In an octagon $$ABCDEFGH$$ of equal side, what is the sum of $$\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} + \vec{AG} + \vec{AH}$$, if, $$\vec{AO} = 2\hat{i} + 3\hat{j} - 4\hat{k}$$
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In an octagon $$ABCDEFGH$$ of equal side, what is the sum of $$\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} + \vec{AG} + \vec{AH}$$, if, $$\vec{AO} = 2\hat{i} + 3\hat{j} - 4\hat{k}$$
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Match List - I with List - II:
| List - I | List - II |
|---|---|
| (a) $$h$$ (Planck's constant) | (i) $$[MLT^{-1}]$$ |
| (b) $$E$$ (kinetic energy) | (ii) $$[ML^2 T^{-1}]$$ |
| (c) $$V$$ (electric potential) | (iii) $$[ML^2 T^{-2}]$$ |
| (d) $$P$$ (linear momentum) | (iv) $$[ML^2 I^{-1} T^{-3}]$$ |
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An engine of a train, moving with uniform acceleration, passes the signal-post with velocity $$u$$ and the last compartment with velocity $$v$$. The velocity with which middle point of the train passes the signal post is:
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A solid sphere of radius $$R$$ gravitationally attracts a particle placed at $$3R$$ from its centre with a force $$F_1$$. Now a spherical cavity of radius $$\frac{R}{2}$$ is made in the sphere (as shown in figure) and the force becomes $$F_2$$. The value of $$F_1 : F_2$$ is:
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Two satellites $$A$$ and $$B$$ of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively. If $$T_A$$ and $$T_B$$ are the time periods of $$A$$ and $$B$$ respectively then the value of $$T_B - T_A$$:
[Given: radius of earth = 6400 km, mass of earth = $$6 \times 10^{24}$$ kg]
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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: The escape velocities of planet A and B are same. But A and B are of unequal mass.
Reason R: The product of their mass and radius must be same. $$M_1R_1 = M_2R_2$$
In the light of the above statements, choose the most appropriate answer from the options given below:
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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: When a rod lying freely is heated, no thermal stress is developed in it.
Reason R: On heating, the length of the rod increases.
In the light of the above statements, choose the correct answer from the options given below:
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A diatomic gas, having $$C_P = \frac{7}{2}R$$ and $$C_V = \frac{5}{2}R$$, is heated at constant pressure. The ratio dU : dQ : dW
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If the time period of a two meter long simple pendulum is 2 s, the acceleration due to gravity at the place where pendulum is executing S.H.M. is:
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A student is performing the experiment of the resonance column. The diameter of the column tube is 6 cm. The frequency of the tuning fork is 504 Hz. Speed of the sound at the given temperature is 336 m s$$^{-1}$$. The zero of the meter scale coincides with the top end of the resonance column tube. The reading of the water level in the column when the first resonance occurs is:
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A proton, a deuteron and an $$\alpha$$ particle are moving with same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them is ______ and their speed is ______ in the ratio.
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Magnetic fields at two points on the axis of a circular coil at a distance of 0.05 m and 0.2 m from the centre are in the ratio 8 : 1. The radius of coil is ______.
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The current (i) at time $$t = 0$$ and $$t = \infty$$ respectively for the given circuit is:
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The angular frequency of alternating current in a L-C-R circuit is 100 rad s$$^{-1}$$. The components connected are shown in the figure. Find the value of inductance of the coil and capacity of condenser.
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Two coherent light sources having intensity in the ratio $$2x$$ produce an interference pattern. The ratio $$\frac{I_{max} - I_{min}}{I_{max} + I_{min}}$$ will be:
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An $$\alpha$$ particle and a proton are accelerated from rest by a potential difference of 200 V. After this, their de Broglie wavelengths are $$\lambda_\alpha$$ and $$\lambda_p$$ respectively. The ratio $$\frac{\lambda_p}{\lambda_\alpha}$$ is:
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Two radioactive substances $$X$$ and $$Y$$ originally have $$N_1$$ and $$N_2$$ nuclei respectively. Half life of $$X$$ is half of the half life of $$Y$$. After three half lives of $$Y$$, number of nuclei of both are equal. The ratio $$\frac{N_1}{N_2}$$ will be equal to:
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A 5 V battery is connected across the points X and Y. Assume $$D_1$$ and $$D_2$$ to be normal silicon diodes. Find the current supplied by the battery if the +ve terminal of the battery is connected to point X.
