When a large air bubble rises from the bottom of a lake to the surface, its radius doubles. If the atmospheric pressure is equal to that of a column of water of height H, then depth of the lake is
A wire elongates by i mm when a load W is hanged from it. If the wire goes over a pulley and two weights W each are hung at the two ends, the elongation of the wire will be (in mm)
A planet revolves around the sun in an elliptical orbit of eccentricity e. If T is the time period of the planet, then the time spent by the planed between the ends of the minor axis and major axis close to the sun is
The region between two concentric spheres of radii 'a' and 'b' respectively (see figure), has volume charge density\[\rho =\frac{A}{r}\], where A is a constant and r is the distance from the r centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant, is
A series combination of\[{{n}_{1}}\]capacitors, each of value \[{{C}_{1}}\] is charged by a source of potential difference 4 V. When another parallel combination of \[{{n}_{2}}\]capacitors, each of value \[{{C}_{2}}\], is charged by a source of potential difference V, it has the same (total) energy stored in it, as the first combination has. The value of \[{{C}_{2}}\], in terms of \[{{C}_{2}}\], is then
The supply voltage to room is 120V. The resistance of the lead wires is 60. A 60 W bulb is already switched on. What is the decrease of voltage across the bulb, when a 240 W heater is switched on in parallel to the bulb?
A metro train starts from rest and in five seconds achieves a speed 108 km/h. After that it moves with constant velocity and comes to rest after travelling 45 m with uniform retardation. If total distance travelled is 395 m, find total time of travelling.
A particle starts from origin at \[t=0\] with a velocity \[5/m{{s}^{-1}}\] and moves in \[x-y\] plane under the action of a force which produces a constant acceleration of \[3\hat{i}+2\hat{j}m{{s}^{-2}}\]. The y-coordinate of the particle at the instant when its x-coordinate is 84 m, is
A long horizontal rod has a bead which can slide along its length and initially placed at a distance L from one end A of the rod. The rod is set in angular motion about A with constant angular acceleration a. If the coefficient of friction between the rod and the bead is u, and gravity is neglected, then the time after which the bead starts slipping is:
The half-life period of a radio-active element X is same as the mean life time of another radio-active element Y. Initially they have the same number of atoms. Then
A transmitting antenna of height h and the receiving antenna of height 45 m are separated by a distance of 40 km for satisfactory communication in line of sight mode. Then the value of h is (given radius of earth is 6400 km)
A bimetallic strip is formed out of two identical strips, one of copper and other of brass. The coefficients of linear expansion of the two metals are \[{{\alpha }_{C}}\]and \[{{\alpha }_{B}}\]. On heating the temperature of the strip goes up by AT and the strip bends to form an arc of radius of curvature R. Then R is
A)
independent of \[\Delta T\]
doneclear
B)
independent of \[\Delta T\]
doneclear
C)
proportional to \[\left| {{\alpha }_{B}}-{{\alpha }_{C}} \right|\]
doneclear
D)
inversely proportional to \[\left| {{\alpha }_{B}}-{{\alpha }_{C}} \right|\]
Work done by a system under isothermal change from a volume \[{{V}_{1}}\,to\,{{V}_{1}}\] for a gas which obeys Vander Waal?s equation \[(V-\beta n)\left( P+\frac{\alpha {{n}^{2}}}{V} \right)=nRT\]is
A body executing linear simple harmonic motion has a velocity of 3 m/s when its displacement is 4 cm and a velocity of 4 m/s when its displacement is 3 cm. What is the amplitude of oscillation?
A man is standing on a railway platform listening to the whistle of an engine that passes the man at constant speed without stopping. If the engine passes the man at time to. How does the frequency f of the whistle as heard by the man changes with time.
