A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy \[U(x)=K\,{{\left| x \right|}^{3}},\]where K is a positive constant. If the amplitude of oscillation is a, then its time period is
If \[\vec{A}\] and \[\vec{B}\] are two non-zero vectors such that \[\left| \vec{A}+\vec{B} \right|=\frac{\left| \vec{A}+\vec{B} \right|}{2}\] and \[\left| {\vec{A}} \right|=2\left| {\vec{B}} \right|,\] then the angle between \[\vec{A}\] and \[\vec{B}\] is
Initial acceleration of a particle moving in a straight line is \[{{a}_{0}}\] and initial velocity is zero. The acceleration reduces continuously to half in every \[{{t}_{0}}\] seconds. The terminal velocity of the particle is
The distance between two moving particles at any time is a. If v be their relative velocity, and \[{{v}_{1}}\] and \[{{v}_{2}}\] be the components of v along and perpendicular to a, the minimum distance between them is
An artificial satellite of mass m is moving in a circular orbit at a height equal to the radius R of the earth. Suddenly due to internal explosion the satellite breaks into two parts of equal masses. One part of the satellite stops just after the explosion. The increase in the mechanical energy of the system (satellite + earth) due to explosion will be (Given: acceleration due to gravity on the surface of earth is g)
One mole of an ideal gas at initial temperature T undergoes a quasi-static process during which the volume V is doubled. During the process the internal energy U obeys the equation \[U=a{{v}^{3}},\] where 'a? is a constant. The work done during the process is
The piston is massless and the spring is ideal and initially stretched. The piston cylinder arrangement encloses an ideal gas. If the gas is heated quasistatically, the PV graph is
Three identical large metal plates each of area S are at distance d and 2d from each other as shown. Metal plate A is uncharged, while metal plates B and C have charges \[+Q\] and \[-Q\] respectively. Metal plates A and C are connected by a conducting wire through a switch K. How much electrostatic energy is lost when the switch is closed?
A cylindrical conductor AB of length l and area of cross-section a is connected to a battery having emf E and negligible internal resistance. The specific conductivity of cylindrical conductor varies as \[\sigma ={{\sigma }_{0}}\frac{1}{\sqrt{x}},\]where \[{{\sigma }_{0}}\] is constant and x is distance from end A. What is the electric field just near the end B of cylinder?
A metallic ring (radius R) of negligible resistance has a resistance r connected across its diameter as shown in the figure. It is moving with velocity \[{{v}_{0}}\]in a constant magnetic field B acting perpendicular to the plane of paper in inward direction. The current in the resistance is
The tube shown is of non-uniform cross-section. The cross-section area at A is half of the cross-section area at B, C and D. A liquid is flowing through in steady state. The liquid exerts on the tube is
Statement I: A net force towards right.
Statement II: A net force towards left.
Statement III: A net force in some oblique direction.
A uniform ring of mass M and radius R carries a current I (see figure). The ring is suspended using two identical strings OA and OB. There exists a uniform horizontal magnetic field \[{{B}_{0}}\] parallel to the diameter AB of the ring. The ratio of tensions in the two strings \[\frac{{{T}_{AO}}}{{{T}_{BO}}}\] is (Given \[\theta =60{}^\circ \])
In a beam of neutrons, the particles are having a kinetic energy of \[0.0327\text{ }eV\]If the half-life of neutron is 693 second, what fraction of neutrons will decay before the beam advances by\[10\text{ }m\]? Given mass of neutron \[=1.675\times {{10}^{-27}}kg\]
A point charge q is located at a distance 'r' from the centre O of an uncharged conducting spherical layer whose inside and outside radii are equal to \[{{R}_{1}}\] and \[{{R}_{2}}\] respectively. It is given that \[r<{{R}_{1}}\] Assuming zero potential at infinity, the potential at O is
The maximum intensity in Young's double slit experiment is \[{{I}_{0}}.\] Distance between the slits is J= \[d=5\lambda ,\]where \[\lambda \] is the wavelength of monochromatic light used in the experiment. What will be the intensity of light in front of one of the slits on a screen at a distance \[D=10\,d\]?
A point source of electromagnetic radiation has an average power output of 800 W. The maximum value of electric field at a distance \[4.0\text{ }m\]from the source is
A particle of mass m and charge q is projected into a uniform magnetic field \[\vec{B}=-{{B}_{0}}\hat{k}\] with velocity \[\overset{\to }{\mathop{v}}\,={{v}_{0}}\hat{i}\] from origin. The position vector of the particle at time t is \[\overset{\to }{\mathop{r}}\,\] . Find the impulse of magnetic force on the particle by the time when \[\overset{\to }{\mathop{r}}\,.\overset{\to }{\mathop{v}}\,\]becomes zero for the first time.
