What will be the projection of vector $$\vec{A} = \hat{i} + \hat{j} + \hat{k}$$ on vector $$\vec{B} = \hat{i} + \hat{j}$$?
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What will be the projection of vector $$\vec{A} = \hat{i} + \hat{j} + \hat{k}$$ on vector $$\vec{B} = \hat{i} + \hat{j}$$?
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A bullet of 4 g mass is fired from a gun of mass 4 kg. If the bullet moves with the muzzle speed of 50 ms$$^1$$, the impulse imparted to the gun and velocity of recoil of gun are
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The motion of a mass on a spring, with spring constant $$K$$ is as shown in figure.
The equation of motion is given by, $$x(t) = A\sin\omega t + B\cos\omega t$$ with $$\omega = \sqrt{\frac{K}{m}}$$. Suppose that at time $$t = 0$$, the position of mass is $$x(0)$$ and velocity $$v(0)$$, then its displacement can also be represented as $$x(t) = C\cos(\omega t - \phi)$$, where $$C$$ and $$\phi$$ are:
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A porter lifts a heavy suitcase of mass 80 kg and at the destination lowers it down by a distance of 80 cm with a constant velocity. Calculate the work done by the porter in lowering the suitcase. (take $$g = 9.8$$ ms$$^{-2}$$)
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Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter. The correct statement for this situation is
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A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height $$h$$ is ___ s.
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What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature $$T$$?
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$$T_0$$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $$\frac{1}{16}$$ times of its initial value, the modified time period is
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An electric dipole is placed on $$x$$-axis in proximity to a line charge of linear charge density $$3.0 \times 10^{-6}$$ C m$$^{-1}$$. Line charge is placed on $$z$$-axis and positive and negative charge of dipole is at a distance of 10 mm and 12 mm from the origin respectively. If total force of 4 N is exerted on the dipole, find out the amount of positive or negative charge of the dipole.
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A Copper (Cu) rod of length 25 cm and cross-sectional area 3 mm$$^2$$ is joined with a similar Aluminium (Al) rod as shown in figure. Find the resistance of the combination between the ends A and B.
(Take resistivity of Copper = 1.7 $$\times 10^{-8}$$ $$\Omega$$m, Resistivity of aluminium = 2.6 $$\times 10^{-8}$$ $$\Omega$$m)
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Statement I : The ferromagnetic property depends on temperature. At high temperature, ferromagnet becomes paramagnet.
Statement II : At high temperature, the domain wall area of a ferromagnetic substance increases. In the light of the above statements, choose the most appropriate answer from the options given below:
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Choose the correct option.
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In a circuit consisting of a capacitance and a generator with alternating emf, $$E_g = E_{go}\sin\omega t$$, $$V_C$$ and $$I_C$$ are the voltage and current. Correct phasor diagram for such circuit is:
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Match List-I with List-II.
| List - I | List - II | |
|---|---|---|
| (a) $$\omega L > \frac{1}{\omega C}$$ | (i) | Current is in phase with emf |
| (b) $$\omega L = \frac{1}{\omega C}$$ | (ii) | Current lags behind the applied emf |
| (c) $$\omega L < \frac{1}{\omega C}$$ | (iii) | Maximum current occurs |
| (d) Resonant frequency | (iv) | Current leads the emf |
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Intensity of sunlight is observed as 0.092 Wm$$^{-2}$$ at a point in free space. What will be the peak value of magnetic field at that point? ($$\varepsilon_0 = 8.85 \times 10^{-12}$$ C$$^2$$ N$$^{-1}$$ m$$^{-2}$$)
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A ray of light passes from a denser medium to a rarer medium at an angle of incidence $$i$$. The reflected and refracted rays make an angle of 90$$^\circ$$ with each other. The angle of reflection and refraction are respectively $$r$$ and $$r'$$. The critical angle is given by,
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An electron of mass $$m_e$$ and a proton of mass $$m_P$$ are accelerated through the same potential difference. The ratio of the de-Broglie wavelength associated with the electron to that with the proton is:
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A nucleus with mass number 184 initially at rest emits an $$\alpha$$-particle. If the Q value of the reaction is 5.5 MeV, calculate the kinetic energy of the $$\alpha$$-particle.
