A ball is projected vertically upwards with a certain initial speed. Another ball of the same mass is projected at an angle of \[60{}^\circ \] with the vertical with the same initial speed. At highest points of their journey, the ratio of their potential energies will be
A uniform rod of length L and mass M is held vertical, with its bottom end pivote to the floor. The rod falls under gravity, freely turning about the pivot. If acceleration due to gravity is g, what is the instantaneous angular speed of the rod when it makes an angle \[60{}^\circ \] with the vertical
Two wires A and B are of the same material. Their lengths are in the ratio \[1:2\] and the diameter are in the ratio\[2 :1\]. If they are pulled by the same force, then increase in length will be in the ratio of
Two positive ions, each carrying a charge q, are separated by a distance d. If F is the force of repulsion between the ions, the number of electrons missing from each ion will be (e being the charge of an electron)
Two identical capacitors, have the same capacitance C. One of them is charged to potential \[{{V}_{1}}\] and the other to \[{{V}_{2}}\]. The negative ends of the capacitors are connected together. When the positive ends are also connected, the decrease in energy of the combined system is
When two identical batteries of internal resistance\[1\,\Omega \]. each are connected in series across a resistor R, the rate of heat produced in R is \[{{J}_{1}}\]. When the same batteries are connected in parallel across R, the rate is \[{{J}_{1}}\]. If \[{{J}_{1}}=2.25\,{{J}_{2}}\] then the value of R in \[\Omega \] is
A particle is projected with a velocity v such that its range on the horizontal plane is twice the greatest height attained by it. The range of the projectile is (where g is acceleration due to gravity)
Two identical photocathodes receive light of frequencies\[{{f}_{1}}\,\,and\,\,{{f}_{2}}\]. If the velocities of the photoelectrons (of mass m) coming out are with velocity \[{{\nu }_{1}}\,\,and\,\,{{\nu }_{2}}\] respectively, then
Assume that the nuclear binding energy per nucleon (B/A) versus mass number [A] is as shown in the figure. Use this plot to choose the correct choice(s) given below.
A)
Fusion of two nuclei with mass numbers lying in the range of \[1 < A < 50\] will release energy
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B)
Fusion of two nuclei with mass numbers lying in the range of \[51<A<100\] will release energy
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C)
Fission of a nucleus lying in the mass range of \[100 < A < 200\] will release energy when broken into two equal fragments
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D)
?Fission of a nucleus lying in the mass range of \[200 < A < 360\] will release energy when broken into two equal fragments
A cylindrical metallic rod in thermal contact with two reservoirs of heat at its two ends conducts an amount of heat Q in time t. The metallic rod is melted and the material is formed into a rod of half the radius of the original rod. What is the amount of heat conducted by the new rod, when placed in thermal contact with the two reservoirs in time t?
One mole of an ideal monoatomic gas at temperature \[{{T}_{0}}\] expands slowly according to the law\[\frac{P}{V}=cons\tan t\]. If the final temperature is \[2{{T}_{0}}\] heat supplied to the gas is
An ideal gas is initially at temperature T and volume V. Its volume is increased by A F due to an increase in temperature\[\Delta T\], pressure remaining constant. The quantity \[\delta =\frac{\Delta V}{V\Delta T}\], varies with temperature as
A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is distrubed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be
One conducting C/tube can slide inside another as shown in figure, maintaining electrical contacts between the tubes. The magnetic field B is perpendicular to the plane of the figure. If each tube moves towards the other at a constant speed v then the emf induced in the circuit in terms of B, I and v where / is the width of each tube, will be
An AC voltage source of variable angular frequency \[\omega \] and fixed amplitude \[{{V}_{0}}\] is connected in series with a capacitance C and an electric bulb of resistance R (inductance zero). When \[\omega \] is increased
Two equally charged pith-balls each of mass 10 gm are suspended from the same point by two silk threads each of length 1.2 m. As a result of mutual repulsion the balls are separated by 5 cm. Find the charge (in coulomb) on each ball.