A stretchable conducting ring is in the shape of a circle. It is kept in a uniform magnetic field that is perpendicular to the plane of the ring. The ring is pulled out uniformly from all sides so as to increase its radius at a constant rate \[\frac{dr}{dt}=V\] while maintaining its circular shape. Find the rate of work done by the external agent against the magnetic force when the radius of the ring is \[{{r}_{0}}\]. (Given the resistance of the ring remains constant at R).
A boy throws a ball with speed u in a well of depth \[14\text{ }m\]as shown. On bounce with bottom of the well the speed of the ball gets halved. What should be the minimum value of u in \[(m/s)\] such that the ball may be able to reach his hand again? Assume that his hands are at 1 m height from top of the well while throwing and catching.
A boy of mass \[60\text{ }kg\]is standing over a platform of mass \[40\text{ }kg\]placed over a smooth horizontal surface. He throws a stone of mass \[1\text{ }kg\]with velocity \[u=10\text{ }m/s\]at an angle of \[45{}^\circ \]with respect to ground. The displacement of platform (with boy) on the horizontal when the stone lands on the ground is\[(g=10m/{{s}^{2}})\,5N\,cm.\]Find the value of N?
A cylinder of mass \[{{M}_{C}}\] and sphere of mass \[{{M}_{S}}\] are placed at points A and B of two inclines, respectively (see figure). If they roll on the incline without slipping such that their acceleration is the same, find the ratio of \[\frac{\sin {{\theta }_{C}}}{\sin {{\theta }_{S}}}\].
Two point coherent sources of power \[{{P}_{0}}\] and \[4{{P}_{0}}\] emitting sound of frequency \[150\text{ }Hz\]are kept at points A and B respectively. Both sources are in same phase. A detector is kept at C as shown in figure. The distance of point A and B is r and \[4r\] from detector respectively. The speed of sound in medium is\[300\text{ }m/s\]. Given \[{{P}_{0}}=64\pi \]watt and\[r=1\text{ }m\]. The intensity observed by detector is\[4N\text{ }watt/{{m}^{2}}\]. Find the value of N.
Figure shows a metre bridge, wire AC has uniform cross-section. The length of wire AC is 100 cm. X is a standard resistor of 4 Q and Y is a coil. When Y is immersed in melting ice the null point is at 40 cm from point A. When the coil Y is heated to \[100{}^\circ C,\] a\[78\text{ }\Omega \]. resistor has to be connected in parallel with Fin order to keep the bridge balanced at the same point. The temperature coefficient of resistance of the coil is \[M\times {{10}^{-4}}{{K}^{-1}}\] Find the value of M?
Which of the compounds HCHO (I), \[C{{H}_{3}}C{{H}_{2}}CHO\] (II), \[C{{H}_{3}}COC{{H}_{2}}\] (III) and \[HCOO{{C}_{2}}{{H}_{5}}\] (IV) will give a secondary alcohol on reaction with excess Grignard reagent, followed by hydrolysis? Select the correct answer using the codes given below:
25 ml of 2 N, \[HCl\], 50 ml of 4 N, \[HN{{O}_{3}}\] and x ml of 2 M, \[{{H}_{2}}S{{O}_{4}}\] are mixed together and the total volume is made up to 1 L after dilution. 50 ml of this acid mixture completely reacted with 25 ml of a 1 N, \[N{{a}_{2}}C{{O}_{3}}\] solution. The value of x is;
An aqueous solution containing 1 M each of \[A{{u}^{+3}},C{{u}^{+2}},A{{g}^{+}},L{{i}^{+}}\] is being electrolyzed by using inert electrodes. The value of standard potentials are: \[E{{_{Ag}^{{}^\circ }}_{^{+}/Ag}}=0.80V,E_{C{{u}^{+}}/Cu}^{{}^\circ }=0.34V\] and \[E{{_{Au}^{{}^\circ }}_{^{+3}/Au}}=1.50V,E_{L{{i}^{+}}/Li}^{{}^\circ }=-3.03V\] With increasing voltage, the sequence of deposition of metals on the cathode will be:
The binding energy of the electron in the lowest orbit of the hydrogen atom is 13.6 ev. The energies required in eV to remove an electron from three lowest orbits of the hydrogen are-
