The centre of mass of three particles of masses 1 kg, 2 kg and 3 kg is at \[(3,3,3)\] with reference to a fixed coordinate system. Where should a fourth particle of mass 4 kg be placed so that the centre of mass of the system of all particles shifts to a point \[(1,1,1)\] ?
One end of a uniform wire of length L and of weight W is attached rigidly to a point in the roof and a weight \[{{W}_{1}}\] is suspended from its lower end. If S is the area of cross- section of the wire, the stress in the wire at a height \[3L/4\] from its lower end is
A block C of mass m is moving with velocity \[{{\text{v}}_{0}}\] and collides elastically with block A of mass m and connected to another block B of mass 2m through spring constant k. What is k if \[{{x}_{0}}\]is compression of spring when velocity of A and B is same?
Let \[P(r)=\frac{Q}{\pi {{R}^{4}}}r\] be the charge density distribution for a solid sphere of radius R and total charge Q. For a point 'p' inside the sphere at distance \[{{r}_{1}}\] from the centre of the sphere, the magnitude of electric field is
The potential at a point x (measured in\[\mu \,m\]) due to some charges situated on the x-axis is given by \[V(x)=20/({{x}^{2}}-4)\]Volt The electric field E' at \[x=4\,\mu \,m\] is given by
A)
\[(10/9)\,Volt/\mu \,m\] Volt/ |li m and in the \[+\text{v}e\,x\]x direction
doneclear
B)
\[\left( 5/3 \right)\text{ }Volt/\mu \,m\]and in the \[-\text{v}ex\]direction
doneclear
C)
\[\left( 5/3 \right)\text{ }Volt/\mu \,m\]and in the \[+\text{v}e\text{ }x\]direction
doneclear
D)
\[\left( 10/9 \right)\text{ }Volt/\mu \text{ }m\]and in the \[-\text{v}e\text{ }x\]direction
A material ?B' has twice the specific resistance of \[A'\]. A circular wire made of ?B? has twice the diameter of a wire made of ?A?. Then for the two wires to have the same resistance, the ratio \[{{\ell }_{B}}/{{\ell }_{A}}\] of their respective lengths must be
Figure shows an arrangement of a rod of length \[\ell \]and mass M and a bead of mass m attached to a weightless string passing over a frictionless pulley. At \[t=0,\]the bead is in level with the upper end of the rod. The bead slides down the string with considerable friction, and is opposite to the other end of the rod after t second. Assuming friction between the bead and the string to be constant all through, the frictional force is
From a tower of height H, a particle is thrown vertically upwards with a speed u. The time taken by the particle, to hit the ground, is n times that taken by it to reach the highest point of its path. The relation between H, u and n is:
The activity of a freshly prepared radioactive sample is \[{{10}^{10}}\] disintegrations per second, whose mean life is \[{{10}^{9}}s\]. The mass of an atom of this radioisotope is \[{{10}^{-25}}kg\]. The mass (in mg) of the radioactive sample is
A stationary object at \[4{}^\circ C\]and weighing \[3.5\text{ }kg\]falls from a height of \[2000\text{ }m\]on a snow mountain at\[0{}^\circ C\]. If the temperature of the object just before hitting the snow is \[0{}^\circ C\] and the object comes to rest immediately \[(g=10m/{{s}^{2}})\] and (latent heat of ice \[=3.5\times {{10}^{5}}\] Joule) is, then the mass of ice that will melt is
A source of frequency f is stationary and an observer starts moving towards it at \[t=0\]with constant small acceleration. Then the variation of observed frequency f registered by the observer with time is best represented by
A body executes simple harmonic motion. The potential energy (RE.), the kinetic energy (K.E.) and total energy (T.E.) are measured as a function of displacements. Which of the following statements is true
For a gas at a temperature T the root-mean-square velocity \[{{\text{v}}_{rms}},\] the most probable speed \[{{\text{v}}_{mp}},\] and the average speed \[{{\text{v}}_{a\text{v}}}\] obey the relationship
A series \[R-C\]combination is connected to an AC voltage of angular frequency \[\omega =500\] radian/s. If the impedance of the \[R-C\]circuit is \[R\sqrt{1.25}\] the time constant (in millisecond) of the circuit is
Two polaroids are placed in the path of unpolarized beam of intensity \[{{I}_{0}}\]such that no light is emitted from the second Polaroid. If a third polarioid whose polarization axis makes an angle \[\theta \] with the polarization axis of first polaroids, is placed between these polaroids then the intensity of light emerging from the last polaroid will be
A \[30cm\]long bar magnet is placed in the magnetic meridian with its north pole pointing south. The neutral point is obtained at a distance of \[40cm\] from the center of the magnet. Find the magnetic dipole moment (in \[A{{m}^{2}}\]) The horizontal component of earth's magnetic field is \[0.34\]gauss.
