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Which of the following equations is dimensionally incorrect?
Where $$t$$ = time, $$h$$ = height, $$s$$ = surface tension, $$\theta$$ = angle, $$\rho$$ = density, $$a, r$$ = radius, $$g$$ = the acceleration due to gravity, $$V$$ = volume, $$p$$ = pressure, $$W$$ = work done, $$\tau$$ = torque, $$\epsilon$$ = permittivity, $$E$$ = electric field, $$J$$ = current density, $$L$$ = length.
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Match List - I with List - II.
List - I List - II
(a) Torque (i) MLT$$^{-1}$$
(b) Impulse (ii) MT$$^{-2}$$
(c) Tension (iii) ML$$^2$$ T$$^{-2}$$
(d) Surface Tension (iv) MLT$$^{-2}$$
Choose the most appropriate answer from the option given below:
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A helicopter is flying horizontally with a speed $$v$$ at an altitude $$h$$ has to drop a food packet for a man on the ground. What is the distance of helicopter from the man when the food packet is dropped?
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A body of mass $$M$$ moving at speed $$V_0$$ collides elastically with a mass $$m$$ at rest. After the collision, the two masses move at angles $$\theta_1$$ and $$\theta_2$$ with respect to the initial direction of motion of the body of mass $$M$$. The largest possible value of the ratio $$\frac{M}{m}$$, for which the angles $$\theta_1$$ and $$\theta_2$$ will be equal, is:
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Angular momentum of a single particle moving with constant speed along circular path:
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The masses and radii of the earth and moon are $$(M_1, R_1)$$ and $$(M_2, R_2)$$ respectively. Their centres are at a distance $$r$$ apart. Find the minimum escape velocity for a particle of mass $$m$$ to be projected from the middle of these two masses:
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A uniform heavy rod of weight 10 kg m s$$^{-2}$$, cross-sectional area 100 cm$$^2$$ and length 20 cm is hanging from a fixed support. Young modulus of the material of the rod is $$2 \times 10^{11}$$ N m$$^{-2}$$. Neglecting the lateral contraction, find the elongation of rod due to its own weight:
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A reversible engine has an efficiency of $$\frac{1}{4}$$. If the temperature of the sink is reduced by 58°C, its efficiency becomes double. Calculate the temperature of the sink:
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For an ideal gas the instantaneous change in pressure $$P$$ with volume $$V$$ is given by the equation $$\frac{dP}{dV} = -aP$$. If $$P = P_0$$ at $$V = 0$$ is the given boundary condition, then the maximum temperature one mole of gas can attain is: (Here $$\mathcal{R}$$ is the gas constant)
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Two particles $$A$$ and $$B$$ having charges 20 $$\mu$$C and $$-5$$ $$\mu$$C respectively are held fixed with a separation of 5 cm. At what position a third charged particle should be placed so that it does not experience a net electric force?
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Consider a galvanometer shunted with 5 $$\Omega$$ resistance and 2% of current passes through it. What is the resistance of the given galvanometer?
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A coil having $$N$$ turns is wound tightly in the form of a spiral with inner and outer radii $$a$$ and $$b$$ respectively. Find the magnetic field at centre, when a current $$I$$ passes through coil:
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A small square loop of side $$a$$ and one turn is placed inside a larger square loop of side $$b$$ and one turn $$(b \gg a)$$. The two loops are coplanar with their centres coinciding. If a current $$I$$ is passed in the square loop of side $$b$$, then the coefficient of mutual inductance between the two loops is:
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In an ac circuit, an inductor, a capacitor and a resistor are connected in series with $$X_L = R = X_C$$. Impedance of this circuit is:
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An object is placed at the focus of concave lens having focal length $$f$$. What is the magnification and distance of the image from the optical centre of the lens?
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Two plane mirrors $$M_1$$ and $$M_2$$ are at right angle to each other as shown. A point source $$P$$ is placed at $$a$$ and $$2a$$ meter away from $$M_1$$ and $$M_2$$ respectively. The shortest distance between the images thus formed is: (Take $$\sqrt{5} = 2.3$$)
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A moving proton and electron have the same de-Broglie wavelength. If $$K$$ and $$P$$ denote the K.E. and momentum respectively. Then choose the correct option:
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A sample of a radioactive nucleus $$A$$ disintegrates to another radioactive nucleus $$B$$, which in turn disintegrates to some other stable nucleus $$C$$. Plot of a graph showing the variation of number of atoms of nucleus $$B$$ versus time is: (Assume that at $$t = 0$$, there are no $$B$$ atoms in the sample)
Choose the correct waveform that can represent the voltage across $$R$$ of the following circuit, assuming the diode is ideal one:
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In the following logic circuit the sequence of the inputs $$A$$, $$B$$ are $$(0, 0), (0, 1), (1, 0)$$ and $$(1, 1)$$. The output $$Y$$ for this sequence will be:
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A car is moving on a plane inclined at 30° to the horizontal with an acceleration of 10 m s$$^{-2}$$ parallel to the plane upward. A bob is suspended by a string from the roof of the car. The angle in degrees which the string makes with the vertical is _________. (Take $$g = 10$$ m s$$^{-2}$$)
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A block moving horizontally on a smooth surface with a speed of 40 m s$$^{-1}$$ splits into two equal parts. If one of the parts moves at 60 m s$$^{-1}$$ in the same direction, then the fractional change in the kinetic energy will be $$x : 4$$ where $$x$$ = _________.
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When a rubber ball is taken to a depth of _________ m in deep sea, its volume decreases by 0.5%.
(The bulk modulus of rubber = $$9.8 \times 10^8$$ N m$$^{-2}$$, Density of sea water = $$10^3$$ kg m$$^{-3}$$, g = 9.8 m s$$^{-2}$$)
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A particle of mass 1 kg is hanging from a spring of force constant 100 N m$$^{-1}$$. The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period $$T$$. The time when the kinetic energy and potential energy of the system will become equal, is $$\frac{T}{n}$$. The value of $$n$$ is _________.
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A wire having a linear mass density $$9.0 \times 10^{-4}$$ kg m$$^{-1}$$ is stretched between two rigid supports with a tension of 900 N. The wire resonates at a frequency of 500 Hz. The next higher frequency at which the same wire resonates is 550 Hz. The length of the wire is _________ m.
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A capacitor of 50$$\mu$$F is connected in a circuit as shown in figure. The charge on the upper plate of the capacitor is _________ $$\mu$$C.
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The voltage drop across 15 $$\Omega$$ resistance in the given figure will be _________ V.
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A square-shaped wire with a resistance of each side 3 $$\Omega$$ is bent to form a complete circle. The resistance between two diametrically opposite points of the circle in a unit of $$\Omega$$ will is _________.
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The electric field in an electromagnetic wave is given by
$$E = (50 \text{ N C}^{-1})\sin\omega\left(t - \frac{z}{c}\right)$$
The energy contained in a cylinder of volume $$V$$ is $$5.5 \times 10^{-12}$$ J. The value of $$V$$ is _________ cm$$^3$$.
(given $$\epsilon_0 = 8.8 \times 10^{-12}$$ C$$^2$$ N$$^{-1}$$ m$$^{-2}$$)
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If the sum of the heights of transmitting and receiving antennas in the line of sight of communication is fixed at 160 m, then the maximum range of LOS communication is _________ km (Take radius of Earth = 6400 km)
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