A binary solid \[\left( {{A}^{+}}{{B}^{-}} \right)\] has zinc blende structure with \[{{B}^{-}}\] ions constituting the lattice and \[{{A}^{+}}\] ions occupying 25% tetrahedral holes. The formula of solid is -
Given (i) \[C{{u}^{2+}}+2{{e}^{-}}\xrightarrow{{}}Cu,E{}^\circ =0.337\,V\] (ii) \[C{{u}^{2+}}+{{e}^{-}}\xrightarrow{{}}C{{u}^{+}},E{}^\circ =0.153\,V\] Electrode potential, \[E{}^\circ \]for the reaction, \[C{{u}^{+}}{{e}^{-}}\xrightarrow{{}}Cu\], will be
When heated, ammonium carbamate decomposes as follows: \[N{{H}_{4}}COON{{H}_{2}}(s)\rightleftharpoons 2N{{H}_{3}}(g)+C{{O}_{2}}(g)\] At a certain temperature, the equilibrium pressure of the system is 0.318 atm. \[{{K}_{p}}\]for the reaction is:
Given that for a reaction of order n. the integrated form of the rate equation is \[K=\frac{1}{t(n-1)}\left[ \frac{1}{{{C}^{n-1}}}-\frac{1}{C_{0}^{n-1}} \right]\] where \[{{C}_{0}}\] and C are the values of the reactant concentration at the start and after time t. What is the relationship between \[{{t}_{3/4}}\] and \[{{t}_{1/2}}\] where \[{{t}_{3/4}}\] is the time required for C to become \[1/4\text{ }{{C}_{0}}\]-
Solubility of calcium phosphate (molecular mass, M) in water is W g per 100 mL at \[25{}^\circ C\]. Its solubility product at \[25{}^\circ C\] will be approximately-
For a cubical system the following information are available. Edge length\[=5\overset{{}^\circ }{\mathop{A}}\,\]; density \[=2\text{ }gm/c{{m}^{3}}.\] Atomic wt. = 75. Determine the radius of the atom in pm?
The freezing point of an 0.08 (m) \[NaHS{{O}_{4}}\] is \[-0.345{}^\circ C.\] Calculate the percentage of \[HSO_{4}^{-}\]ions that transfer a proton to water to form \[SO{{_{4}^{2-}}^{~}}\]ion. \[{{K}_{f}}\left( {{H}_{2}}O \right)=1.86\,mo{{l}^{-1}}\,kg\]
A chord AB drawn from the point \[A(0,3)\] on circle \[{{x}^{2}}+4x+{{(y-3)}^{2}}=0\]meets to M in such a way that \[AM=2AB,\]then the locus of point At will be
Let \[f(x)=\left\{ \begin{matrix} (x-1)\sin \frac{1}{x-1} & if\,x\ne 1 \\ 0 & if\,x=1 \\ \end{matrix} \right.\] Then which one of the following is true?
A)
f is differentiable at \[x=0\] and \[x=1\]
doneclear
B)
f is differentiable at \[x=0\] but not at \[x=1\]
doneclear
C)
f is differentiable at \[x=1\] but not at \[x=0\]
doneclear
D)
f is neither differentiable at \[x=0\] nor at \[x=1\]
In a town of 10,000 families it was found that 40% family buy newspaper A, 20% buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers. then number of families which buy A only is
The numbers P, Q and R for which the function\[f(x)=P{{e}^{2x}}+Q{{e}^{x}}+Rx\] satisfies the conditions \[f(0)=-1,\] \[f'(\log \,2)=31\] and \[\int_{0}^{\log \,\,4}{[f(x)-Rx]dx=\frac{39}{2}}\]are given by
If a circles \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]and the rectangular hyperbola \[xy={{c}^{2}}\] intersect m four points, \[\left( c{{t}_{r}},\frac{c}{{{t}_{r}}} \right),\] \[r=1,2,3,4\]then \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}\] is equal to
The equation of the planes passing through the line of intersection of the planes \[x+3y+6=0\]and \[x+3y+\text{ }6=0\]whose distance from the origin is 1, are
If \[\vec{a}=(1,-1,2),\] \[\vec{b}=(-2,3,5),\]\[\vec{c}=(2,-2,4),\]and \[\hat{i}\] is the unit vector in the x-direction, then \[(\vec{a}-2\vec{b}+3\vec{c})\hat{i}=\]
For all complex numbers \[{{z}_{1}},{{z}_{2}}\]satisfying \[|{{z}_{1}}|=12\] and \[|{{z}_{2}}-3-4i|=5,\] the minimum value of \[|{{z}_{1}}-{{z}_{2}}|\] is
Two numbers are selected at random from \[1,2,3.....100\]and are multiplied, then the probability correct to two places of decimals that the product thus obtained is divisible by 3, is