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The angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is ${37}^{\circ }$.If $|\overrightarrow{A}|=15$ and $|\overrightarrow{B}|=12$. Then $|\overrightarrow{A}\times \overrightarrow{B}|$ is

A body traveling along a straight line traversed first one third of the total distance with a velocity $4\text{m/s}$. The remaining part of the distance was covered with a velocity $2\text{m/s}$ for half the time and with velocity $6\text{m/s}$ for the other half of time. The average velocity over the whole time of motion is

In the figure shown, find the tension $T_1$, if all surfaces are smooth. Take $g=10{\text{m/s}}^2$.

A hoop of radius $r$ and mass $m$ rotating with an angular velocity ${\omega }_0$ is placed on a rough horizontal surface. The initial velocity of the center of the hoop is zero. What will be the velocity of the center of the hoop when it ceases to slip?

Two strings $A$ and $B$ made of same material are stretched by same tension. The radius of cross-section of string $A$ is double the radius of cross-section of string $B$. A transverse wave travels through string $A$ with speed $v_A$ and through string $B$ with speed $v_B$. The ratio $\frac{v_A}{v_B}$ is

The figure below shows a source of alternating voltage connected to a capacitor and a resistor. Which of the following phasor diagrams correctly describe the phase relationship between $I_c$ (current between the source and the capacitor) and $I_R$ (the current in the resistor)?

Two point sound sources of same frequency 'f' and amplitude 2A and 4A are situated at $x=0$ and $x=20\lambda$ respectively $(\lambda$ = wavelength of wave) on the $x$ - axis. These sources are vibrating in same phase. Four observers A, B, C and D are situated at $x_A=4\lambda ,x_B=8\lambda ,x_C=12\lambda ,x_D=16\lambda$ respectively. The value of intensity observed by another observer O situated exactly at midpoint of location of two sources is $I_0$. $(\text{Given}{I}_0=\frac{36A^{2}K}{100})$
List - I gives the above four observers.

$\begin{array}{cc}\text{List - I}& \text{List - II}\\ \text{(I) Observer A}& \left(P\right)\frac{9}{16}\\ \text{(II) Observer B}& \left(Q\right)\frac{49}{144}\\ \text{(III) Observer C}& \left(R\right)\frac{4}{9}\\ \text{(IV) Observer D}& \left(S\right)\frac{5}{16}\\ & \left(T\right)\frac{81}{64}\\ & \left(U\right)\frac{10}{36}\end{array}$

List - II contains the values of intensity observed by respective observers in unit of $K{A}^2$.
Match respective intensities observed by the observers.

The relation between internal energy $U$, pressure $P$ and volume $V$ of a gas in an adiabatic process is given as
$U=a+bPV$
where $a$ and $b$ are constants. What is the value of adiabatic constant $\left(\gamma \right)$?

Point charges $Q_1=5mC$ (millicoulomb) and $Q_2=6mC$ (millicoulomb) are at a separation of 100m from each other. What is the magnitude of force on any one of the charge?

If a particle of mass m is moving in a horizontal circle of radius r with a centripetal force (-$\frac{k}{r^2}$) , the total energy is

At what temperature will the speed of sound be double of its value at $0^{\circ }C$ ?

If the radius of curvature of the path of two particles of same masses are in the ratio 1 : 2, then in order to have constant centripetal force, their velocity, should be in the ratio of

A spherical volume contains a uniformly distributed charge of volume density $2.0\times {10}^{-4}C{m}^{-3}$. Find the strength of electric field at a point inside the volume at a distance $4.0cm$ from the centre.

A train moves towards a stationary observer with a speed of 34 m/s. The train sounds a whistle and its frequency registered by the observer is f₁. If the train’s speed is reduced to 17 m/s, the frequency registered is f₂. If the speed of sound is 340 m/s, then the ratio $\frac{f_1}{f_2}$ is

Pressure is exerted by a gas on the walls of the container because

The roots of the equation $3^{2x}-{10.3}^x+9=0$ are

A disc of radius R is cut out from a larger disc of radius 2R in such a way that the edge of the hole touches the edge of the disc.Locate the centre of mass of the residual disc.