Two equal masses of 6.40 kg each are separated by a distance of 0.16 m. A small body is released from a point P equidistant from the two masses and at a distance 0.06 m from the line joining them as shown in fig. Calculate the velocity (in m/s) of this body when it passes through Q.
A horizontal telegraph wire 10m long oriented along the magnetic east-west direction, falls freely under gravity to the ground from a height of 10 m. Find the emf (in volt) induced in the wire at the instant the wire strikes ground. \[({{B}_{H}}= 2.5 \times 1{{0}^{-}}^{5}Wb/{{m}^{2}}, g = 9.8 m/{{s}^{2}})\]
The speed of a wave on a string is 150 m/s when the tension is 120 N. The percentage increase in the tension in order to raise the wave speed by \[20\,%\] is
A star initially has \[{{10}^{40}}\] deutrons. It produces energy by processes, \[_{1}{{H}^{2}}+\,{{\,}_{1}}{{H}^{2}}\,\,\xrightarrow{{}}\,{{\,}_{1}}{{H}^{3}}+p\]and \[_{1}{{H}^{2}}+\,{{\,}_{1}}{{H}^{3}}\,\,\xrightarrow{{}}\,{{\,}_{2}}H{{e}^{4}}+n\] If average power radiated by star is \[{{10}^{16}}W\] find time (in second) in which deutron is exhausted. \[\operatorname{M}{{(}_{1}}{{H}^{2}})=\,\,2.01471\,amu\], \[M{{(}_{2}}H{{e}^{4}})=400388\,\,amu\], \[{{\operatorname{m}}_{p}}= 1.00783 amu\], \[{{\operatorname{m}}_{n}}=1.00866amu\]
In the 'Ring test' of \[N{{O}_{3}}^{\Theta },\] both \[N{{O}_{2}}^{\Theta }\] and \[{{I}^{\Theta }}\] interferes in the 'Ring test of\[N{{O}_{3}}^{\Theta }\]. These are removed respectively by
A)
Boiling the mixture with \[N{{H}_{4}}Cl\] and adding \[HgC{{l}_{2}}\]
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B)
Boiling the mixture with \[N{{H}_{4}}OH\] and adding \[HgC{{l}_{2}}\]
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C)
Boiling the mixture and evaporate to half the solution and adding \[{{H}_{2}}{{C}_{2}}{{O}_{4}}\]
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D)
Boiling the mixture with cone. \[HN{{O}_{3}}\] and adding \[C{{l}_{2}}\] water.
In the process: \[{{H}_{2}}O(s),-10{}^\circ C,1atm)\xrightarrow{{}}{{H}_{2}}O(l,\,\,10{}^\circ C,\,1atm){{C}_{P}}\] for \[ice=9\,cal\,{{\deg }^{-1}}mo{{l}^{-1}},\]\[{{C}_{P}}\] for \[{{H}_{2}}O=18cal{{\deg }^{-1}}mo{{l}^{-1}}.\]Latent heat of fusion of \[ice=1440\,cal\,mo{{l}^{-1}}\] at\[0{}^\circ C\]. The entropy change for the above process in \[cal\,\,{{\deg }^{-1}}mo{{l}^{-1}}\] and the total number of steps in which the third law of thermodynamics used are
Metallic gold crystallises in the fee lattice. The length of the cubic unit cell i\[a=4242\text{ }\overset{o}{\mathop{A}}\,\]. Closest distance between two gold atoms and packing factor are, respectively.
In the reaction, \[2{{H}_{2}}{{O}_{2}}\xrightarrow{{}}2{{H}_{2}}O+{{O}_{2}}.\]The rate of decomposition of \[{{H}_{2}}{{O}_{2}}\] is\[0.68\text{ }kg\text{ }h{{r}^{-1}}.\] What are the rates of formation of \[{{O}_{2}}\] and \[{{H}_{2}}O\] respectively, (in \[kg\,h{{r}^{-1}}\])?
