A solid cylinder rolls down an inclined plane of height 3 m and reaches the bottom of plane with angular velocity of \[2\sqrt{2}\,rad.{{s}^{-1}}\]. The radius of cylinder must be \[\left( Take g = 10 m{{s}^{-}}^{2} \right)\]
In the figure shown, a particle of mass m is released from the position A on a smooth track. When the particle reaches at B, then normal reaction on it by the track is
The electric field in a certain region is given by \[\overrightarrow{\operatorname{E}}=\left( 5\hat{i}\,-3\hat{j} \right)\,k\,V/m\]. The potential difference \[{{V}_{B}}-{{V}_{A}}\] between points A and B, having coordinates (4,0, 3)m and (10,3, 0)m respectively, is equal to
Two long parallel wires P and Q are held perpendicular to the plane of the paper at a separation of 5 m. If P and Q carry currents of 2.5 A and 5 A respectively in the same direction, then the magnetic field at a point midway between P and Q is
Two seconds after projection a projectile is travelling in a direction inclined at \[30{}^\circ \] to the horizontal. After one more second, it is travelling horizontally. The magnitude and direction of its initial velocity are -
A 40 kg slab rests on a frictionless floor as shown in the figure. A 10 kg block rests on the top of the slab. The static coefficient of friction between the block and slab is 0.60 while the coefficient of kinetic friction is 0.40. The 10 kg block is acted upon by a horizontal force 100 N. If \[\operatorname{g} = 9.8 m/{{s}^{2}}\], the resulting acceleration of the slab will be
Two cars P and Q start from a point at the same time in a straight line and their positions are represented by \[{{\operatorname{x}}_{p}}(t)=at\,+b{{t}^{2}}\] and \[{{\operatorname{x}}_{Q}}(t)=ft\,-{{t}^{2}}\]. At what time do the cars have the same velocity
Ultraviolet light of wave length 300 nm and intensity \[1.0 watt/{{m}^{2}}\] falls on the surface of a photosensitive material. If \[1%\] of the incident photons produce photoelectrons, then find the number of photoelectrons emitted from an area of \[1.0\text{ }c{{m}^{2}}\] of the surface.
The concentration of hole - electron pairs in pure silicon at T=300Kis7x 1015 per cubic meter. Antimony is doped into silicon in a proportion of 1 atom in 107 Si atoms. Assuming that half of the impurity atoms contribute electron in the conduction band, calculate the factor by which the number of charge carriers increases due to doping. The number of silicon atoms per cubic meter is 5 x 1028
Shown below are the black body radiation curves at temperatures\[{{\operatorname{T}}_{1}}\,\,and\,\,{{T}_{2}}\] \[({{T}_{2}}>{{T}_{1}})\]. Which one of the following plots is correct?
A forced oscillator is acted upon by a force \[F={{F}_{0}}\,\sin \,\omega t\]. The amplitude of oscillation is given by \[\frac{55}{\sqrt{2{{\omega }^{2}}-36\omega +9}}\].
