Question 1 :
A man swims from a point {tex} A {/tex} on one bank of width {tex} 100 \mathrm { m } {/tex}. When he swims perpendicular to the water current, he reaches the other bank {tex} 50 \mathrm { m } {/tex} downstream. The angle to the bank at which he should swim, to reach the directly opposite point {tex} B {/tex} on the other bank is<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f62820c6cd1b22465389"><br>

Question 2 :
A river flows with a speed more than the maximum speed with which a person can swim in still water. He intends to cross the river by the shortest possible path (i.e, he wants to reach the point on the opposite bank which directly opposite to the starting point). Which of the following is correct?

Question 3 :
A ball is projected from the ground at angle {tex} \theta {/tex} with the horizontal. After 1s it is moving at angle {tex} 45 ^ { \circ } {/tex} with the horizontal and after 2s it is moving horizontally. What is the velocity of projection of the ball?

Question 4 :
Consider a disc rotating in the horizontal plane with a constant angular speed {tex} \omega {/tex} about its centre {tex} \mathrm { O } {/tex}. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles {tex} P {/tex} and {tex} Q {/tex} are simultaneously projected at an angle towards {tex} R {/tex}. The velocity of projection is in the {tex} y {/tex} - {tex} z {/tex} plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed {tex} 1 / 8 {/tex} rotation, (ii) their<br>range is less than half the disc radius, and (iii) {tex} \omega {/tex} remains constant throughout. Then<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5f2cfd22bf25265a28d355fd">

Question 5 :
Two particles {tex} P {/tex} and {tex} Q {/tex} are projected simultaneously away from each other from a point {tex} A {/tex} as shown in figure. The velocity of {tex} P {/tex} relative to {tex} Q {/tex} in {tex} m s ^ { - 1 } {/tex} at the instant when the motion of {tex} P {/tex} is horizontal is<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f47b20c6cd1b2246530e"><br>

Question 6 :
A truck is moving with a constant velocity of {tex} 54 \mathrm { kmh } ^ { - 1 } {/tex}. In which direction (angle with the direction of motion of truck) should a stone be projected up with a velocity of {tex} 20 \mathrm { ms } ^ { - 1 } , {/tex} from the floor of the truck, as to appear at right angles to the truck, for a person standing on earth?<br>

Question 7 :
A particle is projected with a certain velocity at an angle {tex} \alpha {/tex} above the horizontal from the foot of an inclined plane of inclination {tex} 30 ^ { \circ } . {/tex} If the particle strikes the plane normally, then {tex} \alpha {/tex} is equal to

Question 8 :
In Fig, the angle of inclination of the inclined plane is {tex} 30 ^ { \circ } . {/tex} Find the horizontal velocity {tex} V _ { 0 } {/tex} so that the particle hits the inclined plane perpendicularly<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f4903df3381b1c3148a7"><br>

Question 9 :
Two trains having constant speeds of {tex} 40 \mathrm { kmh } ^ { - 1 } {/tex} and {tex} 60 \mathrm { kmh } ^ { - 1 } {/tex} respectively are heading towards each other on the same straight track<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f5078e34721b52b492ab"><br>A bird, when can fly with a constant speed of {tex} 30 \mathrm { kmh } ^ { - 1 } {/tex} flies off from one train when they are {tex} 60 \mathrm { km } {/tex} apart and heads directly for the other train. On reaching the other train, it flies back directly to the first and so forth. What is the total distance travelled by the bird before the two trains crash?

Question 10 :
Two boys {tex} P {/tex} and {tex} Q {/tex} are playing on a river bank. {tex} P {/tex} plans to swim across the river directly and come back. {tex} Q {/tex} plans to swim downstream by a length equal to the width of the river and then come back. Both of them<br>bet each other, claiming that the boy succeeding in less time will win. Assuming the swimming rate of both {tex} P {/tex} and {tex} Q {/tex} to the same, it can be concluded that<br>

Question 11 :
The horizontal range and maximum height attained by a projectile are {tex}R{/tex}and {tex} H , {/tex} respectively. If a constant horizontal acceleration {tex} a = \mathrm { g } / 4 {/tex} is imparted to the projectile due to wind, then its horizontal range and maximum height will be

