Ball A is dropped from the top of a building. At the same instant ball B is thrown vertically upwards from the ground. When the balls collide, they are moving in opposite directions and the speed of A is twice the speed of B. At what fraction of the height of the building did the collision occurs?
A particle of mass 'm' moves on the x-axis under the influence of a force of attraction towards the origin given by \[\vec{F}=\frac{-k}{{{x}^{2}}}\hat{i}.\] If the particle starts from rest at \[x=a,\]the speed it will attain on reaching the point \[x=\frac{a}{2}\]will be
A uniform sphere of mass M and radius R exerts a force F on a small mass m situated at a distance of \[2R\] from the centre O of the sphere. A spherical portion of diameter R is cut from the sphere as shown in figure. The force of attraction between the remaining part of the sphere and the mass m will be
A small amount of water is poured from the top and the water surface assumes the profile as shown in the figure. Assume that water does not wet the vertical part of the tube. The angle of contact \[\alpha \] is \[120{}^\circ \]and the surface tension of water is\[0.070\text{ }N/m\]. The thickness h of the layer of water is (Assume the thickness of the layer of water substantially smaller than the diameter of opening of tube and\[g=10\text{ }m/{{s}^{2}}\])
A particle is performing SHM. Magnitude of its maximum acceleration is a. Find the magnitude of average acceleration for duration of \[\frac{7}{6}\] time period starting from mean position.
A guitar string of length L is stretched between two fixed points P and Q and made to vibrate transversely as shown in the figure. Two particles A and B on the string are separated by a distance s. The maximum kinetic energies of A and B are \[{{K}_{A}}\] and \[{{K}_{B}}\] respectively. Which of the following gives the correct phase difference and maximum kinetic energies of the particles?
A)
Phase difference - \[\left( \frac{3s}{2L} \right)\times 360{}^\circ \] Maximum kinetic energy - \[{{K}_{A}}<{{K}_{B}}\]
doneclear
B)
Phase difference - \[\left( \frac{3s}{2L} \right)\times 360{}^\circ \] Maximum kinetic energy - same
doneclear
C)
Phase difference - \[180{}^\circ \] Maximum kinetic energy - \[{{K}_{A}}<{{K}_{B}}\]
doneclear
D)
Phase difference - \[180{}^\circ \] Maximum kinetic energy - same
The introduction of a metal plate between the plates of a parallel plate eapacitor increases its capacitance by \[4.5\] times. If d is the separation of the two plates of the capacitor, the thickness of the metal plate introduced is
A straight conductor AB lies along the axis of a hollow metal cylinder L, which is connected to earth through a conductor CA The quantity of charge will flow through C is
A)
If a current begins to flow through AB
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B)
If the current through AB is reversed
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C)
If AB is removed, and abeam of electrons flows in its place.
An inductor, three resistors and a battery of 90 volt are connected in circuit as shown. Switch 'S" is closed for a long time, let the current in \[10\Omega \] is\[{{i}_{1}}\]. Now switch 'S" is suddenly opened, let the current in \[10\Omega \] is \[{{i}_{2}}\]. The ratio of \[{{i}_{1}}/{{i}_{2}}\] is
When wave of wavelength \[0.2\text{ }cm\]is made incident normally on a slit of width \[0.004\text{ }m,\] then the semi-angular width of central maximum of diffraction pattern will be
The circuit contains two ideal cells connected as shown in the figure. Initially key is connected to position 1 and then pushed to position 2. Then match the List-1 with appropriate values in List-2.
List-1
List-2
P.
The final charge on capacitor (in \[\mu C\])
1.
2
Q.
The work done by 4 volt cell (in \[\mu J\])
2.
4
R.
The gain in potential energy of capacitor (in \[\mu J\])
In a \[10\text{ }m\]long potentiometer wire the first \[5\text{ }m\]length is of radius 'r' and remaining is of radius\[2r\] The wires are connected to a battery of steady voltage 2 V and negligible internal resistance.
(i) The null point distance for a cell of emf \[1.0\] volt is\[312.5\text{ }cm\].
(ii) The null point distance for a cell of emf \[1.5\] volt is\[512.5\text{ }cm\].