A soap bubble in vacuum has a radius of \[3\text{ }cm\]and another soap bubble in vacuum has a radius of\[4\text{ }cm\]. If the two bubbles coalesce under isothermal condition, then the radius (in cm) of the new bubble is
Carbon monoxide is carried around a closed cycle abc in which bc is an isothermal process as shown in the figure. The gas absorbs \[7000\text{ }J\]of heat as its temperature increases from \[300\text{ }K\]to \[1000\text{ }K\]in going from a to b. The quantity of heat (in joule) rejected by the gas during the process \[ca\]is
Alight of wavelength \[5000\text{ }A{}^\circ \] and intensity \[4.68\text{ }mW/c{{m}^{2}}\] is incident on a light sensitive surface. If only\[~5%\] of incident photons produce photoelectrons. Find the number of electrons emitted per unit area per unit time.
A coil of resistance\[50\Omega \]is connected across a \[5.0\text{ }V\]battery, \[0.1\,s\] after the battery is connected, the current in the coil is\[60\text{ }mA\]. Calculate the inductance of (in henry) the coil.
Aniline is treated with bromine water to give an organic compound 'X' which when treated with \[NaN{{O}_{2}}\]and \[HCl\]at \[0{}^\circ C\] gives a water soluble compound 'Y?. Compound ?Y? on treatment with \[C{{u}_{2}}C{{l}_{2}}\]and HCl gives compound 'Z'. Compound 'Z' is
Consider the following equilibrium in a closed container, \[{{N}_{2}}{{O}_{4(g)}}2N{{O}_{2(g)}}\]. At a fixed temperature, the volume of the reaction mixture is halved. For this change, which of the following statements holds true regarding the equilibrium constant\[({{K}_{p}})\]and degree of dissociation \[(\alpha )\]?
A)
Neither\[({{K}_{p}})\] nor \[\alpha \] changes.
doneclear
B)
Both \[{{K}_{p}}\]and\[\alpha \] change.
doneclear
C)
\[{{K}_{p}}\]changes but\[\alpha \]does not.
doneclear
D)
\[{{K}_{p}}\]does not change but\[\alpha \]changes
A 1.0 M solution with respect to each of metal halides \[A{{X}_{3}},B{{X}_{2}},C{{X}_{3}}\]and \[D{{X}_{2}}\]is electrolysed using platinum electrodes. If \[E_{{{A}^{3+}}/A}^{o}=1.50V,E_{{{B}^{2+}}/B}^{o}=0.34V,\] \[E_{{{C}^{3+}}/C}^{o}=-0.74V,E_{{{D}^{2+}}/D}^{o}=-2.37V,\]the correct sequence in which the various metals are deposited at the cathode, is
In kinetic study of a chemical reaction, slopes are drawn at different times in the plot of concentration of reactants versus time. The magnitude of slopes with increase of time
Phenol is converted into bake lite by heating it with formaldehyde in the presence of an alkali or an acid. Which statement is true regarding this reaction?
A)
The electrophile in both cases is \[C{{H}_{2}}=O\].
doneclear
B)
The electrophile in both cases is \[C{{H}_{2}}=\overset{+}{\mathop{O}}\,H\].
doneclear
C)
The electrophile is \[C{{H}_{2}}=O\]in the presence of an alkali and \[C{{H}_{2}}=\overset{+}{\mathop{O}}\,H\]in the presence of an acid.
