JEE-Advanced 2017
Q26
Allen
5. JEE-Advanced PYQs
MCQ
Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density $\rho$ remains uniform throughout the volume. The rate of fractional change in density $\left(\frac{1}{\rho}\frac{d\rho}{dt}\right)$ is constant. The velocity v of any point on the surface of the expanding sphere is proportional to :
A.
$R^{3}$
B.
$\frac{1}{R}$
C.
R
D.
$R^{2/3}$
JEE-Main 2015
Q27
Allen
3. JEE-Main PYQs
MCQ
Two stones are thrown up simultaneously from the edge of a cliff 240 m high with initial speed of 10 m/s and 40 m/s respectively. Which of the following graph best represents the time variation of relative position of the second stone with respect to the first?
(Assume stones do not rebound after hitting the ground and neglect air resistance, take $g = 10 \, m/s^{2}$ ) (The figure are schematic and not drawn to scale)
A.
B.
C.
D.
JEE-Advanced 2014
Q28
Allen
5. JEE-Advanced PYQs
Numerical
A rocket is moving in a gravity free space with a constant acceleration of $2 \, ms^{-2}$ along +x direction (see figure). The length of a chamber inside the rocket is 4m. A ball is thrown from the left end of the chamber in +x direction with a speed of $0.3 \, ms^{-1}$ relative to the rocket. At the same time, another ball is thrown in -x direction with a speed of $0.2 \, ms^{-1}$ from its right end relative to the rocket. The time in seconds when the two balls hit each other is
Correct Answer: 8 or 2
Explanation:
Ans. (8 or 2)
Assuming open chamber
$V _ {\mathrm{relative}} = 0. 5 \mathrm{m} / \mathrm{s}$
$S _ {\mathrm{relative}} = 4 m$
$\mathrm{Time} = \frac {4}{0 . 5} = 8 m / s$
Alternate:
Assuming closed chamber
In the frame of chamber :
Maximum displacement of ball A from its left end is $\frac{u_{A}^{2}}{2a} = \frac{(0.3)^{2}}{2(2)} = 0.0225 \, m$
This is negligible with respect to the length of chamber i.e. 4m. So, the collision will be very close to the left end.
Hence, time taken by ball B to reach left end will be given by
$S = u _ {B} t + \frac {1}{2} a t ^ {2}$
$4 = (0. 2) (t) + \frac {1}{2} (2) (t) ^ {2}$
Solving this, we get
$t \approx 2 s$
Assuming open chamber
$V _ {\mathrm{relative}} = 0. 5 \mathrm{m} / \mathrm{s}$
$S _ {\mathrm{relative}} = 4 m$
$\mathrm{Time} = \frac {4}{0 . 5} = 8 m / s$
Alternate:
Assuming closed chamber
In the frame of chamber :
Maximum displacement of ball A from its left end is $\frac{u_{A}^{2}}{2a} = \frac{(0.3)^{2}}{2(2)} = 0.0225 \, m$
This is negligible with respect to the length of chamber i.e. 4m. So, the collision will be very close to the left end.
Hence, time taken by ball B to reach left end will be given by
$S = u _ {B} t + \frac {1}{2} a t ^ {2}$
$4 = (0. 2) (t) + \frac {1}{2} (2) (t) ^ {2}$
Solving this, we get
$t \approx 2 s$
JEE-Advanced 2014
Q29
Allen
5. JEE-Advanced PYQs
Numerical
Airplanes A and B are flying with constant velocity in the same vertical plane at angles $30^{\circ}$ and $60^{\circ}$ with respect to the horizontal respectively as shown in figure. The speed of A is $100\sqrt{3}$ ms $^{-1}$ . At time t = 0 s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $t = t_{0}$ , A just escapes being hit by B, $t_{0}$ in seconds is
Correct Answer: 5
Explanation:
Ans. (5)
As observed from A, B moves perpendicular to line of motion of A. It means velocity of B along A is equal to velocity of A
$\begin{array}{l} {V _ {B} \cos 3 0 = 1 0 0 \sqrt {3}} \\ {V _ {B} = 2 0 0} \end{array}$
If A is observer A remains stationary therefore
$t = \frac {5 0 0}{V _ {B} \sin 3 0} = \frac {5 0 0}{1 0 0} = 5$
As observed from A, B moves perpendicular to line of motion of A. It means velocity of B along A is equal to velocity of A
$\begin{array}{l} {V _ {B} \cos 3 0 = 1 0 0 \sqrt {3}} \\ {V _ {B} = 2 0 0} \end{array}$
If A is observer A remains stationary therefore
$t = \frac {5 0 0}{V _ {B} \sin 3 0} = \frac {5 0 0}{1 0 0} = 5$
JEE-Main 2013
Q30
Allen
3. JEE-Main PYQs
MCQ
A projectile is given an initial velocity of $(\hat{t} + 2\hat{j})$ m/s, where $\hat{t}$ is along the ground and $\hat{j}$ is along the vertical. If $g = 10 \, m/s^{2}$ , the equation of its trajectory is :
A.