(iii) The null point distance for a cell of emf \[1.8\] volt is\[750\text{ }cm\].
(iv) The null point distance for a cell of emf \[1.6\] volt is\[500\text{ }cm\].
A charge of magnitude Q is placed at O, the centre of an imaginary sphere, which can be considered over any radius as shown in figure. Consider parts of the spheres over a solid angle \[\Omega =\pi \]steradian through which three areas \[{{A}_{1}},{{A}_{2}},{{A}_{3}}\] are considered at a distance \[r,2r,3r\]from the centre respectively as shown. Then study the following statements:
(i) The flux passing through \[{{A}_{1}}\] is given by \[\frac{Q}{4{{\varepsilon }_{0}}}\].
(ii) The flux passing through \[{{A}_{2}}\] is given by \[\frac{Q}{9{{\varepsilon }_{0}}}\](iii) The flux passing through \[{{A}_{2}}\] within the area equal to \[{{A}_{1}}\] is given by\[\frac{Q}{16{{\varepsilon }_{0}}}\].
(iv) The flux passing through \[{{A}_{3}}\] within the area equal to \[{{A}_{2}}\] is given by \[\frac{Q}{4{{\varepsilon }_{0}}}\]
Figure shows a part of RC circuit and a graph showing the variation of potential difference across resistor \[({{v}_{R}})\] with time. Initially \[(t=0)\]capacitor is uncharged. The variation curve of charge on the plates of capacitor for \[{{t}_{0}}<t<2{{t}_{0}}\] w.r.t. time is
Two small spheres each of mass 'm' and each with charge q lie inside a non-conducting smooth hemispherical bowl of radius R placed on ground. The charge required for equilibrium at separation 2d is given by
Six identical balls are lined up along a straight frictionless groove. Two similar balls moving with speed v along the groove collide elastically with this row on extreme left side end. Then
A)
One ball from the right end will move on with speed 2v, all the other remains at rest.
doneclear
B)
Two balls from the extreme right will move on with speed v each and the remaining balls will be at rest.
doneclear
C)
All the balls will start moving to right with speed \[v/8\]each.
doneclear
D)
All the six balls originally at rest will move on with speed \[v/6\] each and the two incident balls will come to rest.
In the given circuit the reading of voltmeter \[{{V}_{1}}\] and \[{{V}_{2}}\] are 300 volts each. The reading of the voltmeter \[{{V}_{3}}\] and ammeter A are, respectively,
A transparent glass slab (G) of thickness \[6\text{ }cm\]is held perpendicular to the principal axis of a convex lens (L) as shown in the figure. The refractive index of the material 3 of the glass is \[\frac{3}{2}\]and its nearer face is at a distance \[40cm\] from the lens. Focal length of the lens is\[20\text{ }cm\]. Find the thickness of the glass slab as observed through the lens.
Addition of two vectors \[\vec{A}\] and \[\vec{B}\] is \[4\hat{i}\] and subtraction of these two vectors is \[2\hat{j}\]. Then scalar product of vectors \[\vec{A}\] and \[\vec{B}\] is
A particle is moving in x-y plane. At certain instant of time, the components of its velocity and acceleration are as follows: \[{{v}_{x}}=3m/s,\] \[{{v}_{y}}=4m/s,\]\[{{a}_{x}}=2m/{{s}^{2}}\]and \[{{a}_{y}}=1m/{{s}^{2}}\]. What is the rate of change of speed in \[(m/{{s}^{2}})\] at this moment?
A force acts on a \[3\text{ }g\]particle in such a way that the position of the particle as a function of time is given by \[x=3t-4{{t}^{2}}+{{t}^{3}},\]where x is in metres and t is in seconds. What is the work done (in\[mJ\]) during the first 4 second?
A system of two blocks A and B, and a wedge C is released from rest as shown in figure. Masses of the blocks and me wedge are m, \[1m\]and \[2m\]respectively. What is the displacement (in cm) of wedge C when block B slides down the plane at a distance 10 cm? (neglect friction)
Two mole of monoatomic gas undergoes a cyclic process ABCDA. Initial temperature of gas is 300 K. The P-V graph of process is shown in the figure. The work done by the gas in cyclic process is \[N{{P}_{0}}{{V}_{0}}\]. Find the value of N.