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Consider a situation in which reverse biased current of a particular P-N junction increases when it is exposed to a light of wavelength $$\le$$ 621 nm. During this process, enhancement in carrier concentration takes place due to generation of hole-electron pairs. The value of band gap is nearly.
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What should be the height of transmitting antenna and the population covered if the television telecast is to cover a radius of 150 km? The average population density around the tower is 2000 km$$^{-2}$$ and the value of $$R_e = 6.5 \times 10^6$$ m.
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Three particles P, Q and R are moving along the vectors $$\vec{A} = \hat{i} + \hat{j}$$, $$\vec{B} = \hat{j} + \hat{k}$$ and $$\vec{C} = -\hat{i} + \hat{j}$$, respectively. They strike on a point and start to move in different directions. Now particle P is moving normal to the plane which contains vector $$\vec{A}$$ and $$\vec{B}$$. Similarly particle Q is moving normal to the plane which contains vector $$\vec{A}$$ and $$\vec{C}$$. The angle between the direction of motion of P and Q is $$\cos^{-1}\left(\dfrac{1}{\sqrt{x}}\right)$$. Then the value of $$x$$ is ___.
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Three students $$S_1$$, $$S_2$$ and $$S_3$$ perform an experiment for determining the acceleration due to gravity $$(g)$$ using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.
(Least count of length = 0.1 m, least count for time = 0.1 s)
If $$E_1$$, $$E_2$$ and $$E_3$$ are the percentage errors in $$g$$ for students 1, 2 and 3, respectively, then the minimum percentage error is obtained by student no ___.
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The position of the centre of mass of a uniform semi-circular wire of radius $$R$$ placed in $$x-y$$ plane with its centre at the origin and the line joining its ends as $$x$$-axis is given by $$\left(0, \frac{xR}{\pi}\right)$$. Then, the value of $$|x|$$ is ___.
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The centre of a wheel rolling on a plane surface moves with a speed $$v_0$$. A particle on the rim of the wheel at the same level as the centre will be moving at a speed $$\sqrt{x}v_0$$. Then the value of $$x$$ is ___.
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The area of cross-section of a railway track is 0.01 m$$^2$$. The temperature variation is 10 $$^\circ$$C. Coefficient of linear expansion of material of track is $$10^{-5}$$ $$^\circ$$C$$^{-1}$$. The energy stored per meter in the track is J m$$^{-1}$$. (Young's modulus of material of track is $$10^{11}$$ N m$$^{-2}$$)
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In 5 minutes, a body cools from 75 $$^\circ$$C to 65 $$^\circ$$C at room temperature of 25 $$^\circ$$C. The temperature of body at the end of next 5 minutes is ___ $$^\circ$$C.
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The total charge enclosed in an incremental volume of $$2 \times 10^{-9}$$ m$$^3$$ located at the origin is ___ nC, if electric flux density of its field is found as $${D} = e^{-x}\sin y \hat{i} - e^{-x}\cos y \hat{j} + 2z\hat{k}$$ C m$$^{-2}$$
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In an electric circuit, a call of certain emf provides a potential difference of 1.25 V across a load resistance of 5 $$\Omega$$. However, it provides a potential difference of 1 V across a load resistance of 2 $$\Omega$$. The emf of the cell is given by $$\frac{x}{10}$$ V. Then the value of $$x$$ is ___.
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A ray of light passing through a prism $$\left(\mu = \sqrt{3}\right)$$ suffers minimum deviation. It is found that the angle of incidence is double the angle of refraction within the prism. Then, the angle of prism is ___ (in degrees).
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In a given circuit diagram, a 5 V zener diode along with a series resistance is connected across a 50 V power supply. The minimum value of the resistance required, if the maximum zener current is 90 mA will be ___ $$\Omega$$.
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