$y = x - 5x^{2}$
B.
$y = 2x - 5x^{2}$
C.
$4y = 2x - 5x^{2}$
D.
$4y = 2x - 25x^{2}$
Q31
Allen
1. JEE-Main Pattern
MCQ
Find ratio of average speed and magnitude of average velocity of tip of minute hand in 20 minutes:
A.
$\frac{2\pi}{3\sqrt{3}}$
B.
$\frac{2\pi}{3}$
C.
$\frac{3\sqrt{3}}{2\pi}$
D.
None
Q32
Allen
1. JEE-Main Pattern
MCQ
The motion of a particle is defined by the position vector $\vec{r} = A(\cos t + t\sin t)\hat{\imath} +A(\sin t - t\cos t)\hat{j}$
Where t is expressed in seconds. Determine the value of t for which positions vectors and acceleration vector are perpendicular.
A.
0
B.
1
C.
3
D.
5
Q33
Allen
1. JEE-Main Pattern
MCQ
Now the particle is moving only along the z-axis, and its position is given by, $(t^{2}-2t-3)\hat{k}$ . At what time does the particle stand still?
A.
1
B.
3
C.
5
D.
7
Q34
Allen
1. JEE-Main Pattern
MCQ
A particle is moving towards East with velocity $8 \, ms^{-1}$ and acceleration $4 \, ms^{-2}$ directed towards West. Find the distance travelled in 6 seconds.
A.
40 m
B.
24 m
C.
32 m
D.
16 m
Q35
Allen
1. JEE-Main Pattern
MCQ
A car moving at 160 km/h when crosses the marker-0, driver applies brake and reduces its speed uniformly to 40 km/h at marker-2. The markers are spaced at equal distances along the road as shown below.
At which part of the track would the car have been traveling at instantaneous speed of 100 km/h?
A.
At marker-1
B.
Between marker-0 and marker-1
C.
Between marker-1 and marker-2
D.
More information required to decide.
Q36
Allen
1. JEE-Main Pattern
MCQ
Two balls are projected simultaneously from the top of a tall building, one vertically upward and other vertical downwards with the same speed of 60 m/s. Then time interval between the balls striking the ground is :-
A.
6 sec.
B.
10 sec.
C.
12 sec.
D.
9 sec.
Q37
Allen
1. JEE-Main Pattern
MCQ
A particle moves on x-axis as per equation $x = (t^{3} - 9t^{2} + 15t + 2)m$ . Distance travelled by the particle between t = 0 and t = 5 s is
A.
25 m
B.
39 m
C.
23 m
D.
52 m
Q38
Allen
1. JEE-Main Pattern
MCQ
The $\mathrm{XI^{th}}$ class students of ALLEN designed a rocket. The rocket was launched from cycle stand of ALLEN straight up into the air. At $t = 0$ , the rocket is at $y = 0$ with $V_{y}(t = 0) = 0$ . The velocity of the rocket is given by: $V_{y} = (24t - 3t^{2})m / s$ for $0 \leq t \leq t_{b}$ where $t_{b}$ is the time at which fuel burns out. Vertically upward direction is taken as positive. $(g = 10m / s^2)$ The time taken by rocket to reach its maximum height is
A.
4 sec.
B.
8 sec.
C.
8.8 sec.
D.
9.6 sec.
Q39
Allen
1. JEE-Main Pattern
MCQ
Acceleration-velocity graph of a moving particle is shown in figure. The particle is
A.
speeding up at P
B.
speeding up at Q
C.
speeding up at S
D.
speeding down at R
Q40
Allen
1. JEE-Main Pattern
MCQ
A body of mass 5 kg starts from the origin with an initial velocity $\vec{u} = (30\hat{i} + 40\hat{j})ms^{-1}$ . If a constant force $(-6\hat{i} - 5\hat{j})N$ acts on the body, the time in which the y component of the velocity becomes zero, is:
A.