In a coil of resistance, the induced current developed by changing magnetic flux through it, is shown in figure as a function of time. The magnitude of change in flux through the coil in Weber is
Figure shows a circuit that contains three identical resistors with resistance \[R=9.0\Omega \] each, two identical inductors with inductance \[L=2.0mH\] each, and an ideal battery with \[emf\varepsilon =18V\] . The current 'i' through the battery just after the switch closed is
The focal length of a piano convex lens is f and its refractive index is 1.5. It is kept over a plane glass plate with its curved surface touching the glass plate. The gap between the lens and the glass plate is filled by a liquid. As a result, the effective focal length of the combination becomes 2f. Then the refractive index of the liquid is
The distance between the plates of a parallel-plate capacitor is 0.05 m. A field of \[3\times {{10}^{4}}V/m\] is established between the plates. It is disconnected from the battery and an uncharged metal plate of thickness 0.01 m is inserted. What would be the potential difference (in kv) if a plate of dielectric constant \[K=2\] is introduced in place of metal plate?
It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is\[{{p}_{d}}\]; while for its similar collision with carbon nucleus at rest, fractional loss of energy is\[{{P}_{c}}\]. Then ratio of \[{{P}_{d}}\,to\,{{P}_{c}}\] will be:
An observer can see through a pin-hole the top end of a thin rod of height h, placed as shown in the figure. The beaker height is 3h and its radius h. When the beaker is filled with a liquid up to a height 2h, he can see the lower end of the rod. Then the refractive index of the liquid is
Calculate the work done (in joule) when 1 mole of a perfect gas is compressed adiabatically. The initial pressure and volume of the gas are \[{{10}^{5}}N/{{m}^{2}}\]gand 6 litre respectively. The final volume of the gas is 2 litre. Molar specific heat of the gas at constant volume is 3R/2. \[\left[ {{3}^{5/3}}=6.19 \right]\]
Using light of wavelength \[\text{6000}\overset{\text{o}}{\mathop{\text{A}}}\,\]stopping potential is obtained 2.4 volt for photoelectric cell. If light of wavelength \[\text{4000}\overset{\text{o}}{\mathop{\text{A}}}\,\]is used then stopping potential (in volt) would be
\[4\text{ }mol\]of a mixture of Mohr's salt and \[F{{e}_{2}}{{\left( S{{O}_{4}} \right)}_{3}}\]requires 500 ml of 1 M, \[{{K}_{2}}C{{r}_{2}}{{O}_{7}}\] for complete oxidation in acidic medium. The mole % of the Mohr's salt in the mixture is -
Element X crystallizes in a 12 co-ordiantion fee lattice. On applying high temperature it changes to 8 co-ordination bcc lattice. Find the ratio of the density of the crystal lattice before and after applying high temperature -
What is the heat of formation of \[HCl\left( g \right)\] from reaction? \[{{H}_{2}}\left( g \right)+C{{l}_{2}}\left( g \right)\to 2HCl\left( g \right).\Delta H=44\]
Sulphide ion in alkaline solution reacts with solid sulphur to form polysulphide ions having formula,\[S_{2}^{2-},S_{3}^{2-},S_{4}^{2-}\], etc. if \[{{K}_{1}}=12\] for \[S+{{S}^{2-}}\rightleftharpoons S_{2}^{2-}\]and \[{{K}_{2}}=132\] for\[2S+{{S}^{2-}}\rightleftharpoons S_{3}^{2-}\], calculate \[{{K}_{3}}\]for \[S+S_{2}^{2-}\rightleftharpoons S_{3}^{2-}\] Give your answer by dividing 5.5