Question 12 :
A man holds an umbrella at {tex} 30 ^ { \circ } {/tex} with the vertical to keep himself dry. He, then, runs at a speed of {tex} 10 \mathrm { ms } ^ { - 1 } {/tex}, and find the rain drops to be hitting vertically. Study the following statements and find the correct options<br>I. Velocity of rain w.r.t. Earth is {tex} 20 \mathrm { ms } ^ { - 1 } {/tex} <br>II. Velocity of rain w.r.t. man is {tex} 10 \sqrt { 3 } \mathrm { ms } ^ { - 1 } {/tex} <br>III. Velocity of rain w.r.t. Earth is {tex} 30 \mathrm { ms } ^ { - 1 } {/tex} <br>IV. Velocity of rain w.r.t. man is {tex} 10 \sqrt { 2 } \mathrm { ms } ^ { - 1 } {/tex}<br>

Question 13 :
A tube of length {tex} L {/tex} is filled completely with an incompressible liquid of mass {tex} M {/tex} and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity {tex} \omega {/tex}. The force exerted by the liquid at the other end is<br>

Question 14 :
A particle is ejected from the tube at {tex} A {/tex} with a velocity {tex} v {/tex} at an angle {tex} \theta {/tex} with the vertical {tex} y {/tex} -axis. A strong horizontal wind gives the particle a constant horizontal acceleration a in the {tex} x {/tex} -dircction. If the particle strikes the ground at a point directly under its released position and the downward {tex} y {/tex} -acceleration is taken as {tex} g {/tex} then<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5f2beeca36e8973c718889ff"><br>

Question 15 :
An elevator is moving upwards with constant acceleration. The broken curve shows the position {tex} y {/tex} of the ceiling of the elevator as a function of time {tex} t {/tex}. A bolt breaks loose and drops from the ceiling<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f74c3df3381b1c314984"><br>Which curve best represents the position of the bolt as a function of time?

Question 16 :
A piece of wire is bent in the shape of a parabola {tex} y = k x ^ { 2 }{/tex} ( y -axis vertical) with a bead of mass {tex} m {/tex} on it. The bead can side on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the {tex} x {/tex} -axis with a constant acceleration {tex} a {/tex}. The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire, from the {tex} y {/tex} - axis is<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f4403df3381b1c31488c">

Question 17 :
Consider a disc rotating in the horizontal plane with a constant angular speed m about its centre O. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles P and Q are simultaneously projected at an angle towards R The velocity of projection is in the y-z plane and is same for both pebbles With respect to the disc. Assume that (i) they land back on the before the disc has completed 1/8 rotation, (ii) their range is less than half the disc radius, and (iii) {tex} \omega {/tex} remains constant throughout. Then<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e2049b53c34554a24698fc6">

Question 18 :
A ball rolls off the top of a staircase with a horizontal velocity {tex} u \mathrm { ms } ^ { - 1 } {/tex}. If the steps are {tex} h {/tex} metre high and {tex} b {/tex} metre wide, the ball will hit the edge of the nth step, if

Question 19 :
An object moves along the {tex} x {/tex} -axis. Its {tex} x {/tex} -coordinates is given as a function of time as {tex} x = 7 t - 3 t ^ { 2 } \mathrm { m } , {/tex} where {tex} x {/tex} is in metres and {tex} t {/tex} is in seconds. Its average speed over the interval {tex} t = 0 {/tex} to {tex} t = 4 \mathrm { s } {/tex} is<br>

Question 20 :
A motor cyclist is trying to jump across a path as shown in fig by driving horizontally off a cliff {tex} A {/tex} at a speed of {tex} 5 \mathrm { ms } ^ { - 1 } {/tex}. Ignore air resistance and take {tex} \mathrm { g } = 10 \mathrm { ms } ^ { - 2 } {/tex}. The speed with which he touches peak {tex} B {/tex} is<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f4f420c6cd1b2246532d"><br>

