A spherical solid ball of volume Vis made of a material of density \[{{\rho }_{1}}\]. It is falling through a liquid of density \[{{\rho }_{2}}({{\rho }_{2}}<{{\rho }_{1}})\]. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed v, i.e., \[{{F}_{\text{viscous}}}=-k{{v}^{2}}(k>0)\]. The terminal speed of the ball is
A disc of mass m and radius R is free to rotate in horizontal plane about a vertical smooth fixed axis passing through its centre. There is a smooth groove along the diameter of the disc and two small balls of mass \[m/2\]each are placed in it on either side of the centre of the disc as shown in figure. The disc is given initial angular velocity \[{{\omega }_{0}}\] and released. The net work done by forces exerted by disc on one of the balls (for the duration ball remains on the disc) is
Radius of moon is \[1/4\] times that of earth and mass is \[1/81\] times that of earth. The point at which gravitational field due to earth becomes equal and opposite to that of moon, is (Distance between centres of earth and moon is 60R, where R is radius of earth)
A disk of radius \[a/4\] having a uniformly distributed charge 6 C is placed in the \[x-y\] plane with its centre at \[(-a/2,0,0)\]. A rod of length a carrying a uniformly distributed charge 8 C is placed on the x - axis from \[x=a/4\] to \[x=5a/4\]. Two point charges \[-7C\] and 3 C are placed at \[(a/4,-a/4,0)\] and \[(-3a/4,\,3a/4,\,0),\] respectively. Consider a cubical surface formed by six surfaces \[x=\pm a/2,\] \[y=\pm a/2,\] \[z=\pm a/2\]. The electric flux through this cubical surface is
A potentiometer wire is 100 cm long and a constant potential difference is maintained across it. Two cells are connected in series first to support one another and then in opposite direction. The balance points are obtained at 50 cm and 10 cm from the positive end of the wire in the two cases. The ratio of emf's is:
A box of mass 2 kg is placed on the roof of a car. The box would remain stationary untill the car attains a maximum acceleration. Coefficient of static friction between the box and the roof of the car is \[0.2\] and \[g=10m{{s}^{-2}}\]The maximum acceleration of the car, for the box to remain stationary is
The dimension of \[\frac{{{e}^{2}}}{4\pi {{\varepsilon }_{0}}hc},\]where e, \[{{\varepsilon }_{0}},\] h and c are electric charge, electric permittivity, Planck's constant and velocity of light in vacuum respectively, is
Two stones are thrown up simultaneously from the edge of a cliff 240 m high with initial speed of \[10\text{ }m/s\]and \[40\text{ }m/s\] respectively. Which of the following graph best represents the time variation of relative position of the second stone with respect to the first?
(Assume stones do not rebound after hitting the ground and neglect air resistance, take \[g=10\,m/{{s}^{2}}\])
(The figures are schematic and not drawn to scale)
In an experiment with NPN transistor amplifier in common emitter configuration, the current gain of the transistor is 100. If the collector current changes by \[1m\text{ }A,\]what will be the change in emitter current?
