Two blocks ($m = 0.5 \mathrm {~kg}$ and $M = 4.5 \mathrm {~kg}$) are arranged on a horizontal frictionless table as shown in the figure. The coefficient of static friction between the two blocks is $\frac { 3 } { 7 }$. Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is $N$. (Round off to the Nearest Integer) [Take $g$ as $9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$]

A person is swimming with a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $120 ^ { \circ }$ with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is $x \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The value of $x$ to the nearest integer is $\_\_\_\_$.

In a spring gun having spring constant $100 \mathrm {~N} \mathrm {~m} ^ { - 1 }$ a small ball $B$ of mass 100 g is put in its barrel (as shown in figure) by compressing the spring through 0.05 m . There should be a box placed at a distance $d$ on the ground so that the ball falls in it. If the ball leaves the gun horizontally at a height of 2 m above the ground. The value of $d$ is $\underline{\hspace{1cm}}$ m.
$$\left( g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } \right)$$

A swimmer wants to cross a river from point $A$ to point $B$. Line AB makes an angle of $30 ^ { \circ }$ with the flow of the river. The magnitude of the velocity of the swimmer is the same as that of the river. The angle $\theta$ with the line $AB$ should be $\_\_\_\_$ ${ } ^ { \circ }$, so that the swimmer reaches point $B$.

The average translational kinetic energy of $N _ { 2 }$ gas molecules at $\_\_\_\_$ ${ } ^ { \circ } \mathrm { C }$ becomes equal to the K.E. of an electron accelerated from rest through a potential difference of 0.1 volt. (Given $k _ { B } = 1.38 \times 10 ^ { - 23 } \mathrm {~J} \mathrm {~K} ^ { - 1 }$) (Fill the nearest integer).

A boy pushes a box of mass 2 kg with a force $\vec { F } = ( 20 \hat { \mathrm { i } } + 10 \hat { \mathrm { j } } ) \mathrm { N }$ on a frictionless surface. If the box was initially at rest, then $\_\_\_\_$ m is displacement along the $x$-axis after 10 s

A body of mass 2 kg moving with a speed of $4\mathrm{~m~s}^{-1}$ makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is $\frac{x}{10}$ m/s. Find the value of $x$.

Suppose two planets (spherical in shape) of radii $R$ and $2R$, but mass $M$ and $9M$ respectively have a centre to centre separation $8R$ as shown in the figure. A satellite of mass $m$ is projected from the surface of the planet of mass $M$ directly towards the centre of the second planet. The minimum speed $v$ required for the satellite to reach the surface of the second planet is $\sqrt { \frac { a } { 7 } \frac { G M } { R } }$, then the value of $a$ is. [Given: The two planets are fixed in their position]

The acceleration due to gravity is found up to an accuracy of $4\%$ on a planet. The energy supplied to a simple pendulum of known mass $m$ to undertake oscillations of time period $T$ is being estimated. If time period is measured to an accuracy of $3\%$, the accuracy to which $E$ is known as $\_\_\_\_$ \%

If the velocity of a body related to displacement $x$ is given by $v = \sqrt{5000 + 24x}\mathrm{~m~s}^{-1}$, then the acceleration of the body is $\_\_\_\_$ $\mathrm{m~s}^{-2}$.

1 mole of rigid diatomic gas performs a work of $\frac { Q } { 5 }$ when heat $Q$ is supplied to it. The molar heat capacity of the gas during this transformation is $\frac { x R } { 8 }$. The value of $x$ is [ $R$ universal gas constant]

A swimmer can swim with velocity of $12 \mathrm {~km} / \mathrm { h }$ in still water. Water flowing in a river has velocity $6 \mathrm {~km} / \mathrm { h }$. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his starting point is $\_\_\_\_$. (Round off to the Nearest Integer) (find the angle in degree)

The angular speed of truck wheel is increased from 900 rpm to 2460 rpm in 26 seconds. The number of revolutions by the truck engine during this time is $\_\_\_\_$. (Assuming the acceleration to be uniform).

A bullet of mass 0.1 kg is fired on a wooden block to pierce through it, but it stops after moving a distance of 50 cm into it. If the velocity of the bullet before hitting the wood is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and, it slows down with uniform deceleration, then the magnitude of effective retarding force on the bullet is $x \mathrm {~N}$. The value of $x$ to the nearest integer is

A body having specific charge $8 \mu \mathrm { C } \mathrm { g } ^ { - 1 }$ is resting on a frictionless plane at a distance 10 cm from the wall (as shown in the figure). It starts moving towards the wall when a uniform electric field of $100 \mathrm {~V} \mathrm {~m} ^ { - 1 }$ is applied horizontally towards the wall. If the collision of the body with the wall is perfectly elastic, then the time period of the motion will be $\_\_\_\_$ s.

A small block slides down from the top of hemisphere of radius $R = 3 \mathrm {~m}$ as shown in the figure. The height $h$ at which the block will lose contact with the surface of the sphere is $\_\_\_\_$ m. (Assume there is no friction between the block and the hemisphere)

An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of $40,000 \mathrm {~kg}$ is moving with a speed of $72 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by 1.0 m. If $90 \%$ of energy of the wagon is lost due to friction, the spring constant is $\_\_\_\_$ $\times 10 ^ { 5 } \mathrm {~N} \mathrm {~m} ^ { - 1 }$.

A person standing on a spring balance inside a stationary lift measures 60 kg. The weight of that person if the lift descends with uniform downward acceleration of $1.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ will be $\_\_\_\_$ $\mathrm { N } . \left[ g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } \right]$

A particle of mass $m$ is moving in time $t$ on a trajectory given by, $$\vec{r} = 10\alpha t^{2}\hat{\mathrm{i}} + 5\beta(t - 5)\hat{\mathrm{j}}$$ where $\alpha$ and $\beta$ are dimensional constants. The angular momentum of the particle becomes the same as it was for $t = 0$ at time $t =$ \_\_\_\_ seconds.

A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section $10 ^ { - 6 } \mathrm {~m} ^ { 2 }$ stretched by an amount 0.04 m. The velocity of the projected stone is $\mathrm { m } \mathrm { s} ^ { - 1 }$. (Young's modulus of rubber $= 0.5 \times 10 ^ { 9 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$)

The coefficient of static friction between two blocks is 0.5 and the table is smooth. The maximum horizontal force that can be applied to move the blocks together is $\_\_\_\_$ N (take $g = 10\text{ m s}^{-2}$)

The volume $V$ of a given mass of monoatomic gas changes with temperature $T$ according to the relation $V = K T ^ { \frac { 2 } { 3 } }$. The workdone when temperature changes by 90 K will be $x R$. The value of $x$ is [ $R$ universal gas constant]

A force $\vec { F } = 4 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } + 4 \widehat { \mathrm { k } }$ is applied on an intersection point of $x = 2$ plane and $x$-axis. The magnitude of torque of this force about a point $( 2,3,4 )$ is $\_\_\_\_$. (Round off to the Nearest Integer)

The following bodies,
(1) a ring
(2) a disc
(3) a solid cylinder
(4) a solid sphere, of same mass $m$ and radius $R$ are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is [Mark the body as per their respective numbering given in the question]

As shown in the figure, a particle of mass 10 kg is placed at a point $A$. When the particle is slightly displaced to its right, it starts moving and reaches the point $B$. The speed of the particle at $B$ is $x \mathrm {~m} \mathrm {~s} ^ { - 1 }$. (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$) The value of $x$ to the nearest integer is [Figure]