A swimmer swims a distance d upstream in 4 s and swims an equal distance downstream in 2 s. The ratio of swimmer's speed in still water to the speed of river water will be:

1.  $\frac{6}{5}$

2.  $\frac{3}{1}$

3.  $\frac{5}{3}$

4.  $\frac{4}{3}$

A particle starts from (-1, 0) in the x-y plane along a straight path making an angle of 60° with the X-axis. Which of the following will not be the coordinate through which the particle will pass?

(1) (0, $\sqrt{3}$)

(2) (1, 2$\sqrt{3}$)

(3) (2, 3$\sqrt{3}$)

(4) (4, 4$\sqrt{3}$)

An aeroplane flies \(400\) m north and then \(300\) m west and then flies \(1200\) m upwards. Its net displacement is:

A projectile is thrown with an initial velocity (${\mathrm{v}}_{\mathrm{x}}\hat{\mathrm{i}} + {\mathrm{v}}_{\mathrm{y}}\hat{\mathrm{j}}$)m/s. Consider horizontal direction and vertical direction as x and y axes respectively. If the range of the projectile is double the maximum height, then $\frac{{\mathrm{v}}_{\mathrm{y}}}{{\mathrm{v}}_{\mathrm{x}}}$ is

(1) 4

(2) 3

(3) 2

(4) 1

A car is moving on a circular track at a speed of 20 m/s having a radius of 100 m. If it starts decelerating uniformly at the rate of $3 \mathrm{m}/{\mathrm{s}}^2$, the angle made by the resultant acceleration with the direction of its instantaneous velocity is:

(1) 37°

(2) 127°

(3) 53°

(4) 143°

An object is moving on a circular path of the radius r with constant speed v. The average acceleration of the object, after it has travelled three and a half rounds, is

1.  $\frac{2{\mathrm{v}}^2}{\mathrm{\pi r}}$

2.  $\frac{4{\mathrm{v}}^2}{3\mathrm{\pi r}}$

3.  $\frac{4{\mathrm{v}}^2}{7\mathrm{\pi r}}$

4.  $\frac{2{\mathrm{v}}^2}{7\mathrm{\pi r}}$

A swimmer can swim in still water at a speed of 5 m/s and takes ${\mathrm{t}}_1 \mathrm{and} {\mathrm{t}}_2$, time to cross a river in the shortest time and shortest path respectively. If the speed of the river increases, the time is taken to cross the river in the shortest time and the shortest path be $\mathrm{t}{\text{'}}_1 \mathrm{and} \mathrm{t}{\text{'}}_2$ respectively. Then

1.  ${\mathrm{t}}_1 < {\mathrm{t}}_1^{\text{'}}, {\mathrm{t}}_2 < {\mathrm{t}}_2^{\text{'}}$

2.  ${\mathrm{t}}_1 > {\mathrm{t}}_1^{\text{'}}, {\mathrm{t}}_2 > {\mathrm{t}}_2^{\text{'}}$

3.  ${\mathrm{t}}_1 = {\mathrm{t}}_1^{\text{'}}, {\mathrm{t}}_2 < {\mathrm{t}}_2^{\text{'}}$

4.  ${\mathrm{t}}_1 = {\mathrm{t}}_1^{\text{'}}, {\mathrm{t}}_2 > {\mathrm{t}}_2^{\text{'}}$

A sprinkler is deployed to irrigate the garden. The speed of water-jet from the sprinkler is u. The maximum area which can be irrigated by the sprinkler is

1.  $\frac{{\mathrm{\pi }}^2{\mathrm{u}}^2}{\mathrm{g}}$

2.  $\frac{{\mathrm{\pi u}}^2}{\mathrm{g}}$

3.  $\frac{{\mathrm{\pi u}}^2}{{\mathrm{g}}^2}$

4.  $\frac{{\mathrm{\pi u}}^4}{{\mathrm{g}}^2}$

Which of the following is an appropriate expression for the radius of curvature of a projectile at the highest point? (R $\to$ Range of projectile)

(1) $\frac{\mathrm{R}}{2} \cot \mathrm{\theta }$

(2) $\frac{\mathrm{R}}{2} \tan \mathrm{\theta }$

(3) 2R cot $\mathrm{\theta }$

(4) 2R tan $\mathrm{\theta }$

The coordinates of a particle moving in the x-y plane vary with time according to the following relation:

$\mathrm{x} = \frac{\mathrm{B}}{\mathrm{t}}$, $\mathrm{y} = \mathrm{A}\sqrt{\mathrm{t}}$.

The locus of the particle is:

(1) Parabola

(2) Circle

(3) Eclipse

(4) Hyperbola