A wire has a mass \[0.3\pm 0.003\text{ }g,\] radius \[0.5\pm 0.005\text{ }mm\] and length \[6\pm 0.06\text{ }cm\]. The maximum percentage error in the measurement of its density is
A particle is moving in a straight line. The particle was initially at rest. Acceleration versus time graph is shown in the figure. Acceleration of particle is given by \[a=3\text{ }sin\,\,\pi t\]in\[m/{{s}^{2}}\]. The time instant (s) when the particle comes to rest is
The velocity of a boat in still water is \[\eta \] times less than the velocity of flow of the river \[(\eta >1)\]. The angle with the stream direction at which the boat must move to minimize drifting is
A block of mass 20 kg is lying on a frictionless table. A block of 5 kg is kept on a block of 20 kg. If a variable force F given by \[F=kx\]is applied on the block of mass 20 kg and initially the mass of 20 kg is lying at \[x=1\text{ }m\]and \[\mu =0.2\] and \[k=5\text{ }N/m,\] find the distance after which 5 kg mass starts slipping from the starting point.
An ideal massless spring s can be compressed \[1.0\text{ }m\] by a \[100\text{ }N\]force. It is placed as shown at the bottom of a frictionless inclined plane which makes an angle of \[\theta =30{}^\circ \] with the horizontal. A 10 kg block is released from rest from the top of the incline and is brought to rest momentarily after compressing the spring\[2.0\text{ }m\]. Through what distance does the mass slide before coming to rest along the inclined surface?
A spool is pulled at an angle of \[\theta \]with the horizontal on a rough horizontal surface as shown in the figure. If the spool remains at rest, then the angle \[\theta \] is equal to
Three stars each of mass M and radius R are initially at rest and the distance between centres of any two stars is d and they form an equilateral triangle. They start moving towards the centroid due to mutual force of attraction. What are the velocities of the stars just before the collision? Radius of each star is R.
Water is coming out from a tap of cross-section area A with speed\[{{v}_{0}}\]. Area of cross-section of water stream changes as it moves down. Which of the following graph can best represent cross-section area 'a' of stream with depth h?
A coaxial cylinder made of glass is immersed in a liquid of surface tension 'S? Radius of inner and outer surface of cylinder is \[{{R}_{1}}\] and \[{{R}_{2}},\] respectively. Height till which liquid will rise is (density of liquid is \[\rho \])
The equation of state for a gas is given by \[PV=nRT+\alpha V,\] where n is the number of moles and \[\alpha \] is a positive constant. The initial temperature and pressure of one mole of the gas contained in a cylinder are \[{{T}_{0}}\] and , respectively. \[{{P}_{0}}\]The work done by the gas when its temperature doubles isobarically will be
A particle is executing a simple harmonic motion of period 2 s. When it is at its extreme displacement from its mean position, it receives an additional energy equal to what it its mean position had. Due to this, in its subsequent motion
A)
Its amplitude becomes \[\sqrt{2}\] times of its initial value
The magnetic field inside a solid conducting long wire at distance r from its axis is given as \[B={{B}_{0}}{{r}^{3}}\] where \[{{B}_{0}}\] is constant. Which of the following relations correctly represents current enclosed in the loop of radius r shown in the figure?
A magnet is suspended in such a way that it oscillates in the horizontal plane if it makes 20 oscillations per minute at a place-1 where dip angle is \[30{}^\circ \]and 15 oscillations per minute at a place-2 where dip angle is \[60{}^\circ \]. The ratio of earth's total magnetic field \[\left( \frac{{{B}_{1}}}{{{B}_{2}}} \right)\]at the two places is
White light of wavelength \[400\text{ }nm\]to \[700\text{ }nm\]is incident normally on a Young's double slit experiment apparatus. The distance between slits is \[d=1\text{ }mm\]and distance between plane of the slits and the screen is\[0.8\text{ }m\]. At a point on the screen just in front of one of the slits, the missing wavelengths is (are)
The ratio of de-Broglie wavelengths of molecules of hydrogen and helium which are at temperature \[27{}^\circ C\]and \[127{}^\circ C,\] respectively, is
A radioactive sample consists of two distinct species having equal number of atoms initially. The mean life time of one species is x and that of the other is 5x. The decay products, in both cases are stable. A plot is made of the total number of radioactive nuclei as a function of time. Which of the following figures best represents the form of this plot?
A point charge q is fixed at point A. Two identical balls of mass m having charge \[+q\] and \[-q\] are attached to the ends of a light rod of length la and are free to rotate about their centre of mass which is fixed. The system is released from the situation shown. The angular velocity of the rod when the rod turns through \[90{}^\circ \] is
Three resistors are connected in series across a \[12\text{ }V\]battery. The first resistor has a value \[1\Omega \] second resistor has a voltage drop of \[4V\] and the third has a power dissipation of 12 W. Possible value of circuit current is
In one-dimensional collision between two identical particles A and B. B is stationary and A has momentum \[5\,kg-m/sec\]before impact. During impact 'B' gives the impulse \[3\text{ }kg-m/sec\]to 'A? What is the coefficient of restitution between A and B?
