Consider a collection of large number of particles each with speed v. The direction of velocity is randomly distributed in the collection. The magnitude of relative velocity between a pair of particles averaged over all the pairs is
Two particles start moving from the same point along the same straight line. The first moves with a constant velocity v and the second with constant acceleration a. During the time that elapses before the second catches the first, the greatest distance between the particle is
A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration \[{{a}_{c}}\] is varying with time t as \[{{a}_{c}}={{k}^{2}}r{{t}^{2}},\] where A; is a constant. The power delivered to the particle by the force acting on it is
Two particles of equal mass m are projected from the ground with speed \[{{v}_{1}}\] and \[{{v}_{2}}\] at angles \[{{\theta }_{1}}\] and \[{{\theta }_{2}}\] as shown in figure. The centre of mass of the two particles
A)
Will move in a parabolic path for any values of \[{{v}_{1}},{{v}_{2}},{{\theta }_{1}}\] and \[{{\theta }_{2}}\]
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B)
Can move in a vertical line
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C)
Can move in a horizontal line
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D)
Will move in a straight line for any values of \[{{v}_{1}},{{v}_{2}},{{\theta }_{1}}\] and \[{{\theta }_{2}}\].
A solid sphere of radius R made of a material of Bulk modulus K is surrounded by a liquid in a cylindrical container. A light piston of area. A floats on the surface of the liquid. When a mass m is placed on the piston to compress the liquid, the fractional change in the radius of the sphere will be
Three identical rods A, B and C are placed end to end. A temperature difference is maintained between the free ends of A and C. The thermal conductivity of B is THRICE that of C and HALF that of A. The effective thermal conductivity of the system will be: (\[{{K}_{A}}\] is the thermal conductivity of rod A).
The graphs in figure show how the displacement x, velocity v and the acceleration a of a body vary with time t when it is oscillating with simple harmonic motion. What is the value of T?
The grid (each square of\[1\text{ }m\times 1\text{ }m\]), represents a region in space containing a uniform electric field. If potentials at points O, A, B, C, D, E, F, G, H are respectively \[0,-1,-2,1,2,0,-1,1\] and O volts, find the electric field intensity.
A particle of charge q, mass m starts moving from the origin under the action of an electric field \[\vec{E}={{E}_{0}}\hat{i}\]and magnetic field\[\vec{B}={{B}_{0}}\hat{k}\]. Its velocity at \[(x,0,0)\] is \[6\hat{i}+8\hat{j}\]. The value of x is
A light is incident on face AB of an equilateral glass prism ABC. After refraction at AB, the ray is incident on face BC at an angle slightly greater than critical angle so that it gets reflected from face BC and finally emerges out from face AC. Net deviation angle of the ray is \[112{}^\circ \]anticlockwise. The angle of incidence 't? has value
Measure of two quantities along with the precision of respective measuring instrument is \[A=2.5\,m{{s}^{-1}}\pm 0.5m{{s}^{-1}}\] \[B=0.10s\pm 0.01s\] The value of AB will be
The total energy of electron in the ground state of hydrogen atom is\[-13.6\text{ }eV\]. The kinetic energy of an electron in the first excited state is
The half-life of a radioactive nucleus is 50 days. The time interval \[({{t}_{2}}-{{t}_{1}})\]between the time \[{{t}_{2}}\] when it has decayed two-third of nucleus and the time \[{{t}_{1}}\] when it had decayed one-third of nucleus is
A wheel is rolling straight on ground without slipping. If the centre of mass of the wheel has speed v, the instantaneous velocity of a point P on the rim, defined by angle \[\theta ,\] relative to the ground will be
An ideal gas heat engine, operates in a Camot's cycle between \[227{}^\circ C\]and\[127{}^\circ C\]. It absorbs \[6\times {{10}^{4}}J\] at high temperature. The amount of heat converted into work is\[N\times {{10}^{4}}J\]. Find the value of N?
In the given figure, string, spring and pulleys are massless. Block A, performing SHM of amplitude 1 metre and time period \[\pi /2\] sec. If block B remains at rest, find the minimum value of co-efficient of friction between block B and surface. \[(g=10\,m/{{s}^{2}})\]
The rectangular loop with a sliding connector of length 10 cm is situated in uniform magnetic field perpendicular to the plane of loop. The magnetic induction is \[0.1\] Tesla and resistance of connector (R) is 1 ohm. The sides AB and CD have resistances 2 ohm and 3 ohm respectively. Find the current (in A) in the connector during motion with constant velocity one metre/sec.