An organic compound 'X' on treatment with pyridiniumchlorochromate in dichloromethane gives compound T'. Compound 'V reacts with \[{{I}_{2}}\] and alkali to form triiodomethane. The compound 'X' is
The standard reduction potential values of three metallic cations, X, Y and Z are 0.52, - 3.03 and -1.18 V respectively. The order of reducing power of the corresponding metals is
Among the following, the total number of alkyl halides that would react by \[{{S}_{N}}1\]mechanism is__. \[C{{H}_{3}}Br,C{{H}_{3}}C{{H}_{2}}Br,C{{H}_{3}}C{{H}_{2}}C{{H}_{2}}I,\] \[{{(C{{H}_{3}})}_{3}}CBr,C{{H}_{3}}CH=C{{H}_{2}},{{C}_{6}}{{H}_{5}}C{{H}_{2}}Br,\] \[{{(C{{H}_{3}})}_{3}}CC{{H}_{3}}Br,{{C}_{6}}{{H}_{5}}-CHBr-C{{H}_{3}}\], \[C{{H}_{3}}CH=CHC{{H}_{2}}Cl\]
2 moles of a perfect gas at \[27{}^\circ C\]is compressed reversibly and isothermally from a pressure of \[1.01\times {{10}^{5}}\text{N}{{\text{m}}^{-2}}\]to\[5.05\times {{10}^{6}}\text{N}{{\text{m}}^{-2}}\]. The free energy change is \[x\times {{10}^{4}}\] joule. The value of x is
In a compound C, H and N are present in 9 : 1 : 3.5 by weight. If molecular weight of the compound is 108, the number of N atoms present in the molecular formula will be _______.
Let \[y=f(x)\] be an invertible function such that x-intercept of the tangent at any point \[P(x,y)\] on \[y=f(x)\] is equal to the square of abscissa of the point of tangency. If \[f(2)=1,\] then\[{{f}^{-1}}\left( \frac{5}{8} \right)\]- equals
When a missile is fired from a ship, the probability that it is intercepted is 1/3. The probability that the missile hits the target, given that it is not intercepted is 3/4. If three missiles are fired independently from the ship, the probability that all three hits their targets .is
The eccentricity of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is reciprocal to that of the ellipse\[{{x}^{2}}+4{{y}^{2}}=4.\]. If the hyperbola passes through a focus of the ellipse, then
A)
The equation of the hyperbola is \[\frac{{{x}^{2}}}{3}-\frac{{{y}^{2}}}{2}=1\]
doneclear
B)
A focus of the hyperbola is \[(\sqrt{3},0)\]
doneclear
C)
The eccentricity of the hyperbola is \[\sqrt{\frac{5}{3}}\]
doneclear
D)
The equation of the hyperbola is \[{{x}^{2}}-3{{y}^{2}}=3\]
If \[f(x)\] is a differentiable function such that \[f'(1)=4\] and \[f'(4)=\frac{1}{2},\] then value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f({{x}^{2}}+x+1)-f(1)}{f({{x}^{4}}-{{x}^{2}}+2x+4)-f(4)}\] is
Let a and b be two arbitrary real numbers. The smallest natural number b for which the equation \[{{x}^{2}}+2(a+b)x+(a-b+8)=0\] has unequal real roots \[\forall \,\,a\in R\] is
Let \[A=[{{a}_{ij}}]\]be a \[3\times 3\] invertible matrix. If determinant value of matrix A is 3, then the value of det. \[({{(adj\,.{{A}^{T}})}^{T}})+\det .({{(adj.{{A}^{-1}})}^{-1}})\]is
Let \[A=\{1, 2, 3, 4\}\]. The number of different ordered pairs \[(B,C)\] that can be formed such that \[B\subseteq A,\] \[C\subseteq A\] and \[B\cap C\] is empty, is
Let \[f(x)=x({{x}^{2}}+mx+n)+2,\] for all \[x\in R\] and m, \[n\in R\] . If Rollers Theorem holds for \[f(x)\] at \[x=4/3\]in \[x\in [1,\,\,2],\] then \[(m+n)\] equals
If the area bounded by the parabolas \[{{y}^{2}}=4\alpha (x+\alpha )\] and \[{{y}^{2}}=-4\alpha (x-\alpha ),\] where \[\alpha >0,\] is 48 square units, then a is equal to
\[\vec{a},\,\vec{b},\,\vec{c}\] are unit vectors such that \[\left| \vec{a}+\vec{b}+3\vec{c} \right|=\sqrt{14}\] . Angle between \[\vec{a}\] and \[\vec{b}\] is\[\alpha ,\] angle between \[\vec{b}\] and \[\vec{c}\] is \[\beta ,\]and angle between \[\vec{a}\] and \[\vec{c}\] is \[\gamma \]. If \[\gamma \in \left[ \frac{\pi }{6},\frac{2\pi }{3} \right],\]then maximum value of \[\cos \alpha +3\cos \beta \] is ____.