects heat isA mass of diatomic gas ($\gamma=1.4$) at a pressure of $2\text{ atm}$ is compressed adiabatically so that its temperature rises from $27^\circ\text{C}$ to $927^\circ\text{C}$. The pressure of the gas in the final state is:
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GRAVITATION
The radii of the circular orbits of two satellites A and B of the earth are 4R and R, respectively. If the speed of the satellite A is 3v, then the speed of the satellite B will be:A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth (mass = 5.98 × 10²⁴ kg) have to be compressed to be a black hole?A body projected vertically from the earth reaches a height equal to earth's radius before returning to the earth. The power exerted by the gravitational force is greatest:The figure shows elliptical orbit of a planet m about the sun S. The shaded area SCD is twice the shaded area SAB. If $t_1$ is the time for planet to move from C to D and $t_2$ is the time to move from A to B, then:If the mass of the sun were ten times smaller and the universal gravitational constant were ten times larger in magnitude, which of the following statements would not be correct?The kinetic energies of a planet in an elliptical orbit around the Sun, at positions A, B and C are $K_A$, $K_B$ and $K_C$ respectively. AC is the major axis and SB is perpendicular to AC at the position of the Sun S, as shown in the figure. Then:At what height from the surface of the earth, are the gravitation potential and the value of $g$ are: $-5.4 \times 10^7 \text{ J/kg}$ and $6.0 \text{ ms}^{-2}$ respectively? (Take, the radius of the earth as $6400 \text{ km}$)A particle is released from a height $S$ above the surface of the earth. At a certain height, its kinetic energy is three times its potential energy. The height from the earth's surface and the speed of the particle at that instant are respectively:
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ATOMS
The Bohr model for the spectra of a H-atom:
(a) will not apply to hydrogen in the molecular form.
(b) will not be applicable as it is for a He-atom.
(c) is valid only at room temperature.
(d) predicts continuous as well as discrete spectral lines.
Choose the correct option:Let En = -(me^4)/(8ε_0^2 n^2 h^2) be the energy of the nth level of H-atom. If all the H-atoms are in the ground state and radiation of frequency (E2 - E1)/h falls on it, then:
(a) it will not be absorbed at all.
(b) some of the atoms will move to the first excited state.
(c) all atoms will be excited to the n=2 state.
(d) no atoms will make a transition to the n=3 state.
Choose the correct option:The wavelength of the first spectral line in the Balmer series of hydrogen atom is 6561 Å. The wavelength of the second spectral line in the Balmer series of singly ionized helium atom is:Match List-I (Spectral Series) with List-II (corresponding wave number expressions).
List-I (Series)
A. Balmer series
B. Lyman series
C. Brackett series
D. Pfund series
List-II (Wave number in cm⁻¹)
I. R(1/1² - 1/n²)
II. R(1/4² - 1/n²)
III. R(1/5² - 1/n²)
IV. R(1/2² - 1/n²)
Choose the correct answer from the options given below:In the Bohr's model of a hydrogen atom, the centripetal force is furnished by the Coulomb attraction between the proton and the electron. If a₀ is the radius of the ground state orbit, m is the mass and e is the charge on the electron, ε₀ is the vacuum permittivity, the speed of the electron is:If the nucleus $^{27}_{13}\mathrm{Al}$ has a nuclear radius of about 3.6 fermis, then $^{125}_{52}\mathrm{Te}$ would have its radius approximately as:An alpha nucleus of energy 1/2 mv² bombards a heavy nuclear target of charge Ze. Then the distance of closest approach for the alpha nucleus will be proportional toThe ionisation energy of hydrogen atom is 13.6 eV, the ionisation energy of helium atom would be:
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Thermal Properties of Matter
The power radiated by a black body is $P$ and it radiates maximum energy at wavelength $\lambda_0$. Temperature of the black body is now changed so that it radiates maximum energy at the wavelength $\frac{3}{4}\lambda_0$. The power radiated by it now becomes $nP$. The value of $n$ is:A piece of ice falls from a height $h$ so that it melts completely. Only one-quarter of the heat produced is absorbed by the ice. The value of $h$ is: [Latent heat of ice is $3.4 \times 10^5 \text{ J/kg}$ and $g = 10 \text{ N/kg}$]A black body at $227^\circ\text{C}$ radiates heat at the rate of $7 \text{ cal cm}^{-2}\text{s}^{-1}$. At a temperature of $727^\circ\text{C}$, the rate of heat radiated in the same units will be:A copper rod of $88 \text{ cm}$ and an aluminium rod of an unknown length have an equal increase in their lengths independent of an increase in temperature. The length of the aluminium rod is: ($\alpha_{\text{Cu}} = 