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Two buses $$P$$ and $$Q$$ start from a point at the same time and move in a straight line and their positions are represented by $$x_P(t) = \alpha t + \beta t^2$$ and $$x_Q(t) = ft - t^2$$. At what time, both the buses have same velocity?
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Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Two identical balls A and B thrown with same velocity 'u' at two different angles with horizontal attained the same range R. If A and B reached the maximum height $$h_1$$ and $$h_2$$ respectively, then $$R = 4\sqrt{h_1 h_2}$$
Reason R: Product of said heights. $$h_1 h_2 = \frac{u^2\sin^2\theta}{2g} \cdot \frac{u^2\cos^2\theta}{2g}$$
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A disc with a flat small bottom beaker placed on it at a distance $$R$$ from its center is revolving about an axis passing through the center and perpendicular to its plane with an angular velocity $$\omega$$. The coefficient of static friction between the bottom of the beaker and the surface of the disc is $$\mu$$. The beaker will revolve with the disc if :
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For a particle in uniform circular motion, the acceleration $$\vec{a}$$ at any point $$P(R, \theta)$$ on the circular path of radius $$R$$ is (when $$\theta$$ is measured from the positive $$x$$-axis and $$v$$ is uniform speed):
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A solid metallic cube having total surface area $$24$$ m$$^2$$ is uniformly heated. If its temperature is increased by $$10°$$C, calculate the increase in volume of the cube. (Given $$\alpha = 5.0 \times 10^{-4}$$ °C$$^{-1}$$).
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A copper block of mass $$5.0$$ kg is heated to a temperature of $$500°$$C and is placed on a large ice block. What is the maximum amount of ice that can melt?
[Specific heat of copper : $$0.39$$ J g$$^{-1}$$ °C$$^{-1}$$ and latent heat of fusion of water : $$335$$ J g$$^{-1}$$]
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The ratio of specific heats $$\frac{C_p}{C_v}$$ in terms of degree of freedom $$f$$ is given by :
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Two metallic plates form a parallel plate capacitor. The distance between the plate is '$$d$$'. A metal sheet of thickness $$\frac{d}{2}$$ and of area equal to area of each plate is introduced between the plates. What will be the ratio of the new capacitance to the original capacitance of the capacitor?
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Two cells of the same EMF $$E$$ but different internal resistances, $$r_1$$ and $$r_2$$ are connected in series with an external resistance $$R$$ as shown in the figure. The terminal potential difference across the second cell is found to be zero. The external resistance $$R$$ must then be:
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If $$n$$ represents the actual number of deflections in a converted galvanometer of resistance $$G$$ and shunt resistance $$S$$. Then the total current $$I$$ when its figure of merit is $$K$$ will be
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Given below are two statements :
Statement - I : Susceptibilities of paramagnetic and ferromagnetic substances increase with decrease in temperature.
Statement - II : Diamagnetism is a result of orbital motions of electrons developing magnetic moments opposite to the applied magnetic field.
Choose the correct answer from the options given below
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A long solenoid carrying a current produces a magnetic field $$B$$ along its axis. If the current is doubled and the number of turns per cm is halved, the new value of magnetic field will be equal to
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A sinusoidal voltage $$V(t) = 210\sin 3000t$$ volt is applied to a series LCR circuit in which $$L = 10$$ mH, $$C = 25$$ $$\mu$$F and $$R = 100\Omega$$. The phase difference $$\Phi$$ between the applied voltage and resultant current will be
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The electromagnetic waves travel in a medium at a speed of $$2.0 \times 10^8$$ m s$$^{-1}$$. The relative permeability of the medium is $$1.0$$. The relative permittivity of the medium will be
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The interference pattern is obtained with two coherent light sources of intensity ratio $$4:1$$. And the ratio $$\frac{I_{max} + I_{min}}{I_{max} - I_{min}}$$ is $$\frac{5}{x}$$. Then, the value of $$x$$ will be equal to :
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A light whose electric field vectors are completely removed by using a good polaroid, allowed to incident on the surface of the prism at Brewster's angle. Choose the most suitable option for the phenomenon related to the prism.
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A proton, a neutron, an electron and an $$\alpha$$-particle have same energy. If $$\lambda_p, \lambda_n, \lambda_e$$ and $$\lambda_\alpha$$ are the de Broglie's wavelengths of proton, neutron, electron and $$\alpha$$ particle respectively, then choose the correct relation from the following
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Which of the following figure represents the variation of $$\ln\frac{R}{R_0}$$ with $$\ln A$$ (if $$R$$ = radius of a nucleus and $$A$$ = its mass number)?
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Identify the logic operation performed by the given circuit
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Match List I with List II
| List I | List II |
|---|---|
| (A) Facsimile | (I) Static Document Image |
| (B) Guided media Channel | (II) Local Broadcast Radio |
| (C) Frequency Modulation | (III) Rectangular wave |
| (D) Digital Signal | (IV) Optical Fiber |
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For $$z = a^2 x^3 y^{\frac{1}{2}}$$, where '$$a$$' is a constant. If percentage error in measurement of '$$x$$' and '$$y$$' are $$4\%$$ and $$12\%$$, respectively, then the percentage error for '$$z$$' will be ______ %.
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A curved in a level road has a radius $$75$$ m. The maximum speed of a car turning this curved road can be $$30$$ m s$$^{-1}$$ without skidding. If radius of curved road is changed to $$48$$ m and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be ______ m s$$^{-1}$$.
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A block of mass $$200$$ g is kept stationary on a smooth inclined plane by applying a minimum horizontal force $$F = \sqrt{x}$$ N as shown in figure. The value of $$x$$ = ______.
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Moment of Inertia (M.I.) of four bodies having same mass $$M$$ and radius $$2R$$ are as follows
$$I_1$$ = M.I. of solid sphere about its diameter
$$I_2$$ = M.I. of solid cylinder about its axis
$$I_3$$ = M.I. of solid circular disc about its diameter
$$I_4$$ = M.I. of thin circular ring about its diameter
If $$2(I_2 + I_3) + I_4 = xI_1$$ then the value of $$x$$ will be ______.
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Two satellites $$S_1$$ and $$S_2$$ are revolving in circular orbits around a planet with radius $$R_1 = 3200$$ km and $$R_2 = 800$$ km respectively. The ratio of speed of satellite $$S_1$$ to the speed of satellite $$S_2$$ in their respective orbits would be $$\frac{1}{x}$$ where $$x =$$ ______.
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When a gas filled in a closed vessel is heated by raising the temperature by $$1°$$C, its pressure increases by $$0.4\%$$. The initial temperature of the gas is ______ K.
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$$27$$ identical drops are charged at $$22$$ V each. They combine to form a bigger drop. The potential of the bigger drop will be ______ V.
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The length of a given cylindrical wire is increased to double of its original length. The percentage increase in the resistance of the wire will be ______ %.
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In a series LCR circuit, the inductance, capacitance and resistance are $$L = 100$$ mH, $$C = 100$$ $$\mu$$F and $$R = 10$$ $$\Omega$$ respectively. They are connected to an AC source of voltage $$220$$ V and frequency of $$50$$ Hz. The approximate value of current in the circuit will be ______ A.
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In an experiment of CE configuration of $$n-p-n$$ transistor, the transfer characteristics are observed as given in figure. If the input resistance is $$200\Omega$$ and output resistance is $$60\Omega$$, the voltage gain in this experiment will be ______.
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