Sinusoidal carrier voltage of frequency \[1.5\text{ }MHz\] and amplitude 50 V is amplitude modulated by sinusoidal voltage of frequency 10 kHz producing 50% modulation. The lower and upper side-band frequencies in kHz are
An ideal gas enclosed in a Vertical cylindrical container supports a freely moving piston of mass M. The piston and the cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is \[{{V}_{0}}\] and its pressure is \[{{P}_{0}}\]. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency
A body of mass M, executes vertical SHM with periods \[{{t}_{1}}\] and \[{{t}_{2}},\] when separately attached to spring A and spring B respectively. The period of SHM, when the body executes SHM, as shown in the figure is to. Then
Let \[\bar{v},\] \[{{v}_{rms}}\] and \[{{v}_{p}}\] respectively denote the mean speed, root mean square speed and most probable speed of the molecules in an ideal monatomic gas at absolute temperature T. The mass of a molecule is m. Then
A)
No molecule can have speed greater than \[\sqrt{2}{{v}_{rms}}\]
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B)
No molecule can have speed less than \[{{v}_{p}}/\sqrt{2}\]
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C)
\[{{v}_{p}}=\bar{v}<{{\bar{v}}_{rms}}\]
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D)
The average kinetic energy of a molecule is \[\frac{3}{4}mv_{p}^{2}\]
The fundamental frequency of a sonometer wire of length \[\ell \]is \[{{f}_{0}}\]. Abridge is now introduced at a distance of \[\Delta \ell \] from the centre of the wire \[(\Delta \ell <<\ell )\]. The number of beats heard if both sides of the bridges are set into vibration in their fundamental modes are-
In a uniform magnetic field of induction B a wire in the form of a semicircle of radius r rotates about the diameter of the circle with an angular frequency \[\omega \]. The axis of rotation is perpendicular to the field. If the total resistance of the circuit is R, the mean power generated per period of rotation is
A light ray from air is incident (as shown in figure) at one end of a glass fiber (refractive index \[\mu =1.5\]) making an incidence angle of \[60{}^\circ \] on the lateral surface, so that it undergoes a total internal reflection. How much time would it take to traverse the straight fiber of length 1 km
In an electrical circuit R, L, C and an a.c. voltage source are all connected in series. When L is removed from the circuit, the phase difference between the voltage the current in the circuit is \[\pi /3\]. If instead, C is removed from the circuit, the phase difference is again \[\pi /3\]. The power factor of the circuit is
A long, straight wire is turned into a loop of radius 10 cm (as shown in figure). If a current of 8 ampere is passed through the loop, then the value of the magnetic field B at the centre C of the loop will be (in\[Wb/{{m}^{2}}\])
A mass of \[2.9\text{ }kg\] suspended from a string of length 50 cm is at rest. Another body of mass 100 g which is moving horizontally with a velocity of \[150\text{ }m/s\] strikes and sticks to it. What is the tension (in newton) in the string when it makes an angle of \[60{}^\circ \] with the vertical?
Figure shows two, identical narrow slits \[{{S}_{1}}\] and \[{{S}_{2}}\]. A very small completely absorbing strip is placed at distance 'y' from the point C. 'C' is the point on the screen equidistant from \[{{S}_{1}}\] and \[{{S}_{2}}\]. Assume \[\lambda <<d<<D,\] where \[\lambda ,\,d\] and D have usual meaning. When \[{{S}_{2}}\] is covered the force due to light acting on strip is 'f and when both slits are opened the force acting on strip is 2f. The minimum positive 'y' \[(<<D)\] coordinate is \[\frac{\lambda D}{xd}\]. Find the value of x.
The mass of \[_{7}{{N}^{15}}\] is \[15.00011\text{ }amu,\] mass of \[_{8}{{O}^{16}}\] is \[15.99492\text{ }amu\] and \[{{m}_{p}}=1.00783\text{ }amu\]. Determine binding energy (in MeV) of last proton of \[_{8}{{O}^{16}}\].
A calorimeter of water equivalent \[5\times {{10}^{-3}}\text{ }kg\] contains \[25\times {{10}^{-3}}\text{ }kg\] of water. It takes 3 minutes to cool from \[28{}^\circ C\] to \[21{}^\circ C\]. When the same calorimeter is filled with \[30\times {{10}^{-3}}\text{ }kg\] of turpentine oil then it takes 2 minutes to cool from \[28{}^\circ C\] to \[21{}^\circ C\]. Find out the specific heat (in \[cal/g{}^\circ C\]) of turpentine oil.
Arrange the carnations, \[{{(C{{H}_{3}})}_{3}}\overline{C},\overline{C}C{{l}_{3}},{{(C{{H}_{3}})}_{2}}\overline{C}H,{{C}_{6}}{{H}_{5}}\overline{C}{{H}_{2}}\]in order of their decreasing stability.
photons come out the metal when hit by a beam of electrons
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B)
photons come out of the nucleus of an atom under the action of an electric field
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C)
electrons come out of metal with a constant velocity which depends on frequency and intensity of incident light
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D)
electrons come out of metal with different velocities not greater than a certain value which depends upon frequency of incident light and not on intensity.
