If the dimensions of length are expressed as \[{{G}^{x}}{{c}^{y}}{{h}^{z}};\] where G, c and h are the universal gravitational constant, speed of light and Planck's constant respectively, then
The x-y plane is the boundary between two transparent media. Medium-1 with \[z\ge 0\] has refractive index \[\sqrt{2}\] and medium-2 with \[z\le 0\]has a refractive index \[\sqrt{3}\] . A ray of light in medium given by vector \[\vec{A}=\sqrt{3}\,\,\hat{i}-\hat{k}\]is incident on the plane of separation. The unit vector in the direction of the refracted ray in medium-2 is
A boy throws a ball upwards with velocity\[{{v}_{0}}=20\text{ }m/s\]. The wind imparts a horizontal acceleration of \[4\text{ }m/{{s}^{2}}\]to the left. The angle \[\theta \] at which the ball must be thrown so that the ball returns to the boy's hand is \[(g=10\,m/{{s}^{2}})\]
In the arrangement shown in the figure, block B of mass M rests on a weighing scale. Ball A is released from a position where spring is in its natural length and the scale shows the correct weight of block B. Find the mass of ball A so that the minimum reading shown by the scale subsequently is half the true weight of B.
A block of mass m is placed over a plank of mass M, which is placed over a smooth horizontal surface as shown in the figure. Friction exists between the block and plank. A horizontal force (which increases with time according to law\[F=\alpha t\]) is applied horizontally on the plank. Then which of the following graphs shows acceleration a of the block?
A particle is projected from a point A, which is at a distance 4R from the centre of the earth and with speed \[{{V}_{1}}\] in a direction making \[30{}^\circ \] with the \[{{V}_{1}}\] line joining the centre of the earth and point A, as shown. Find the speed \[{{V}_{1}}\] if particle passes grazing the surface of the earth. Consider gravitational interaction only between these two. (Use \[\frac{GM}{R}=6.4\times {{10}^{7}}{{m}^{2}}/{{s}^{2}}\])
An upright U-tube manometer with its limbs \[0.6\text{ }m\]high and spaced \[0.3\text{ }m\]apart contains a liquid to a height of \[0.4\text{ }m\]on each limb. If the U-tube is rotated at 10 radians/second about a vertical axis at \[0.1\text{ }m\]from one limb. \[{{z}_{1}}\] and \[{{z}_{2}}\] are heights of liquid columns on rotation. Choose the correct option. (Use\[g=10\text{ }m/{{s}^{2}}\])
A cone of height h and base of radius r is submerged in a fluid as shown in the figure such that half of its height is outside the fluid. The angle of contact between the fluid and cone is 6 and surface tension for the fluid is T. The net force experienced by cone due to surface tension is
A particle is acted upon by force \[F={{F}_{0}}\cos (\omega t)\] along X-axis. The amplitude of its velocity is given by \[v=\sqrt{\frac{1}{X{{\omega }^{2}}-Y\omega +Z}}\]Choose the CORRECT condition for resonance.
A parallel plate air capacitor is charged by a battery of emf \[\varepsilon \], then it is disconnected from the battery and allowed to discharge through a dielectric slab of dielectric constant K and resistivity \[\rho ,\] as shown in figure. There is a uniform outward magnetic field B at the location of the capacitor. The total momentum attained by the slab after complete discharge is
In the given figure, a wire loop has been bent so that it has three segments AB (a quarter circle), BC (a square comer), and CA (straight). The following are the three choices for a magnetic field through the loop.
Where B is in millitesia and t is in second if the induced current in the loop due to \[{{\vec{B}}_{1}},\,{{\vec{B}}_{2}}\] and \[{{\vec{B}}_{3}}\] are \[{{i}_{1}},{{i}_{2}}\] and \[{{i}_{3}}\] respectively, then
A convex lens of focal length \[20\text{ }cm\]is placed in front of a plane mirror at \[20\text{ }cm\] A point object 'O' is placed at \[30\text{ }cm\]towards left from convex lens as shown. The position of final image is
In the Young's double slit experiment using a monochromatic light of wavelength A, the path difference (in terms of an integer \[\lambda \]) corresponding to any point having half the peak intensity is
Mass spectrometric analysis of potassium and argon atoms in a Moon rock sample shows that the ratio of the number of (stable) \[^{40}Ar\] atoms present to the number of (radioactive) \[^{40}K\] atoms is\[10.3\]. Assume that all the argon atoms were produced by the decay of potassium atoms, with a half-life of\[1.25\times {{10}^{9}}yr\]. How old is the rock?
In the given figure, the electric lines of forces shaped like arcs of concentric circles with their centre at point O in the certain confined region of a field \[[a<r<b],\] where r is the radius of concentric arcs. The intensity of electric field in this section should be
A)
Directly proportional to distance from \[O\]
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B)
Directly proportional to square of distance from \[O\]
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C)
Inversely proportional to distance from \[O\]
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D)
Inversely proportional to square of distance from \[O\]
A conducting circular loop of radius a is placed in a uniform magnetic field B which is perpendicular to loop. A rod OA touches the loop as shown. A resistor R is also connected between 0 and circumference of loop (loop and rod are resistance less). Rod is rotated with constant angular velocity \[\omega \] in anticlockwise direction. Torque of the external force needed to keep the rod rotating with constant angular velocity \[\omega \] is
A point object \[(O)\] lies at a distance of \[20\text{ }cm\]on the principal axis of a convex lens of focal length\[f=10\text{ }cm\]. The object begins to move in a direction making an angle of \[45{}^\circ \] with the principal axis. At what angle with the principal axis does the image begin to move?