Three closed vessels A, B and Care at the same temperature T and contain gases which obey the Maxwellian distribution of velocities. Vessel A contains only \[{{O}_{2}}\], B only \[{{N}_{2}}\] and C a mixture of equal quantities of \[{{O}_{2}}\,and\,{{N}_{2}}\]. If the average speed of the \[{{O}_{2}}\] molecules in vessel A is \[{{V}_{1}}\], that of the \[{{N}_{2}}\] molecules in vessel B is \[{{V}_{2}}\], the average speed of the \[{{O}_{2}}\] molecules in vessel C is
In the figure shown a source of sound of frequency 510 Hz moves with constant velocity \[{{v}_{s}} = 20 m/s\] in the direction shown. The wind is blowing at a constant velocity \[{{\operatorname{v}}_{w}}=20m/s\] towards an observer who is at rest at point B. Corresponding to the sound emitted by the source at initial position A, the frequency detected by the observer is equal to (speed of sound relative to air = 330 m/s)
In fig, CODF is a semicircular loop of a conducting wire of resistance R and radius r. It is placed in a uniform magnetic field B, which is directed into the page (perpendicular to the plane of the loop). The loop is rotated with a constant angular speed \[\omega \] about an axis passing through the centre O, and perpendicular to the page. Then the induced current in the wire loop is
If \[\operatorname{E} = 100 sin \left( 100\,t \right) volt\] and \[I=100sin\,\left( 100\,t\,+\frac{\pi }{3} \right)mA\] are the instantaneous values of voltage and current, then the r.m.s. values of voltage and current are respectively
A plane electromagnetic wave is incident on a plane surface of area A, normally and is perfectly reflected. If energy E strikes the surface in time t then average pressure exerted on the surface is \[\left( c = speed of light \right)\]
An inclined plane making an angle of \[30{}^\circ \] with the horizontal is placed in a uniform electric field of intensity 100 V/m. A particle of mass 1 kg and charge 0.01 C is allowed to slide down from rest on the plane from a height of 1 m. If the coefficient of friction is 0.2, then find the time taken (in second) by the particle to reach the bottom.
A satellite is to be placed in equatorial geostationary orbit around earth for communication. The height (in metre) of such a satellite is \[[{{M}_{E}}= 6 \times 1{{0}^{24}}\,kg,\,\,{{R}_{E}}=6400\,km,\,\,T=24\,h\]\[\operatorname{G}=6.67\times {{10}^{-11}}\,N{{m}^{2}}\,k{{g}^{-\,2}}]\]
A simple electric motor has an armature resistance of \[1\,\,\Omega \] and runs from a dc source of 12 volt. When running unloaded it draws a current of 2 amp. When a certain load is connected, its speed becomes one-half of its unloaded value. What is the new value of current drawn (in ampere)?
A gas can be taken from A to B via two different processes ACB and ADB. When path ACB is used 60 J of heat flows into the system and 30J of work is done by the system. If path ADB is used work done by the system is 10 J. The heat flow (in joule) into the system in path ADB is:
If 200 MeV energy is released per fission of U235 nuclei. Find the mass of U235 consumed (in mg) per day in a reactor of power 1MW assuming its efficiency is \[80\,%\].
In the volumetric estimation of Fe (II) with \[C{{r}_{2}}O_{7}^{2-}\]in acidic medium,\[{{K}_{3}}[Fe{{(CN)}_{6}}]\] is used as an external indicator. The end point will be reached when the solution of iron salt
A white crystalline solid on boiling with caustic soda solution gave a gas which when passed through an alkaline solution of potassium mercuric iodide gave a brown ppt. The substance on heating gave a gas which rekindled a glowing splinter but did not give brown fumes with nitric oxide. The gases, and the substance respectively are
The equilibrium, \[{{P}_{4(g)}}+6C{{l}_{2(g)}}4PC{{l}_{3(g)}}\]is attained by mixing equal moles of \[{{P}_{4}}\]and \[C{{l}_{2}}\]in an evacuated vessel. Then at equilibrium
At a certain temperature and pressure, a 500 mL flask contains 25 moles of nitrogen gas. A different flask at the same temperature and pressure contains 100 mL of helium gas. The moles of helium present in the second flask is ___.
\[\Delta G\]for the reaction,\[\frac{4}{3}Al+{{O}_{2}}\xrightarrow[{}]{{}}\frac{2}{3}A{{l}_{2}}{{O}_{3}}\] is \[-772\text{ kJ mo}{{\text{l}}^{-}}^{1}.\] The minimum EMF in volts required to carry out an electrolysis of \[A{{l}_{2}}{{O}_{3}}\]is ___.