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 and the current in the circuit is \[\pi /3\]. If instead, C is removed from the circuit, the phase difference is again\[\pi /3\]. What is the power factor of the circuit?
A man of mass m is standing on one end of a plank of mass 1m floating on a river. The other end just touches the bank of the river as shown in figure. With what minimum speed (in m/s) w.r.t. the plank should the man jump to get out of the river. All surfaces are smooth and plank is always in level with the bank. Given that the length of the plank is 15 m.
Which method of purification is represented by the following equations \[\underset{\left( Impure \right)}{\mathop{Ti+2{{I}_{2}}}}\,\xrightarrow{523K}Ti{{I}_{4}}\xrightarrow{1700K}\underset{\left( pure \right)}{\mathop{Ti}}\,+2{{I}_{2}}\]
For the redox reaction, \[MnO_{4}^{-}>+{{C}_{2}}O_{4}^{-2}+{{H}^{+}}\to M{{n}^{+2}}+C{{O}_{2}}+{{H}_{2}}O\] the correct coefficient of reactants \[MnO_{4}^{-},{{C}_{2}}O_{4}^{-2},\]\[{{H}^{+}}\] for the balanced reaction are respectively:
The density of gas A is twice that to B. At the same temperature the molecular weight of gas B is twice that of A. The ratio of pressure of gas A and B will be:
\[{{I}_{2}}\left( s \right)/{{I}^{-}}\left( 0.1\text{ }M \right)\] half-cell is connected to a \[{{H}^{+}}(aq)/{{H}_{2}}(1\,bar)/Pt\] half cell and e.m.f. is found to be \[0.7714\text{ }V.\text{ }If\,E_{{{I}_{2}}/{{I}^{-}}}^{{}^\circ }=0.535\text{ }V,\] find the pH of \[{{H}^{+}}/{{H}_{2}}\] half-cell.
I. The ligand thiosulphato, \[{{S}_{2}}{{O}_{3}}^{2-}\] can give rise to linkage isomers.
II. In metallic carbonyls the ligand CO molecule acts both as donor and acceptor.
III. The complex \[\left[ Pt\left( Py \right)\left( N{{H}_{3}} \right)\left( N{{O}_{2}} \right)ClBrI \right]\] exists in eight different geometrical isomeric forms
IV. The complex ferricyanide ion does not follow effective atomic number (BAN) rule.
A complex of molecular formula \[CrC{{l}_{3}}.6{{H}_{2}}O\] has green colour. 0.1 mole of the complex when treated with excess of \[AgN{{O}_{3}}\] gave 28.7 g of white precipitate of \[AgCl\] (Molar mass = 143.5 g). The formula of the complex would be
A solid is formed and it has three types of atoms X, Y and Z. X forms a fee lattice with Y atoms occupying all tetrahedral voids and Z atoms occupying half of octahedral voids. The formula of solid is-
The heat of formation of \[N{{H}_{3}}\left( g \right)\] is \[-46kJ\text{ }mo{{l}^{-1}}.\] The \[\Delta H\] (in kJ) of the reaction is: \[2N{{H}_{3}}\left( g \right)\xrightarrow{{}}{{N}_{2}}\left( g \right)+3{{H}_{2}}\left( g \right)\]
\[N{{H}_{4}}HS(s)\rightleftharpoons N{{H}_{3}}(g)+{{H}_{2}}S(g)\] The equilibrium pressure at \[25{}^\circ C\] is 0.660 atm. What is \[{{K}_{p}}\] for the reaction?
Calculate the wavelength in \[\overset{{}^\circ }{\mathop{A}}\,\] of the photon that is emitted when an electron in Bohr orbit with n = 2 returns to orbit with n = 1 in H atom. The ionisation potential of the ground state of H-atom is \[2.17\times {{10}^{-11}}erg.\]
The bond distance between \[C-Cl\] in \[CCl\] is\[1.76\overset{{}^\circ }{\mathop{A}}\,\]. If atomic radius of C is\[0.77\overset{{}^\circ }{\mathop{A}}\,\], determine the atomic radius of Cl.
An atom of element 'X' is 1.02 times heavier than that of an atom of 'Y'. An atom of 'Y' is 0.1809 times heavier than that of an atom of oxygen. What is the atomic weight of 'X'?
What mass of \[{{N}_{2}}{{H}_{4}}\]can be oxidized to \[{{N}_{2}}\] by 24.0 gm of \[{{K}_{2}}Cr{{O}_{4}},\] which is reduced to\[Cr\left( OH \right)_{4}^{-}\]?
