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There are two vectors $\overrightarrow{A}=3\hat{i}+\hat{j}$ and $\overrightarrow{B}=\hat{j}+2\hat{k}.$ Find the magnitude of component of $\overrightarrow{A}$ along $\overrightarrow{B}$

A particle is placed at the lowest point of a smooth wire frame in the shape of a parabola, lying in the vertical $xy-$ plane having equation $x^2=5y$, where $(x,y)$ are in $\text{metres}$. After slight displacement, the particle is set free to move. Find the angular frequency of oscillation $\text{in rad/s}$.

[Take $g=10{\text{m/s}}^2)$]

From rest you suddenly stand up and start walking. But according to Newton's first law there has to be a force for change of state. What force made you move?

A locomotive of mass $m$ starts moving so that its velocity varies according to the law $v=k\sqrt{s}$ where $k$ is constant and $s$ is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first $t$ seconds after the beginning of motion.

All the graphs below are intended to represent the same motion. One of them does it incorrectly. Choose the incorrect one.

The translational kinetic energy of $1\text{mole}$ of an ideal gas at standard temperature is close to

An earth satellite has angular momentum $\alpha$ and its areal velocity about earth is $\beta$, then mass of satellite is

At a depth of $1000\text{m}$ in an ocean, what is the gauge pressure, when density of sea water is given as $1.03\times {10}^3{\text{kg/m}}^3$? (Take $g=10{\text{m/s}}^2$)

The density of a material in CGS system of units is $4{\text{g/cm}}^3$. In a system of units in which unit of length is $10\text{cm}$ and unit of mass is $100\text{g}$, the value of density of material will be

Two strings are attached to each other as shown in the figure. The linear mass density of the second string is four times that of the first string and the boundary between the two strings is at $x=0$. The wave equation of the incident wave at the boundary is $y_i=A_i\cos (k_{1}x-{\omega }_{1}t)$. Then, which of the following correctly represents the equation of the transmitted wave? Assume, tension in the two strings are same.

A block of mass $0.1\text{kg}$ is held against a wall applying a horizontal force of $5\text{N}$ on the block. If the coefficient of friction between the block and the wall is $0.5$, the magnitude of the frictional force acting the block is

Find the accelerations of the three blocks shown in the figure, if a horizontal force of $10\text{N}$ is applied on $2\text{kg}$ block.

(Take $g=10m/s^2$)

Given $\overrightarrow{P}=\overrightarrow{A}+\overrightarrow{B}$ and $\overrightarrow{Q}=\overrightarrow{A}-\overrightarrow{B}$. If the magnitudes of vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ are equal, what is the angle between vectors $\overrightarrow{A}$ and $\overrightarrow{B}$?

A ray incident at an angle of incidence ${60}^{\circ }$ enters a glass sphere of refractive index $\mu =\sqrt{3}$. This ray is reflected and refracted at the farther surface of the sphere. The angle between reflected and refracted rays at this surface is

A boy standing on a building of height $20\text{m}$ throws a stone with a velocity of $15{\text{ms}}^{-1}$. What should be the angle of projection so that the stone has a range of $30\text{m}$ ?

(Take $g=10m/s^2$)

Assume that the temperature remains essentially constant in the upper part of the atmosphere. Obtain an expression for the variation in pressure with height in the upper atmosphere. The mean molecular weight of air is M. Assume a variable $P_O$ for the pressure at a height h = 0, where h = 0 depends on the choice of origin (could very well be at the beginning of the stratosphere, for example).

From the top of a tower, a stone is thrown up and reaches the ground in time $t_1$ . A second stone is thrown down with the same speed and reaches the ground in time $t_2$. A third stone is released from rest and reaches the ground in time $t_3$ -

A small particle of mass $0.36\text{g}$ rests on a horizontal turntable at a distance $25\text{cm}$ from the axis of the spindle. The turntable is accelerated at a rate of $\alpha =\frac{1}{3}{\text{rad s}}^{-2}$. The frictional force that the table exerts on the particle $2\text{s}$ after the startup is:

(Here, $\mu$ in the options represents micro)

The process
of superimposing signal frequency (i.e.
audio wave) on the carrier

wave is known as [AIIMS 1987]

A magnetic field exists in a smoke chamber shown perpendicular to the plane of paper (either upwards or downwards). A beam coming out of a radioactive material [consisting of $\alpha ,{\beta }^{-},\gamma$ particles, proton $\left(H^{+}\right)$, Neutron] enters into the chamber through a fine hole, the four rays are named as $A,B,C$ and $D$.

[Assume $\alpha ,{\beta }^{-},$ Neutron and proton have nearly the same velocity.]

$\begin{array}{cccc}& \text{Column-I}& & \text{Column-II}\\ \text{(A)}& A& \text{(P)}& \alpha \\ \text{(B)}& B& \text{(Q)}& {\beta }^{-}\\ \text{(C)}& C& \text{(R)}& \gamma \\ \text{(D)}& D& \text{(S)}& H^{+}\\ & & \text{(T)}& \text{Neutron}\end{array}$

Which of the following option has the correct combination considering column-I and column-II.

Which of these are not real vectors?