Number of points of non-differentiability of the function \[f(x)=x[x]\,\{x\}\] in the interval \[[-1,2]\]is (where \[[x]\] and \[\{x\}\] denote the greatest integer function and fractional part function, respectively)
Let \[{{A}_{1}},{{A}_{2}},{{A}_{3}},.....{{A}_{n}}\] be the vertices of a polygon of side n. If the number of pentagons that can be constructed by joining these vertices such that no side of the polygon is a side of the pentagon is 36, then the value of n is equal to
From the point \[P(-1,-2),\] PQ and PR are the tangents drawn to the circle \[{{x}^{2}}+{{y}^{2}}-6x-8y=0.\]. Then angle subtended by QR at the centre of the circle is
Let \[y=cosec\text{ }x-cot\text{ }x\]such that \[\frac{\frac{{{d}^{2}}y}{d{{x}^{2}}}}{y}=\lambda {{\sec }^{2}}\frac{x}{2}.\] Then \[\lambda \] is equal to
If m is a non-zero real numer such that \[\int{\frac{{{x}^{5m-1}}+2{{x}^{4m-1}}}{{{({{x}^{2m}}+{{x}^{m}}+1)}^{3}}}}dx=f(x)+C,\]where C is Constant of integration, then \[f(x)\] is
Let \[f(x)\] be a continuous function which satisfies \[f({{x}^{2}}+1)=\frac{2}{f({{2}^{x}})-1}\] and \[f(x)>0\,\forall \,x\,\in R.\] Then \[\underset{x\to 1}{\mathop{\lim }}\,f(x)\] is
\[\vec{p},\vec{q}\] and \[\vec{r}\] are unit vectors such that \[\vec{p}\] is perpendicular to \[\vec{q}\] and \[\vec{r}\] . If \[|\vec{p}+\vec{q}+\vec{r}|=1,\] then angle between \[\vec{q}\]and \[\vec{r}\] is
If \[f(x)\] is a differentiable function such that \[f(1)=\sin 1,\] \[f(2)=\sin 4\]and \[f(3)=\sin 9,\] then the minimum number of distinct solutions of equation \[f'(x)=2x\cos {{x}^{2}}\] in \[(1,3)\] is
For a variable ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1,\] if distance between focus and extremity of minor axis is fixed, then locus of one end of its latus rectum will be
The tangent at the point (h, k) to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] cuts the circle \[{{x}^{2}}+{{y}^{1}}={{c}^{2}}\]at points whose ordinates are \[{{y}_{1}}\] and\[{{y}_{2}}\]. Then \[{{y}_{1}},\] k, \[{{y}_{2}}\] are in
If \[m=\sum\limits_{r=0}^{\infty }{{{a}^{r}},}\] \[n=\sum\limits_{r=0}^{\infty }{{{b}^{r}},}\]where \[0<a,\text{ }b<1,\] then which of the following equations has roots a and b?
Let \[f:R\to R\] satisfies the relation \[f(x)\,f(y)-f(xy)=x+y\forall x,\] \[y\in R\] and \[f(1)>0\]. Then number of roots of the equation \[f(x)=\cos x\] is ______.
In a building A, somewhere in New Delhi, there are 9 healthy people and 1 dengue infected person. In another building B, there are 10 healthy people. 9 people from building A move to building B and then 9 people from building B move to building The probability that the dengue infected person is still in building A is ______.
If \[\tan \theta =\frac{1}{2+\frac{1}{2+\frac{1}{2+\infty }}},\]where \[\theta \in (0,2\pi ),\] then the number of possible values of \[\theta \] is ______.
In a triangle ABC, if sides a, b and c are roots of the equation \[{{x}^{3}}-11{{x}^{2}}+38x-40=0,\] then \[\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}\]is equal to ____________.
Let \[\lambda \] be a non-zero real number. Matrix A of size \[2\times 2\] is formed with entries \[\lambda \] or \[-\lambda \] only. If a matrix is randomly chosen, then probability that \[{{T}_{r}}(A)\] is zero is equal to____________.