A flask of \[10\text{ }d{{m}^{3}}\]capacity contains \[{{O}_{2}}\] gas at \[100\text{ }kPa\]and\[300\text{ }K\]. The gas pressure is reduced to \[10\text{ }Pa\]by attaching the flask to a pump. If now \[2.8\text{ }g\]of \[{{N}_{2}}\] is introduced, assuming ideal behaviour, the final pressure in the flask is ______. \[(R\approx 8.3\,kPa\,d{{m}^{3}}{{K}^{-1}}mo{{l}^{-1}})\]
In water \[(H-O-H)\] bond angle is\[105{}^\circ \]. The distance between \[(O-H)\] is \[1.0A{}^\circ \,{{u}_{{{H}_{2}}O}}=1.827D\]. The magnitude of charge on the H-atom in water molecule in esu is______. (Given\[\cos \,\,52.5{}^\circ =0.609\])
\[20\text{ }mL\]of \[x\text{ }M\text{ }HCl\]neutralises completely \[10\text{ }mL\]of \[0.1\text{ }M\text{ }NaHC{{O}_{3}}\]solution and a further \[5\text{ }mL\]of \[0.2\,M\,N{{a}_{2}}C{{O}_{3}}\] solution to methyl orange end point. The value of x is ______.
Elevation in boiling point studies of \[Ca{{(N{{O}_{3}})}_{2}}\] gives molar mass as\[131.2\]. The degree of dissociation of \[Ca{{(N{{O}_{3}})}_{2}}\] is ______.
A solution of zirconium phosphate contains \[8\times {{10}^{-5}}\,M\,\,P{{O}_{4}}^{3-}\]ions. The \[{{K}_{sp}}\]of zirconium phosphate is \[x\times {{10}^{-3}}{{M}^{7}}\]. The value of x is ______.
If two tangents drawn from the point \[P\left( \alpha ,\beta \right)\] to the parabola \[{{y}^{2}}=4x\] be such that the slope of one tangent is double of the other then -
If \[\alpha ,\beta \] are the roots of \[a{{x}^{2}}+bx+c=0\] and \[\alpha +\beta ,{{\alpha }^{2}}+{{\beta }^{2}},{{\alpha }^{3}}+{{\beta }^{3}}\] are in G.P., where \[\Delta ={{b}^{2}}-4ac,\] then -
If \[{{a}_{1}},{{a}_{2}},.....{{a}_{n}}\] are positive real numbers whose product is a fixed number c, then the minimum value of \[{{a}_{1}}+{{a}_{2}}+....+{{a}_{n-1}}+2{{a}_{n}}\] is -
The solution of differential equation \[({{x}^{2}}+{{y}^{2}})dy=xy.dx\] is \[y=y\left( x \right).\text{ }If\,y\left( 1 \right)=1\] and \[y\left( {{x}_{0}} \right)=e\] then \[{{x}_{0}}\] is -
Let h(x) be differentiable for all x and let\[f\left( x \right)=\left( kx+{{e}^{x}} \right)h\left( x \right)\], where k is some constant. If \[h\left( 0 \right)=5,\text{ }h'\left( 0 \right)=-2\] and\[f'\left( 0 \right)=18\], then the value of k is -
Vectors \[\vec{a}=\hat{i}+2\hat{j}+3\hat{k};\,\,\vec{b}=2\hat{i}-\hat{j}+\hat{k}\] and \[\vec{c}=3\hat{i}+\hat{j}+4\hat{k}\] are so placed that the end point of one vector is the starting point of the next vector. Then the vectors are-
If \[a\ne b\ne c\] and if \[ax+by+c=0;bx+cy+a=0,\]\[cx+ay+b=0\] are concurrent then \[{{2}^{{{a}^{2}}{{b}^{-1}}{{c}^{-1}}}}{{2}^{{{b}^{2}}{{c}^{-1}}{{a}^{-1}}}}{{2}^{{{c}^{2}}{{a}^{-1}}{{b}^{-1}}}}=\]
If \[{{S}_{1}},{{S}_{2}},{{S}_{3}}\] are the sums of first n natural numbers, their squares, their cubes respectively, then find the value of \[\frac{{{S}_{3}}(1+8{{S}_{1}})}{{{S}_{2}}^{2}}.\]