If mass of $A=10kg$, coefficient of static friction ${\mu }_s=0.2$, coefficient of kinetic friction ${\mu }_k=0.2$ and the mass of $B\text{is}20kg$. Then find the acceleration of the masses.

A plane wall is $25\text{cm}$ thick with an area of $1{\text{m}}^2$ and has a thermal conductivity of $0.5\text{W/mK}$. If a temperature difference of ${60}^{\circ }\text{C}$ is maintained across it, then find the rate of heat flow.

Which of the following equations can be that of an SHM? Here, x represents displacement from mean position.

t represents time

k is a positive constant

v represents velocity

A projectile is fired with some velocity making certain angle with the horizontal. Which of the following graphs is the best representation for the total kinetic energy of a projectile $\left(\text{KE}\right)$ versus its horizontal displacement $\left(x\right)$?

Two lenses with powers; 2D & -4D are kept together. What is the effective focal length of the combination?

Consider a particle moving in simple harmonic motion according to the equation $x=2.0cos(50\Pi t+ta{n}^{-1}0.75)$

where x is in centimeter and t in second. The motion is started at t = 0. (a) When does the particle come to rest for the second time?

Force acting upon a charged particle kept between the plates of a charged condenser is F. If one plate of the condenser is removed, then the force acting on the same particle will become:

A particle is projected with a speed $V$ from a point O making an angle of ${30}^{\circ }$ with the vertical. At the same instant, a second particle is thrown vertically upwards with a speed $v$ from a point A. The two particles reach $H$, the highest point of the parabolic path of particle one simultaneously. Then ratio $V/v$ is

Some liquid is filled in a cylindrical vessel of radius $R$. Let $F_1$ be the force applied by the liquid on the bottom of the cylinder. Now the same liquid is poured into a vessel of uniform square cross-section of side $R$. Let $F_2$ be the force applied by the liquid on the bottom of this new vessel. Then:

An electron (mass $m$) with an initial velocity, $\overrightarrow{v}=v_0\hat{i}(v_0>0)$ is in an electric field, $\overrightarrow{E}=-E_0\hat{i}(E_0=\text{constant}>0)$ . Its de broglie wavelength at time $t$ is given by (${\lambda }_0$ is the initial wavelength of electron)

A man can row a boat with a speed of $4\frac{km}{hr}$ in still water. If he is crossing a river where the current is $2\frac{km}{hr},$ and the width of the river is 12 km, the time taken by him to cross with zero drift is

A boat, which has a speed of $5km/h$ in still water, crosses a river of width $1km$ along the shortest possible path in $15\text{minutes}$. What is the magnitude of velocity of the river water?

A capacitor of capacitance $C_1=1\mu F$ charged up to a voltage $V=110V$ is connected in parallel to the terminals of a circuit consisting of two uncharged capacitors connected in series and possessing the capacitance $C_2=2\mu F$ and $C_3=3.0\mu F$. Then what is the charge flowing through the wire

A dancer is rotating on a smooth horizontal floor with an angular momentum $L$. The dancer folds her hands so that her moment of inertia decreases by $25\%$. The new angular momentum is

A particle is aimed at a mark which is in the same horizontal plane as that of point of projection. If it falls $10m$ short of the target when it is projected with in elevation of ${75}^o$ and fall $10m$ ahead of the target when it is projected with an elevation of ${45}^o$. If $\theta$ is the correct elevation so that it exactly hits the target, find the value of sin $2\theta$ = (for the same velocity of projection)

A pulley mass arrangement is shown is the figure. A force of $50\text{N}$ is applied on the $2\text{kg}$ block. Find the acceleration of COM of the system.

A mass of $100\text{gram}$ is attached to the end of a rubber string $49\text{cm}$ long having an area of cross-section $20{\text{mm}}^2$. The string is whirled around horizontally at a constant speed of $40\text{rps}$ in a circle of radius $51\text{cm}$. Find the ratio of longitudinal stress and longitudinal strain in the rubber string.

Equation of a plane progressive wave is given by $y=0.5\text{sin}3\pi (t-\frac{x}{3})$. On reflection from a denser medium its amplitude becomes $\frac{2}{5}$ of the amplitude of the incident wave. The equation of the reflected wave is

A ball is moving with uniform speed in a circular path of radius $3\text{m}$ at a rate of $5\text{revolution/minute}$. Find the average speed of the ball.

In two pulley - particle figures (i) and (ii), the acceleration and force imparted by the string on the pulley and tension in the strings are, $(a_1,a_2),(N_1,N_2)$, and $(T_1,T_2)$ respectively. Ignoring friction in all contacting surfaces, which of the following relations is true?

For $V-T$ diagrams of two process $ab$ and $cd$ shown in figure for same gas, match the following two columns and find the correct option.

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(a) Work done}& \text{(p) is more in process ab}\\ \text{(b) Change in internal energy}& \text{(q) is more in process cd}\\ \text{(c) Heat exchange}& \text{(r) is same in both process}\\ \text{(d) Molar heat capacity}& \text{(s) can't say anything}\end{array}$

Three rectangular plate of dimension each $(l\times h)$ are arranged as shown in the figure.



If the masses of plates ($A,B,C$) are in the ratio $1:2:1$, then find the coordinates of the $\text{COM}$ of the system.