Question 21 :
A body is projected up along a smooth inclined plane with velocity {tex} u {/tex} from the point {tex} A {/tex} as shown in Fig. The angle of inclination is {tex} 45 ^ { \circ } {/tex} and the top is connected to a well of diameter {tex} 40 \mathrm { m } {/tex}. If the body just manages to cross the well, what is the value of {tex} u {/tex} ? Length of inclined plane is {tex} 20 \sqrt { 2 } \mathrm { m } {/tex}<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f6888e34721b52b49328"><br>

Question 22 :
The coordinates of a particle moving in a plane are given by {tex} x ( t ) = a \cos ( \mathrm { pt } ) {/tex} and {tex} y ( t ) = b \sin ( \mathrm { pt } ) {/tex} where {tex} a , b ( < a ) {/tex} and {tex} p {/tex} are positive constants of appropriate dimensions. Then

Question 23 :
A train is moving slowly on a straight track with a constant speed of $2$ $ms^{-1}$. A passenger in that train starts walking at a steady speed of $2$ $ms^{-1}$ to the back of the train in the opposite direction of the motion of the train. So to an observer standing on the platform directly in front of that passenger, the velocity of the passenger appears to be?

Question 24 :
Rain appears to fall vertically to a man walking at {tex} 3 \mathrm { kmh } ^ { - 1 } {/tex}, but when he changes his speed to double, the rain appears to fall at {tex} 45 ^ { \circ } {/tex} with vertical. Study the following statements and find which of them are correct<br>1. Velocity of rain is {tex} 2 \sqrt { 3 } \mathrm { kmh } ^ { - 1 } {/tex}<br> 2. The angle of fall of rain (with vertical) is {tex} \theta = \tan ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right) {/tex}<br>3. The angle of fall of rain (with vertical) is {tex} \theta = \sin ^ { - 1 } \left( \frac { 1 } { \sqrt { 2 } } \right) {/tex} <br>4. Velocity of rain is {tex} 3 \sqrt { 2 } \mathrm { kmh } ^ { - 1 } {/tex}

Question 25 :
Three vectors {tex} \overrightarrow { \mathrm { P } } , \overrightarrow { \mathrm { Q } } {/tex} and {tex} \overrightarrow { \mathrm { R } } {/tex} are shown in the figure. Let {tex} \mathrm { S } {/tex} be anypoint on the vector {tex} \overrightarrow { \mathrm { R } } {/tex}. The distance between the points {tex} \mathrm { P } {/tex} and {tex} \mathrm { S } {/tex} is {tex} \mathrm { b } | \overrightarrow { \mathrm { R } } | {/tex}. The general relation among vectors {tex} \overrightarrow { \mathrm { P } } , \overrightarrow { \mathrm { Q } } {/tex} and {tex} \overrightarrow { \mathrm { S } } {/tex} is<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e2049bb3c34554a24698fcb">

Question 26 :
A projectile is thrown in the upward direction making an angle of {tex} 60 ^ { \circ } {/tex} with the horizontal direction with a velocity of {tex} 150 \mathrm { ms } ^ { - 1 } {/tex}. Then the time after which its inclination with the horizontal is {tex} 45 ^ { \circ } {/tex} is

Question 27 :
The displacement {tex} ( x ) {/tex} of a particle depends on time {tex} ( t ) {/tex} as {tex} x = \alpha t ^ { 2 } - \beta t ^ { 3 } {/tex}

Question 28 :
Wind is blowing in the north direction at speed of {tex} 2 \mathrm { ms } ^ { - 1 } {/tex}, which causes the rain to fall at some angle with the vertical. With what velocity should a cyclist drive so that the rain appears vertical to him:

Question 29 :
The acceleration of a particle which moves along the positive {tex} x {/tex} -axis varies with its position as shown in Fig. If the velocity of the particle is {tex} 0.8 \mathrm { ms } ^ { - 1 } {/tex} at {tex}x = 0{/tex} , then velocity of the particle at {tex} x = 1.4 \mathrm { m } \text { is (in } \mathrm { ms } ^ { - 1 }) {/tex}<br><img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e37f5623df3381b1c3148e6"><br>

Question 30 :
The friction of air causes a vertical retardation equal to {tex} 10 \% {/tex} of the acceleration due to gravity (take {tex} \mathrm { g } = 10 \mathrm { ms } ^ { - 2 } {/tex} ). The maximum height will be decreased by