A buffer with pH 9 is to be prepared by mixing \[N{{H}_{4}}Cl\] and \[N{{H}_{4}}OH.\] Calculate the number of mole of \[N{{H}_{4}}Cl\]that should be added to one litre of 1 M \[N{{H}_{4}}OH.\left( {{K}_{b}}=1.8\times {{10}^{-5}} \right)\]
If \[y=f(x)\] is discontinuous at exactly three points \[{{\alpha }_{1}},{{\alpha }_{2}}\] and \[{{\alpha }_{3}}\] where \[{{\alpha }_{i}}<0\,\forall \,i=1,2,3,\]then function \[y=f\left( -|3x-1| \right)\]is discontinuous at \[x={{\beta }_{1}},{{\beta }_{2}},......{{\beta }_{n}}\]The value of \[\sum\limits_{i=1}^{n}{{{\beta }_{i}}}\] is
If \[{{b}^{2}}-4ac\le 0\](where \[a\ne 0\] and a, b, c, x, \[y\in R\]) satisfies the system \[a{{x}^{2}}+x(b-3)+c+y=0\] and \[a{{y}^{2}}+y(b-1)+c+3x=0,\] then value of is
Normals at \[({{x}_{1}},{{y}_{1}}),\,({{x}_{2}},{{y}_{2}})\] and \[({{x}_{3}},{{y}_{3}})\] to the parabola \[{{y}^{2}}=4x\]are concurrent at point P. If \[{{y}_{1}}{{y}_{2}}+{{y}_{2}}{{y}_{3}}+{{y}_{3}}{{y}_{1}}={{x}_{1}}{{x}_{2}}{{x}_{3}},\]then locus of point P is part of a parabola, length of whose latus rectum is
\[\vec{u},\,\vec{v}\] and \[\vec{w}\] are three vectors of magnitude \[\sqrt{3},1\] and 2, respectively, such that \[\vec{u}\times (\vec{u}\times \vec{w})+3\vec{v}=0.\]. If \[\theta \] is the angle between \[\vec{u}\] and \[\vec{w}\], then \[{{\cos }^{2}}\theta \] is equal to
Let \[Y=SX\]and \[Z=tX\] such that\[\left| \begin{matrix} X & Y & Z \\ {{X}_{1}} & {{Y}_{1}} & {{Z}_{1}} \\ {{X}_{2}} & {{Y}_{2}} & {{Z}_{2}} \\ \end{matrix} \right|+\left| \begin{matrix} {{S}_{1}} & {{t}_{1}} \\ {{S}_{2}} & {{t}_{2}} \\ \end{matrix} \right|={{X}^{n}},\]all the variables being differentiable functions of x and lower suffices denote the derivative with respect to x. Then n =
Let \[{{A}_{1}},{{A}_{2}},{{A}_{3}}\] me three arithmetic means between two positive numbers a and b (where \[a>b\]). If the equation \[{{A}_{1}}{{x}^{2}}+2{{A}_{2}}x+{{A}_{3}}=0\] has imaginary roots, then the range of \[\frac{a}{b}\]is
Let \[A=[{{a}_{ij}}]\] be a matrix of order \[3\times 3\] and \[B=[{{b}_{ij}}]\] be another matrix of order \[3\times 3\] such that \[{{b}_{ij}}\] is the sum of the elements of \[{{i}^{th}}\] row of A except\[{{a}_{ij}}\]. If det. \[(A)=3,\] then the value of det. \[(B)\] is equal to
In \[\Delta ABC,\] circumradius is 3 and inradius is 1. Let I be the in centre of the triangle. If D, E and F are the feet of the perpendiculars from I to BC, CA and AB, respectively, then the value of \[\frac{IA.IB.IC}{ID.IE.IF}=\]
Water is being emptied from a spherical jar of radius 10 cm. If the depth of the water in the tank is 4 cm and it is decreasing at the rate of 2 cm/sec, then the radius of the top surface of water is decreasing at the rate (cm/sec) of
From a point on the line \[x-y+2=0\] tangents are drawn to the hyperbola \[\frac{{{x}^{2}}}{6}-\frac{{{y}^{2}}}{2}=1\] such that the chord of contact passes through a fixed point (h, k). Then \[\frac{h}{k}\] is equal to ________.
Let \[g:R\to R\] be given by \[g(x)={{e}^{2x}}+3x+\sin x+1.\].If \[{{g}^{-1}}\] is the inverse function of g, then the value of \[\frac{1}{({{g}^{-1}})'(2)}\] is ________.
If area of square circumscribing the ellipse E is 10 square units and maximum distance of a normal from the centre of ellipse is 1 unit, then eccentricity of the ellipse is ________.
A straight line L intersects perpendicularly both the lines \[\frac{x+2}{2}=\frac{y+6}{3}=\frac{z-34}{-10}\] and \[\frac{x+6}{4}=\frac{y-7}{-3}=\frac{z-7}{-2}\] . The perpendicular distance of origin from L is ________.