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Given below are two statements:
Statement I: A speech signal of 2 kHz is used to modulate a carrier signal of 1 MHz. The bandwidth requirement for the signal is 4 kHz.
Statement II: The side band frequencies are 1002 kHz and 998 kHz.
In the light of the above statements, choose the correct answer from the options given below:
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The pitch of the screw gauge is 1 mm and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while 72$$^{nd}$$ division on circular scale coincides with the reference line. The radius of the wire is
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A small bob tied at one end of a thin string of length 1 m is describing a vertical circle so that the maximum and minimum tension in the string is in the ratio 5 : 1. The velocity of the bob at the highest position is ______ m s$$^{-1}$$. (Take $$g = 10$$ m s$$^{-2}$$)
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The potential energy (U) of a diatomic molecule is a function dependent on $$r$$ (interatomic distance) as $$U = \frac{\alpha}{r^{10}} - \frac{\beta}{r^5} - 3$$ where, $$\alpha$$ and $$\beta$$ are positive constants. The equilibrium distance between two atoms will be $$\left(\frac{2\alpha}{\beta}\right)^{\frac{a}{b}}$$, where $$a$$ = ______.
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In a certain thermodynamical process, the pressure of a gas depends on its volume as $$kV^3$$. The work done when the temperature changes from 100°C to 300°C will be $$xnR$$ where $$n$$ denotes number of moles of a gas, find $$x$$.
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A monoatomic gas of mass 4.0 u is kept in an insulated container. The container is moving with velocity 30 m s$$^{-1}$$. If the container is suddenly stopped then a change in temperature of the gas ($$R$$ = gas constant) is $$\frac{x}{3R}$$. Value of $$x$$ is,
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The electric field in a region is given by $$\vec{E} = \left(\frac{3}{5}E_0\hat{i} + \frac{4}{5}E_0\hat{j}\right)$$ N C$$^{-1}$$. The ratio of flux of reported field through the rectangular surface of area 0.2 m$$^2$$ (parallel to $$y-z$$ plane) to that of the surface of area 0.3 m$$^2$$ (parallel to $$x-z$$ plane) is $$a : b = a : 2$$, where $$a$$ = ______?[Here $$\hat{i},\hat{j}$$ and$$\hat{k}$$ are unit vectors along $$x,y$$ and $$z$$-axes respectively]
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512 identical drops of mercury are charged to a potential of 2 V each. The drops are joined to form a single drop. The potential of this drop is ______ V in Volt.
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In the given circuit of potentiometer, the potential difference $$E$$ across $$AB$$ (10 m length) is larger than $$E_1$$ and $$E_2$$ as well. For key $$K_1$$ (closed), the jockey is adjusted to touch the wire at point $$J_1$$ so that there is no deflection in the galvanometer. Now the first battery ($$E_1$$) is replaced by second battery ($$E_2$$) for working by making $$K_1$$ open and $$K_2$$ closed. The galvanometer gives then null deflection at $$J_2$$. The value of $$\frac{E_1}{E_2}$$ is $$\frac{a}{2}$$, where $$a$$ = ______.
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A coil of inductance 2 H having negligible resistance is connected to a source of supply whose voltage is given by $$V = 3t$$ volt. (where $$t$$ is in second). If the voltage is applied when $$t = 0$$, then the energy stored in the coil after 4 s in J.
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A transmitting station releases waves of wavelength 960 m. A capacitor of 2.56 $$\mu$$F is used in the resonant circuit. The self-inductance of coil necessary for resonance is $$x \times 10^{-8}$$ H. Find $$x$$.
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The same size images are formed by a convex lens when the object is placed at 20 cm or at 10 cm from the lens. The focal length of convex lens is ______
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