Fundamental frequency of a stretched sonometer wire is \[{{f}_{0}}\]. When its tension is increased by \[96%\] and length decreased by \[35%,\] its fundamental frequency becomes \[{{\eta }_{1}}\,\,{{f}_{0}}\]. When its tension is decreased by \[36%\] and its length is increased by \[30%,\] its fundamental frequency becomes\[{{\eta }_{2}}\,\,{{f}_{0}}\]. Find \[\frac{{{\eta }_{1}}}{{{\eta }_{2}}}\].
A uniform wire of resistance 20 ohm having resistance \[1\,\Omega /m\] is bent in the form of a circle as shown in figure. If the equivalent resistance between P and Q is \[1.8\text{ }ohm,\]then the length of the shorter section is (in m)?
The circuit shown in the figure continues to infinity. The potential difference between points 1 and 2 is \[\frac{V}{2},\] that between points 3 and 4 is \[\frac{V}{4},\] and so on, i.e., the potential difference becomes \[\frac{1}{2}\] after every step of the ladder. Find the ratio\[\frac{{{C}_{1}}}{{{C}_{2}}}\].
The oscillating magnetic field of a plane electromagnetic wave is given as: \[B=4\times {{10}^{-6}}\sin [200\pi x-30\times {{10}^{9}}\pi t]\] tesla. What is the amplitude of electric field (in v/m)?
Specific volume of cylindrical virus particle is \[6.02\times {{10}^{-2}}cc/g.\]whose radius and length \[7\overset{o}{\mathop{A}}\,\]& \[10\overset{o}{\mathop{A}}\,\] respectively. If \[{{N}_{A}}=6.02\times {{10}^{23}},\] find molecular weight of virus
The equivalent conductivity of \[N/10\]solution of acetic acid at \[25{}^\circ C\]is\[14.3\text{ }oh{{m}^{-1}}\text{ }c{{m}^{2}}\text{ }e{{q}^{-1}}\]. What will be the degree of dissociation of acetic acid? \[({{\wedge }^{\infty }}_{C{{H}_{3}}COOH}=390.71\,oh{{m}^{-1}}c{{m}^{2}}e{{q}^{-1}})\]
The radius of hydrogen atom in the ground state is\[0.53\overset{o}{\mathop{A}}\,\]. Calculate the radius of \[L{{i}^{2+}}\] ion (atomic number = 3) in a similar state.
Vapour density of the equilibrium mixture of the reaction \[S{{O}_{2}}C{{l}_{2}}(g)S{{O}_{2}}(g)+C{{l}_{2}}(g)\] is \[50.0\]Calculate the percent dissociation of\[S{{O}_{2}}C{{l}_{2}}\].
If \[\alpha \]and \[\beta \]are solutions of \[{{\sin }^{2}}x+a(sin\,x)+b=0\]as well as that of \[{{\cos }^{2}}x+c(cos\,x)+d=0\], then \[\sin (\alpha +\beta )\] is equal to
If \[{{\overline{x}}_{1}}\]and \[{{\overline{x}}_{2}}\]are means of two distribution such that \[{{\overline{x}}_{1}}<{{\overline{x}}_{2}}\]and \[\overline{x}\]is the mean of the joint distribution then
The solution of the differential equation \[y\frac{dy}{dx}=x\left[ \frac{{{y}^{2}}}{{{x}^{2}}}+\frac{\phi \left( \frac{{{y}^{2}}}{{{x}^{2}}} \right)}{\phi '\left( \frac{{{y}^{2}}}{{{x}^{2}}} \right)} \right]\]is (where c is a constant)
Let \[P({{x}_{1}},{{y}_{1}})\]and \[Q({{x}_{2}},{{y}_{2}})\]are two points such that their abscissa \[{{x}_{1}}\]and \[{{x}_{2}}\]are the roots of the equation \[{{x}^{2}}+2x-3=0\]while the ordinates \[{{y}_{1}}\] and \[{{y}_{2}}\]are the roots of the equation \[{{y}^{2}}+4y-12=0\]. The centre of the circle with PQ as diameter is
If number of numbers greater than 3000, which can be formed by using the digits 0, 1, 2, 3, 4, 5 without repetition, is n then \[\frac{n}{230}\]is equal to
Suppose a, x, y, z and b are in A.P. when x + y + z = 15, and \[a,\alpha ,\beta ,\gamma ,b\] are in H.P. when \[\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=\frac{5}{3}\]. Find a if a > o.