A)
At angle \[45{}^\circ \] with the principal axis in downward direction.
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B)
At angle \[75{}^\circ \] with the principal axis in downward direction.
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C)
At angle \[45{}^\circ \] with the principal axis in upward direction.
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D)
At angle \[75{}^\circ \] with the principal axis in upward direction.
A block of mass M is placed on top of a hole in a horizontal table. A spring of force constant k is connected to the block through the hole. The other end of the massless spring has a particle of mass m connected to it. With what maximum amplitude can the particle oscillate up and down such that the block does not lose contact with the table?
Carbon monoxide is carried around a closed cycle abc, in which be is an isothermal process, as shown in figure. The gas absorbs \[7000\text{ }J\] of heat, as its temperature increased from \[300\text{ }K\]to \[1000\text{ }K\]in going from a to b. The quantity of heat ejected by the gas during the process ca is \[98\times {{10}^{x}}J.\] Find the value of x.
Two particles A and B have de-Broglie's wavelengths \[30\overset{o}{\mathop{A}}\,\] and \[20\overset{o}{\mathop{A}}\,,\] combined to form a particle C. Momentum is conserved in this process. What is the possible de-Broglie's wavelength of C (in\[A{}^\circ \])?
The p-n junction diode used in the given figure has a cut-in voltage of 1 volt and forward resistance of \[2\,k\Omega \]. If diode can dissipate a maximum power of \[200\text{ }mW,\] find the maximum possible value of the battery voltage \[{{V}_{B}}\] (in volt)
A current of 8 A is to be sent through a resistor of \[5\Omega \] Calculate the least number of cells in a mixed combination required for the purpose when each cell has emf 2 V and internal resistance \[0.5\Omega \].
The resistance of an 8 m long potentiometer wire is 80 A high resistance box and a 2 V battery are connected in series with it. What should be the value of the resistance (in Q.) in the box, if potentiometer wire is desired to have a potential drop of \[1\,\mu V/mm\]?
If \[0.5\text{ }mol\]of \[BaC{{l}_{2}}\] is mixed with \[0.2\]mole of \[N{{a}_{3}}P{{O}_{4}},\] find the maximum amount of \[B{{a}_{3}}{{(P{{O}_{4}})}_{2}}\] that can be formed.
A litre of solution is saturated with\[AgCl\]. To this solution if \[1.0\times {{10}^{-4}}\]mole of solid \[NaCI\]is added, what will be the \[[A{{g}^{+}}],\] assuming no volume change?
At \[80{}^\circ C,\] the vapour pressure of pure liquid 'A' is \[520\text{ }mm\text{ }Hg\]and that of pure liquid 'B' is\[1000\text{ }mm\text{ }Hg\]. If a mixture solution of 'A' and 'B' boils at \[80{}^\circ C\] and 1 atm pressure, the amount of 'A' in the mixture is \[(1\,atm=760\,mm\,Hg)\]
Given the reaction at \[975{}^\circ C\]and 1 atm. \[CaC{{O}_{3}}(s)\xrightarrow{{}}CaO(s)+C{{O}_{2}}(g);\] \[\Delta H=176\,kJ.\] Calculate the value of\[\Delta E\].
The uncertainties in the velocities of two particles, A and B are \[0.05\] and \[0.02\,m{{s}^{-1}},\] respectively. The mass of B is five times to that of the mass of A. What is the ratio of uncertainties \[\left( \frac{\Delta {{x}_{A}}}{\Delta {{x}_{B}}} \right)\] in their positions?
How many compounds gives positive iodoform test? \[{{(C{{H}_{3}})}_{2}}CHC{{H}_{2}}OH;2-Pentanone\] \[{{C}_{6}}{{H}_{5}}-OH;3-Pentanone;C{{H}_{3}}C{{H}_{2}}OH\] \[C{{H}_{3}}-C{{H}_{2}}-\underset{OH}{\mathop{\underset{|}{\mathop{\overset{H}{\mathop{\overset{|}{\mathop{C}}\,}}\,}}\,}}\,-C{{H}_{2}}-C{{H}_{3}};\,\,Ethanal\]
Aluminum oxide may be electrolysed at \[1000{}^\circ C\]to furnish aluminum metal (At. Mass \[=27\text{ }amu;\]1 Faraday = 96,500 Coulombs). The cathode reaction is \[-A{{l}^{3+}}+3{{e}^{-}}\to Al\] To prepare \[5.12\text{ }kg\]of aluminum metal by this method we require electricity of____ C
A is one of 6 horses entered for a race, and is to be ridden by one of two jockeys B and C. It is 2 to 1 that B rides A, in which case all the horses are equally likely to win. If C rides A, his chances of winning is tripled. What are the odds against winning of A?
If \[{{z}_{1}},{{z}_{2}}\]and \[{{z}_{3}}\]are complex numbers such that \[|{{z}_{1}}|=|{{z}_{2}}|=|{{z}_{3}}|=\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|=1\], then\[|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}|\]is