In a face-centred cubic arrangement of A and B atoms in which A atoms are at the corners of the unit cell and B atoms are at the face centres, one of the A atoms is missing from one corner in unit cell. The simplest formula of the compound is
An organic aromatic compound having molecular formula \[{{C}_{7}}{{H}_{8}}O\]does not give characteristic colour with neutral \[FeC{{l}_{3}}\]but bubbles of hydrogen gas are formed when it is treated with metallic sodium. The compound is
A mixture of 1.57 moles of \[{{N}_{2}}\] , 1.92 moles of \[{{H}_{2}}\] and 8.13 moles of \[N{{H}_{3}}\] is introduced into a 20 L reaction vessel at 500 K. At this temperature, the equilibrium constant, \[{{K}_{c}}\] for the reaction, \[{{N}_{2(g)}}+3{{H}_{2(g)}}\rightleftharpoons 2N{{H}_{3(g)}}\] is \[1.7\times {{10}^{2}}\] . Select the correct statement.
The equivalent conductances of two strong electrolytes at infinite dilution in \[{{H}_{2}}O\] (where ions move freely through a solution) at \[25{}^\circ C\] are given below: \[{{\Lambda }^{o}}_{C{{H}_{3}}COONa}=91.0Sc{{m}^{2}}/equiv\] . \[{{\Lambda }^{o}}_{HCl}=426.2S\,c{{m}^{2}}/equiv\] What additional information/quantity one needs to calculate \[{{\Lambda }^{o}}\] of an aqueous solution of acetic acid?
A)
\[{{\Lambda }^{o}}\] of chloroacetic acid \[(ClC{{H}_{2}}COOH)\]
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B)
\[{{\Lambda }^{o}}\] of NaCl
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C)
\[{{\Lambda }^{o}}\] of \[C{{H}_{3}}COOK\]
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D)
the limiting equivalent conductance of \[{{H}^{+}}({{\lambda }^{o}}_{{{H}^{+}}})\] .
Amount of of potassium dichromate in grams required to oxidise 20.0 g of \[F{{e}^{2+}}\] in \[FeS{{O}_{4}}\] to \[F{{e}^{3+}}\] if the reaction is carried out in an acidic medium is ________. (Molar masses of \[{{K}_{2}}C{{r}_{2}}{{O}_{7}}\] and \[FeS{{O}_{4}}\] are 294 and 152 respectively.)
The mass of a non-volatile solute (in g) which should be dissolved in 114 g octane to reduce its vapour pressure to 80% is ________. (Given : Molar mass of solute \[40g\,\text{mo}{{\text{l}}^{-1}}\] )
An open vessel contains 200 mg of air at \[17{}^\circ C\] . The weight percent of air that would be expelled if the vessel is heated to \[117{}^\circ C\] is _________.
The standard heat of formation of \[C{{H}_{4}}_{(g)},C{{O}_{2(g)}}\] and \[{{H}_{2}}{{O}_{(g)}}\] are -76.2, -394.8 and -241.6 kJ \[\text{mo}{{\text{l}}^{-1}}\] respectively. The amount of heat evolved (in kJ) by burning \[1{{m}^{3}}\] of methane measured at NTP is \[x\times {{10}^{4}}\] . The value of x is ________.
If statements p and q take truth values as TT, TF, FT, FF in order, then the respective truth values of statement \[(p\to q)\leftrightarrow (-p\to -q)\] will be
Tangents are drawn to a circle of radius R from an external point P, to touch the circle at A and B. If C is centre of the circle and mid-point of AB is Q, then the value of \[CP\cdot CQ\] is
If A and B are two non-singular matrices of order 3 such that \[A{{A}^{T}}=2I\] and \[{{A}^{-1}}={{A}^{T}}-A\cdot adj(2{{B}^{-1}})\], then det (B) equals
If \[\left| {\vec{a}} \right|=1,\left| {\vec{b}} \right|=2,\left| {\vec{c}} \right|=3,\,\vec{a},\,\vec{b}\] are perpendicular to \[\vec{c}\] and \[\vec{a}\cdot \vec{b}=1,\] then \[\left[ \frac{\vec{a}\times \vec{b}}{5}\,\,\vec{b}\times \vec{c}\,\,\frac{\vec{a}+\vec{b}+\vec{c}}{\sqrt{3}} \right]=\]
Number of ways in which 7 green bottles and 8 blue bottles can be arranged in a row, if exactly 1 pair of green bottles is side by side, is (assume all bottles to be alike except for the colour)
Let f be a positive continuous function on the interval. is the area of the region bounded by the graph of and the lines and where . If then the value of is.