In Young? double slit experiment, the fringes are displaced by a distance x when a glass plate of refractive index \[1.5\] is introduced in the path of one of the beams. When this plate is replaced by another plate of the same thickness, the shift of fringes is\[(3/2)x\]. What is the refractive index of the second plate?
Four alkali metals A, B, C and D are having respectively standard electrode potential as -3.05, -1.66, -0.40 and 0.80. Which one will be the most reactive?
A 0.5 M NaOH solution offers a resistance of 31.6 ohm in a conductivity cell at room temperature. What shall be the approximate molar conductance of this NaOH solution if cell constant of the cell is ?
A circle is drawn to pass through the extremeties of the latus ractum of the parabola \[{{y}^{2}}=8x\] and touching the directrix, then radius of this circle is-
A man running round a race course notes that the sum of the distances of two flag-posts from him is always 10 metres and the distance between the flag-posts is 8 meters. The area of the path he encloses in square metres is
The line \[2x+y=1\] is tangent to the hyperbola\[\frac{{{x}^{2}}}{{{a}^{2}}}~-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is -
A football match may be either won, drawn or lost by the host country's team. So, there are three ways of forecasting the result of any one match, one correct and two incorrect. Then the probability of forecasting at least three correct results for four matches, is -
Given three vectors \[\vec{a},\vec{b}\] and \[\vec{c}\] each two of which are non-collinear. Further if \[\left( \vec{a}+\vec{b} \right)\] is collinear with \[\vec{c},\left( \vec{b}+\vec{c} \right)\] is collinear with \[\vec{a}\]and\[|\vec{a}|=|\vec{b}|=|\vec{c}|=\sqrt{2}\]. Then the value of \[\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}=\]
Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place is -
Two students while solving a quadratic equation in x, one copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of \[{{x}^{2}}\] correctly as - 6 and 1 respectively. The correct roots are-
\[x+y+z=15\] if 9, x, y, z, a are in A.P. while \[\frac{1}{X}+\frac{1}{Y}+\frac{1}{Z}=\frac{5}{3}\] if 9, X, Y, Z, a are in H.P., then the value of a will be-
If \[f\left( x \right)={{e}^{x}}\], then \[\underset{x\to 0}{\mathop{\lim }}\,{{(f(x))}^{\frac{1}{\{f(x)\}}}}\] (where { } denotes the fractional part of x) is equal to -
If \[f(x)=\left\{ \begin{align} & [x]+[-x];x\ne 2 \\ & \lambda \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;\,x=2 \\ \end{align} \right.\] then f is continuous at x = 2, provided A. is equal to -
Let \[f(x)=\left\{ \begin{matrix} {{x}^{3}}-{{x}^{2}}+10x-5, & x\le 1 \\ -2x+{{\log }_{2}}({{b}^{2}}-2), & x>1 \\ \end{matrix} \right.\] the set of values of b for which f (x) have greatest value at x = 1 is given by -
The equations of the perpendicular bisector of the sides AB and AC of a \[\Delta ABC\] are \[x-y+5=0\] and \[x+2y=0\] respectively. If the point A is \[\left( 1,-2 \right)\] then the equation of the line BC is -
For all complex numbers \[{{z}_{1}},{{z}_{2}}\] satisfying \[|{{z}_{1}}\text{ }\!\!|\!\!\text{ }=12\] and\[|{{z}_{2}}-3-4i|=5\], find the minimum value of \[\left| {{z}_{1}}-{{z}_{2}} \right|\].
An unbiased coin is tossed n times. If the probability that head occurs 6 times is equal to the probability that head occurs 8 times, then find the value of n.
If \[{{D}_{k}}\left| \begin{matrix} 1 & n & n \\ 2k & {{n}^{2}}+n+1 & {{n}^{2}}+n \\ 2k-1 & {{n}^{2}} & {{n}^{2}}+n+1 \\ \end{matrix} \right|\] and \[\sum\limits_{k=1}^{n}{{{D}_{k}}=56}\] then find value of n.