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If the time period $$t$$ of the oscillation of a drop of liquid of density $$d$$, radius $$r$$, vibrating under surface tension $$s$$ is given by the formula $$t = \sqrt{r^{2b}s^c d^{a/2}}$$. It is observed that the time period is directly proportional to $$\sqrt{\frac{d}{s}}$$. The value of $$b$$ should therefore be :
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A 70 kg man leaps vertically into the air from a crouching position. To take the leap the man pushes the ground with a constant force F to raise himself. The center of gravity rises by 0.5 m before he leaps. After the leap the c.g. rises by another 1 m. The maximum power delivered by the muscles is : (Take g = 10 ms$$^{-2}$$)
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A boy of mass 20 kg is standing on a 80 kg free to move long cart. There is negligible friction between cart and ground. Initially, the boy is standing 25 m from a wall. If he walks 10 m on the cart towards the wall, then the final distance of the boy from the wall will be
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A projectile of mass M is fired so that the horizontal range is 4 km. At the highest point the projectile explodes in two parts of masses M/4 and 3M/4 respectively and the heavier part starts falling down vertically with zero initial speed. The horizontal range (distance from point of firing) of the lighter part is :
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A particle of mass 2 kg is moving such that at time t, its position, in meter, is given by $$\vec{r}(t) = 5\hat{i} - 2t^2\hat{j}$$. The angular momentum of the particle at $$t = 2s$$ about the origin in kgm$$^{-2}$$ s$$^{-1}$$ is :
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A body of mass 'm' is tied to one end of a spring and whirled round in a horizontal plane with a constant angular velocity. The elongation in the spring is 1 cm. If the angular velocity is doubled, the elongation in the spring is 5 cm. The original length of the spring is :
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A copper wire of length 1.0 m and a steel wire of length 0.5 m having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by 1 mm. If the Young's modulii of copper and steel are respectively $$1.0 \times 10^{11}$$ Nm$$^{-2}$$ and $$2.0 \times 10^{11}$$ Nm$$^{-2}$$, the total extension of the composite wire is :
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Wax is coated on the inner wall of a capillary tube and the tube is then dipped in water. Then, compared to the unwaxed capillary, the angle of contact $$\theta$$ and the height $$h$$ upto which water rises change. These changes are :
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A thin tube sealed at both ends is 100 cm long. It lies horizontally, the middle 20 cm containing mercury and two equal ends containing air at standard atmospheric pressure. If the tube is now turned to a vertical position, by what amount will the mercury be displaced?
(Given : cross-section of the tube can be assumed to be uniform)
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The ratio of the coefficient of volume expansion of a glass container to that of a viscous liquid kept inside the container is 1 : 4. What fraction of the inner volume of the container should the liquid occupy so that the volume of the remaining vacant space will be same at all temperatures?
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500 g of water and 100 g of ice at 0°C are in a calorimeter whose water equivalent is 40 g. 10 g of steam at 100°C is added to it. Then water in the calorimeter is : (Latent heat of ice = 80 cal/g, Latent heat of steam = 540 cal/g)
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This question has Statement-1 and Statement-2. Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement 1: The internal energy of a perfect gas is entirely kinetic and depends only on absolute temperature of the gas and not on its pressure or volume.
Statement 2: A perfect gas is heated keeping pressure constant and later at constant volume. For the same amount of heat the temperature of the gas at constant pressure is lower than that at constant volume.
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Bob of a simple pendulum of length $$l$$ is made of iron. The pendulum is oscillating over a horizontal coil carrying direct current. If the time period of the pendulum is T then :
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A sonometer wire of length 114 cm is fixed at both the ends. Where should the two bridges be placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio 1 : 3 : 4?
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Consider a finite insulated, uncharged conductor placed near a finite positively charged conductor. The uncharged body must have a potential :
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A liquid drop having 6 excess electrons is kept stationary under a uniform electric field of 25.5 kVm$$^{-1}$$. The density of liquid is $$1.26 \times 10^3$$ kg m$$^{-3}$$. The radius of the drop is (neglect buoyancy).
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A parallel plate capacitor of area 60 cm$$^2$$ and separation 3 mm is charged initially to 90$$\mu$$C. If the medium between the plate gets slightly conducting and the plate loses the charge initially at the rate of $$2.5 \times 10^{-8}$$ C/s, then what is the magnetic field between the plates?
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Which of the four resistances P, Q, R and S generate the greatest amount of heat when a current flows from A to B?
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A rectangular loop of wire, supporting a mass m, hangs with one end in a uniform magnetic field $$\vec{B}$$ pointing out of the plane of the paper. A clockwise current is set up such that $$i > mg/Ba$$, where a is the width of the loop. Then :
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A particle of charge $$16 \times 10^{-16}$$ C moving with velocity 10 ms$$^{-1}$$ along x-axis enters a region where magnetic field of induction $$\vec{B}$$ is along the y-axis and an electric field of magnitude $$10^4$$ Vm$$^{-1}$$ is along the negative z-axis. If the charged particle continues moving along x-axis, the magnitude of $$\vec{B}$$ is :
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The plot given below is of the average power delivered to an LRC circuit versus frequency. The quality factor of the circuit is :
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Select the correct statement from the following :
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This question has Statement-1 and Statement-2. Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement 1: Very large size telescopes are reflecting telescopes instead of refracting telescopes.
Statement 2: It is easier to provide mechanical support to large size mirrors than large size lenses.
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A light ray falls on a square glass slab as shown in the diagram. The index of refraction of the glass, if total internal reflection is to occur at the vertical face, is equal to :
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$$n$$ identical waves each of intensity $$I_0$$ interfere with each other. The ratio of maximum intensities if the interference is (i) coherent and (ii) incoherent is :
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Electrons are accelerated through a potential difference V and protons are accelerated through a potential difference 4 V. The de-Broglie wavelengths are $$\lambda_e$$ and $$\lambda_p$$ for electrons and protons respectively. The ratio of $$\frac{\lambda_e}{\lambda_p}$$ is given by: (given $$m_e$$ is mass of electron and $$m_p$$ is mass of proton).
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In the Bohr's model of hydrogen-like atom the force between the nucleus and the electron is modified as $$F = \frac{e^2}{4\pi\epsilon_0}\left(\frac{1}{r^2} + \frac{\beta}{r^3}\right)$$, where $$\beta$$ is a constant. For this atom, the radius of the $$n^{th}$$ orbit in terms of the Bohr radius $$\left(a_0 = \frac{\epsilon_0 h^2}{m\pi e^2}\right)$$ is :
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A system of four gates is set up as shown. The 'truth table' corresponding to this system is :
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Which of the following statement is NOT correct?
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Correct set up to verify Ohm's law is :
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