Calculate the number of moles of gas present in the container of volume 10 lit at 300 K. If the manometer containing glycerin shows 5m difference in level as shown in diagram. Given: \[{{d}_{giycerin}}=2.72\text{ }gm/ml,\text{ }{{d}_{mercury}}=13.6\text{ }gm/ml.\]
If the normal at the point to the ellipse \[\frac{{{x}^{2}}}{14}+\frac{{{y}^{2}}}{5}=1\] intersects it again at the point \[Q(2\theta )\], then \[\cos \,\theta \] is equal to
If a, b, c, d and p are distinct real numbers such that \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}){{p}^{2}}-2p(ab+bc+cd)+({{b}^{2}}+{{c}^{2}}+{{d}^{2}})\le 0\] then a, b, c, d are in-
In the function \[f(x)=\frac{2x-{{\sin }^{-1}}x}{2x+{{\tan }^{-1}}x},\,\,(x\ne 0)\], is continuous at each point of its domain, then the value of f(0) is
In a certain town \[25\,%\] families own a phone and \[15\,%\] own a car, \[65\,%\] families own neither a phone nor a car. 2000 families own both a car and a phone. Consider the following statements in this regard:
If the functions f(x) and g(x) are defined on \[R\to R\] such that \[f(x)=\left\{ \begin{matrix} 0,\,\,\,x\in rational \\ x,\,\,\,\,x\in irrational\, \\ \end{matrix} \right.\]\[g(x)=\left\{ \begin{matrix} 0,\,\,\,x\in \,\,\,irrational \\ x,\,\,\,\,x\in \,\,rational\, \\ \end{matrix} \right.\] then
In a \[\Delta \,ABC,\,\,\angle B=\pi /3\,\,and\,\,\angle \,C\,=\,\pi /4\]. If D divides BC internally in ratio \[1:3\], then \[\frac{\sin \,\angle BAD}{\sin \angle CAD}\]
Let A be vector parallel to line of intersection of planes \[{{P}_{1}}\,and\,{{P}_{2}}\] Plane \[{{P}_{1}}\] is parallel to the vectors \[2\hat{j}+3\hat{k}\] and \[4\,\hat{j}-3\,\hat{k}\] and that \[{{P}_{2}}\] is parallel to \[\hat{j}-\hat{k}\,\,and\,\,3\hat{i}+3\hat{j}\], then the angle between vector A and a given vector \[2\hat{i}\,+ \hat{j} -2\hat{k}\] is
If \[\operatorname{f}(x) = a\left| sin x \right|\,+\,\,b{{e}^{\left| x \right|}}\,+c{{\left| x \right|}^{3}}\] and if is differentiable at \[\operatorname{x}= 0\], then
\[\vec{a},\,\,\vec{b}\,\,and\,\,\vec{c}\] are three vectors with magnitude\[\left| {\vec{a}} \right|=4\], \[\left| {\vec{b}} \right|=4,\,\,\left| {\vec{c}} \right|=2\] and such that \[\vec{a}\] is perpendicular to \[(\vec{b}\,+\vec{c}),\vec{b}\] is perpendicular to \[(\vec{c}\,\,+\,\,\vec{a})\] and \[\vec{c}\] is perpendicular to \[(\vec{a}\,\,+\,\,\vec{b})\]. It follows that \[(\vec{a}\,\,+\,\,\vec{b}+\vec{c})\] is equal to:
If r, .s, t are prime numbers and p, q are the positive integers such that LCM of \[{{r}^{2}}\,{{s}^{4}}\,{{t}^{2}}\], then the number of ordered pairs (p, q) is
Let \[\omega \ne 1\] be a cube root of unity and S be the set of all non a b singular matrices of the form \[\left[ \begin{matrix} 1 & a & b \\ \omega & 1 & c \\ {{\omega }^{2}} & \omega & 1 \\ \end{matrix} \right]\], where each a, b and c is either \[\omega \,\,or\,\,{{\omega }^{2}}\]. Then the number of distinct matrices in the set S is
Let f be a positive function. Let \[{{I}_{1}}=\int\limits_{1\,-\,k}^{k}{x\,f\left[ x(1-x) \right]dx}\] \[{{I}_{2}}=\,\int\limits_{1\,-\,k}^{k}{\,f\left[ x(1-x) \right]dx}\], where \[2k\,-1 > 0\]. Then \[\frac{{{I}_{1}}}{{{I}_{2}}}\] is
Tangents are drawn from the point (4, 3) to the circle \[{{\operatorname{x}}^{2}}+{{y}^{2}}= 9\]. The area of the triangle formed by them and the line joining their points of contact is
Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is