1.7 \times 10^{-5} \text{ K}^{-1}$ and $\alpha_{\text{Al}} = 2.2 \times 10^{-5} \text{ K}^{-1}$)A black body at $227^{\circ}\text{C}$ radiates heat at the rate of $7\text{ cal cm}^{-2}\text{s}^{-1}$. At a temperature of $727^{\circ}\text{C}$, the rate of heat radiated in the same units will beThe two ends of a rod of length $L$ and a uniform cross-sectional area $A$ are kept at two temperatures $T_1$ and $T_2$ ($T_1 > T_2$). The rate of heat transfer $\frac{dQ}{dt}$ through the rod in a steady state is given by:Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is at $100^{\circ}\text{C}$, while the other one is at $0^{\circ}\text{C}$. If the two bodies are brought into contact, then assuming no heat loss, the final common temperature is -The total radiant energy per unit area per unit time, normal to the direction of incidence, received at a distance $R$ from the centre of a star of radius $r$, whose outer surface radiates as a black body at a temperature $T\text{ K}$ is given by (where $\sigma$ is Stefan's constant):
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WAVE OPTICS
The fundamental frequency in an open organ pipe is equal to the third harmonic of a closed organ pipe. If the length of the closed organ pipe is $20 \text{ cm}$, the length of the open organ pipe is:When a string is divided into three segments of lengths $l_1, l_2$ and $l_3$, the fundamental frequencies of these three segments are $\nu_1, \nu_2$ and $\nu_3$ respectively. The original fundamental frequency ($\nu$) of the string is:The wave described by $y = 0.25\sin(10\pi x - 2\pi t)$, where $x$ and $y$ are in metres and $t$ in seconds, is a wave traveling along the:The two nearest harmonics of a tube closed at one end and open at the other end are $220 \text{ Hz}$ and $260 \text{ Hz}$. What is the fundamental frequency of the system?A wave travelling in the positive x-direction having maximum displacement along y-direction as $1 \text{ m}$, wavelength $2\pi \text{ m}$ and frequency of $1/\pi \text{ Hz}$ is represented byAn air column, closed at one end and open at the other, resonates with a tuning fork when the smallest length of the column is $50 \text{ cm}$. The next larger length of the column resonating with the same tuning fork isIf we study the vibration of a pipe open at both ends, then the following statement is not true:A uniform rope of length $L$ and mass $m_1$ hangs vertically from a rigid support. A block of mass $m_2$ is attached to the free end of the rope. A transverse pulse of wavelength $\lambda_1$ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is $\lambda_2$. The ratio $\frac{\lambda_2}{\lambda_1}$ is:
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ELECTROMAGNETIC INDUCTION
A conducting circular loop of face area $2.5 \times 10^{-3} \text{ m}^2$ is placed perpendicular to a magnetic field which varies as $B=0.5 \sin(100\pi t) \text{ T}$. The magnitude of induced EMF at time $t=0 \text{ s}$ is:A coil of resistance $400\Omega$ is placed in a magnetic field. If the magnetic flux $\phi\;(\text{Wb})$ linked with the coil varies with time $t\;(\text{sec})$ as $\phi=50t^2+4$. The current in the coil at $t=2\text{s}$ is:A wooden stick of length $3l$ is rotated about an end with constant angular velocity $\omega$ in a uniform magnetic field $B$ perpendicular to the plane of motion. If the upper one third of its length is coated with copper, the potential difference across the whole length of the stick is:The primary and secondary coils of a transformer have 50 and 1500 turns respectively. If the magnetic flux $\phi$ linked with the primary coil is given by $\phi = \phi_0 + 4t$, where $\phi$ is in Weber, $t$ is time in seconds, and $\phi_0$ is a constant, the output voltage across the secondary coil is:In which of the following devices, the eddy current effect is not used?The current in an inductor of self-inductance $4 \text{ H}$ changes from $4 \text{ A}$ to $2 \text{ A}$ in $1 \text{ s}$. The emf induced in the coil is:In the above diagram, a strong bar magnet is moving towards solenoid-2 from solenoid-1. The direction of induced current in solenoid-1 and that in solenoid-2, respectively, are through the directions:A rod of length $L$ rotates with a small uniform angular velocity $\omega$ about its perpendicular bisector. A uniform magnetic field $B$ exists parallel to the axis of rotation. The potential difference between the centre of the rod and an end is:
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Mechanical Properties of Fluid