5s
B.
20 s
C.
40 s
D.
80 s
Q41
Allen
1. JEE-Main Pattern
MCQ
A boy throws a ball from shoulder height at an initial velocity of 30 m/s. Spending 4.8 s in air, the ball is caught by another boy as the same shoulder-height level. What is the angle of projection?
A.
$37^{\circ}$
B.
$30^{\circ}$
C.
$53^{\circ}$
D.
$60^{\circ}$
Q42
Allen
1. JEE-Main Pattern
MCQ
A particle is projected from the ground with velocity u at angle $\theta$ with horizontal. The horizontal range, maximum height and time of flight are R, H and T respectively.
They are given by $R = \frac{u^{2} \sin 2\theta}{g} H = \frac{u^{2} \sin^{2} \theta}{2g}$ and $T = \frac{2u \sin \theta}{g}$ Now keeping u fixed, $\theta$ is varied from $30^{\circ}$ to $60^{\circ}$ , then
A.
R will first increase then decrease, H will increase and T will decrease
B.
R will first increase then decrease while H and T both will increase
C.
R will decrease while H and T both will increase
D.
R will increase while H and T both will also increase
Q43
Allen
1. JEE-Main Pattern
MCQ
Suppose a player hits several baseballs. Which baseball will be in the air for the longest time?
A.
The one with the farthest range.
B.
The one which reaches maximum height.
C.
The one with the greatest initial velocity.
D.
The one leaving the bat at $45^{\circ}$ with respect to the ground.
Q44
Allen
1. JEE-Main Pattern
MCQ
A ball is hit by a batsman at an angle of $37^{\circ}$ as shown in figure. The man standing at P should run at what minimum velocity so that he catches the ball before it strikes the ground. Assume that height of man is negligible in comparison to maximum height of projectile.
A.
3 m/s
B.
5 m/s
C.
9 m/s
D.
12 m/s
Q45
Allen
1. JEE-Main Pattern
MCQ
A particle P is projected from a point on the surface of smooth inclined plane (see figure). Simultaneously another particle Q is released on the smooth inclined plane from the same position. P and Q collide after t = 4 s. The speed of projection of P is :-
A.
5 m/s
B.
10 m/s
C.
15 m/s
D.
20 m/s
Q46
Allen
1. JEE-Main Pattern
MCQ
A trolley is moving horizontally with a constant velocity of v with respect to earth. A man starts running from one end of the trolley with a velocity 1.5 v with respect to trolley. After reaching the opposite end, the man returns back and continues running with a velocity of 1.5 v w.r.t. the trolley in the backward direction. If the length of the trolley is L then the displacement of the man with respect to earth during the process will be :-
A.
$2.5L$
B.
$1.5L$
C.
$\frac{5L}{3}$
D.
$\frac{4L}{3}$
Q47
Allen
1. JEE-Main Pattern
MCQ
An elevator car (lift) is moving upward with uniform acceleration of $2 \, m/s^{2}$ . At the instant, when its velocity is $2 \, m/s$ upwards a ball is thrown upward from its floor. The ball strikes back the floor $2 \, s$ after its projection. Find the velocity of projection of the ball relative to the lift.
A.
$10 \, m/s \uparrow$
B.
$10 \, m/s \downarrow$
C.
$12 \, m/s \uparrow$
D.
$12 \, m/s \downarrow$
Q48
Allen
1. JEE-Main Pattern
MCQ
A flag is mounted on a car moving due North with velocity of 20 km/hr. Strong winds are blowing due East with velocity of 20 km/hr. The flag will point in direction :-
A.
East
B.
North-East
C.
South-East
D.
South-West
Q49
Allen
1. JEE-Main Pattern
MCQ
Wind is blowing in the north direction at speed of 2 m/s 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:
A.
2 m/s south
B.
2 m/s north
C.
4 m/s west
D.
4 m/s south
Q50
Allen
1. JEE-Main Pattern
MCQ
A man is crossing a river flowing with velocity of 5 m/s. He reaches a point directly across the river at a distance of 60 m in 5 sec. His velocity in still water should be :-
A.
$12m / s$
B.
$13m / s$
C.
$5m / s$
D.
$10m / s$