When the image is of double size as the object, the magnification is

White light is used in a young's double slit experiment. Find the minimum order of the vioet fringe $(\lambda =400nm)$ which overlaps with a red fringe $(\lambda =700nm.)$

A man wants to cross the river to an exactly opposite point on the other bank. If he can row his boat with twice the velocity of the current, then at what angle to the current he must keep the boat pointed?

After adding no – volatile solute freezing point of water decreases to $-{0.186}^{\circ }C$. Calculate $\Delta T_b$ if $K_f$=1.86 K kg $mo{l}^{-1}$ and $K_b$ = 0.521 K kg $mo{l}^{-1}$

The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m, Guess the expression for its frequency from dimensional analysis.

A 220 V main supply is connected to a resistance of 100k $\Omega$ . The rms current is

Ball A of mass 'm' moving with speed v collides head on with stationary ball B of the same mass.If e is coefficient of restitution, which of the following statements are correct?

Figure shows four paths for a kicked football. Ignoring the effects of air on the flight, rank the paths accoding to initial horizontal velocity component, highest first

A man holds an umbrella at ${30}^{\circ }$ with the vertical to keep himself dry. He, then, runs at a speed of $10{\text{ms}}^{-1}$ and finds the rain drops to be hitting vertically. Speed of the rain drops w.r.t. the running man and w.r.t. earth are

A ceiling fan has a diameter (of the circle through the outer edges of the three blades) of 120 cm and rpm 1500 at full speed. Consider a particle of mass 1 g sticking at the outer end of a blade. How much force does it experience when the fan runs at full speed ? Who exerts this force on the particle? How much force does the particle exert on the blade along its surface?

Velocity of sound in air is 320 m/s. A pipe closed at one end has a length of 1m. Neglecting end correction, the air column in the pipe can resonate for the sound of frequency ;

$\int \frac{x{e}^x}{(x+1{)}^2}dx=$

A block of mass $m$ is placed on a surface with a vertical cross section given by $y=\frac{x^3}{6}$. If the coefficient of friction is $0.5$, the maximum height above the ground at which the block can be placed without slipping is

Two wires of same diameter and made of the same material are having the length $1m$ and $2m$ . If the force $F$ is applied on each end of the wires, the ratio of the work done in the two wires will be

A thin uniform rod AB of mass M and length L is hinged at one end A to the horizontal floor. Initially it stands vertically.It is allowed to fall freely in a vertical plane.If the linear speed of B when it hits the floor is $\sqrt{\frac{xg}{L}}$,what is x?

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ have magnitude $3$ each. If $\overrightarrow{A}\times \overrightarrow{B}=-5\hat{k}+2\hat{i}$, then the angle between $A$ and $B$ is

Moment of inertia of a uniform semicircular disc about an axis passing through center of mass and parallel to diameter as shown in figure is: [take${\pi }^2=10$]

One mole of an ideal gas is kept enclosed under a light piston of area $A={10}^{-2}{\text{m}}^2$ which is connected to a compressed spring of spring constant $100\text{N/m}$. The volume of the gas is $0.83{\text{m}}^3$ and its temperature is $100\text{K}$. The gas is heated so that it compresses the spring further by $0.1\text{m}$. The work done by the gas in the process is: [Take$R=8.3\text{J/K mol}$ and suppose there is no atmosphere].

Starting from rest, a partricle is moving along x-axis. The variation of force $F$ with position $x$ is shown in figure. Find the $KE$ of the particle when it has moved from $x=1\text{m}$ to $x=3\text{m}$.

A particle of mass $m$ moving in the $x-$ direction with speed $2v$ is hit by another particle of mass $2m$ moving in the $y-$ direction with speed $v$. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to

A wire carrying current $I$ has the shape as shown in adjoining figure. Linear parts of the wire are very long and parallel to $X$-axis while semicircular portion of radius $R$ is lying in $Y-Z$ plane. Magnetic field at point $O$ is

The specific heat of a metal at low temperatures varies according to $S=a{T}^3$ where $a$ is a constant and $T$ is absolute temperature. The heat energy needed to raise the temperature of unit mass of the metal from $T=1K$ to $T=2K$ is