A rectangular film of liquid is extended from $(4 \text{ cm} \times 2 \text{ cm})$ to $(5 \text{ cm} \times 4 \text{ cm})$. If the work done is $3 \times 10^{-4} \text{ J}$, then the value of the surface tension of the liquid is:A small hole of an area of cross-section $2 \text{ mm}^2$ is present near the bottom of a fully filled open tank of height $2 \text{ m}$. Taking $g = 10 \text{ m/s}^2$, the rate of flow of water through the open hole would be nearly:In a U-tube, as shown in the figure, the water and oil are in the left side and right side of the tube respectively. The height of the water and oil columns are 15 cm and 20 cm respectively. The density of the oil is: [take ρ_water = 1000 kg/m^3]A U-tube with both ends open to the atmosphere is partially filled with water. Oil, which is immiscible with water, is poured into one side until it stands at a distance of $10 \text{ mm}$ above the water level on the other side. Meanwhile, the water rises by $65 \text{ mm}$ from its original level (see diagram). The density of the oil is:Which of the following statements is not true?The heart of a man pumps $5 \text{ L}$ of blood through the arteries per minute at a pressure of $150 \text{ mm}$ of mercury. If the density of mercury is $13.6 \times 10^3 \text{ kg/m}^3$ and $g = 10 \text{ m/s}^2$, then the power of heart in watt is:Water rises to a height $h$ in a capillary tube. If the length of the capillary tube above the surface of water is made less than $h$, then:A liquid does not wet the solid surface if the angle of contact is:
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ELECTROMAGNETIC WAVES
The energy of the EM wave is of the order of $15 \text{ keV}$. To which part of the spectrum does it belong?Light with an average flux of 20 W/cm² falls on a non-reflecting surface at normal incidence having a surface area of 20 cm². The energy received by the surface during a time span of 1 minute is:The ratio of the amplitude of the magnetic field to the amplitude of electric field for an electromagnetic wave propagating in vacuum is equal to:A parallel plate capacitor is charged by connecting it to a battery through a resistor. If $i$ is the current in the circuit, then in the gap between the plates:In an electromagnetic wave in free space the root mean square value of the electric field is $E_{rms} = 6$ V/m. The peak value of the magnetic field is:The electric field of an electromagnetic wave in free space is given by $\vec{E} = 10 \cos(10^7 t + kx) \hat{j}$ V/m, where $t$ and $x$ are in seconds and meters respectively. It can be inferred that:
(1) The wavelength $\lambda$ is $188.4 \text{ m}$.
(2) The wave number $k$ is $0.33 \text{ rad/m}$.
(3) The wave amplitude is $10 \text{ V/m}$.
(4) The wave is propagating along $+x$ direction.
Which one of the following pairs of statements is correct?$\varepsilon_0$ and $\mu_0$ are the electric permittivity and magnetic permeability of free space respectively. If the corresponding quantities of a medium are $2\varepsilon_0$ and $1.5\mu_0$ respectively, the refractive index of the medium will nearly be:To produce an instantaneous displacement current of $2 \text{ mA}$ in the space between the parallel plates of a capacitor of capacitance $4 \text{ }\mu\text{F}$, the rate of change of applied variable potential difference $\left(\frac{dV}{dt}\right)$ must be:
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Wave Optics
The interference pattern is obtained with two coherent light sources of intensity ratio $n$. In the interference pattern, the ratio $\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$ will beAt the first minimum adjacent to the central maximum of a single slit diffraction pattern, the phase difference between the Huygen's wavelet from the edge of the slit and the wavelet from the midpoint of the slit isA major breakthrough in the studies of cells came with the development of an electron microscope. This is because:Two slits in Young's experiment have widths in the ratio of $1:25$. The ratio of intensity at the maxima and minima in the interference pattern $\frac{I_{max}}{I_{min}}$ is:For a parallel beam of monochromatic light of wavelength $\lambda$, diffraction is produced by a single slit whose width $a$ is much greater than the wavelength of the light. If $D$ is the distance of the screen from the slit, the width of the central maxima will be:In a diffraction pattern due to a single slit of width $a$, the first minimum is observed at an angle $30^\circ$ when light of wavelength $5000\text{ \AA}$ is incident on the slit. The first secondary maximum is observed at an angle ofIn a double-slit experiment, when light of wavelength $400 \text{ nm}$ was used, the angular width of the first minima formed on a screen placed $1 \text{ m}$ away, was found to be $0.2^{\circ}$. What will be the angular width of the first minima, if the entire experimental apparatus is immersed in water? $\left(\mu_{\text{water}} = \frac{4}{3}\right)$For Young's double-slit experiment, two statements are given below:
Statement I: If screen is moved away from the plane of slits, angular separation of the fringes remains constant.
Statement II: If the monochromatic source is replaced by another monochromatic source of larger wavelength, the angular separation of fringes decreases.
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MAGNETISM AND MATTER
If a diamagnetic substance is brought near the north or the south pole of a bar magnet, it isThe variation of susceptibility ($\chi$) with absolute temperature ($T$) for a paramagnetic material is represented as:If $\theta_1$ and $\theta_2$ be the apparent angles of dip observed in two vertical planes at right angles to each other, then the true angle of dip $\theta$ is given by:A magnetic needle suspended parallel to a magnetic field requires $\sqrt{3}$ J of work to turn it through $60^\circ$. The torque needed to maintain the needle in this position will be:An iron bar of length $L$ has a magnetic moment $M$. It is bent at the middle of its length such that the two arms make an angle $60^{\circ}$ with each other. The magnetic moment of this new magnet is:Above Curie temperature:The magnetic susceptibility is negative for:The following figures show the arrangement of bar magnets in different configurations. Each magnet has a magnetic dipole moment $\mathbf{m}$. Which configuration has the highest net magnetic dipole moment?
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Kinetic Theory
The mean free path l for a gas molecule depends upon the diameter, d of the molecule as:The value $\gamma = \frac{C_P}{C_V}$ for hydrogen, helium, and another ideal diatomic gas $X$ (whose molecules are not rigid but have an additional vibrational mode), are respectively equal to:An ideal gas at $0^\circ\text{C}$ and atmospheric pressure $P$ has volume $V$. The percentage increase in its temperature needed to expand it to $3V$ at constant pressure is:The root-mean-square speed of hydrogen molecules at 300 K is 1930 m/s. Then the root mean square speed of oxygen molecules at 900 K will be:The molar specific heats of an ideal gas at constant pressure and volume are denoted by $C_P$ and $C_V$ respectively. If $\gamma = C_P/C_V$ and $R$ is the universal gas constant, then $C_V$ is equal to:In the given V-T diagram, what is the relation between pressure $P_1$ and $P_2$?One mole of an ideal monatomic gas undergoes a process described by the equation $PV^3 = \text{constant}$. The heat capacity of the gas during this process is:The volume occupied by the molecules contained in $4.5 \text{ kg}$ water at STP, if the molecular forces vanish away, is:
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CURRENT ELECTRICITY
Figure shows a potentiometer circuit for comparison of two resistances. The balance point with a standard resistor R = 10.0 Ω is found to be 58.3 cm, while that with the unknown resistance X is 68.5 cm. The value of X is:The potential difference across the 100 Ω resistance in the following circuit is measured by a voltmeter of 900 Ω resistance. The percentage error made in reading the potential difference is:The equivalent resistance of the infinite network given below is:The sliding contact C is at one fourth of the length of the potentiometer wire (AB) from A as shown in the circuit diagram. If the resistance of the wire AB is R₀, then the potential drop (V) across the resistor R is:In a potentiometer arrangement, a cell of emf 1.25 V gives a balance point at 35.0 cm length of the wire. If the cell is replaced by another cell and the balance point shifts to 63.0 cm, then the emf of the second cell is:The V-I graph for a conductor at temperatures $T_1$ and $T_2$ are as shown in the figure. The term $(T_2 - T_1)$ is proportional to:In a meter bridge experiment, the null point is at a distance of 30 cm from A. If a resistance of 16 Ω is connected in parallel with resistance Y, the null point occurs at 50 cm from A. The value of the resistance Y is:The figure shows a 2.0 V potentiometer used for the determination of the internal resistance of a 1.5 V cell. The balance point of the cell in the open circuit is 76.3 cm. When a resistor of 9.5 Ω is used in the external circuit of the cell, the balance point shifts to 64.8 cm length of the potentiometer wire. The internal resistance of the cell is:
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Mechanical Properties of Solids
The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?When a block of mass $M$ is suspended by a long wire of length $L$, the length of the wire becomes $(L+l)$. The elastic potential energy stored in the extended wire is:The maximum elongation of a steel wire of $1 \text{ m}$ length if the elastic limit of steel and its Young's modulus, respectively, are $8 \times 10^8 \text{ N m}^{-2}$ and $2 \times 10^{11} \text{ N m}^{-2}$, is:Two wires are made of the same material and have the same volume. The first wire has a cross-sectional area $A$ and the second wire has a cross-sectional area $3A$. If the length of the first wire is increased by $\Delta l$ on applying a force $F$, how much force is needed to stretch the second wire by the same amount?Let a wire be suspended from the ceiling (rigid support) and stretched by a weight $W$ attached at its free end. The longitudinal stress at any point of the cross-sectional area $A$ of the wire is:
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