Q. No.A cyclist starts from the center \(\mathrm{O}\) of a circular park of radius \(1\) km, reaches the edge \(\mathrm{P}\) of the park, then cycles along the circumference, and returns to the center along \(\mathrm{QO}\) as shown in the figure. If the round trip takes \(10\) min, then the average speed of the cyclist is:
1. \(22.42\) km/h
2. \(23.32\) km/h
3. \(21.42\) km/h
4. \(27.12\) km/h
2. \(23.32\) km/h
3. \(21.42\) km/h
4. \(27.12\) km/h
Three girls skating on a circular ice ground of radius \(200\) m start from a point \(P\) on the edge of the ground and reach a point \(Q\) diametrically opposite to \(P\) following different paths as shown in the figure. The correct relationship among the magnitude of the displacement vector for three girls will be:
1. \(A > B > C\)
2. \(C > A > B\)
3. \(B > A > C\)
4. \(A = B = C\)
Consider the motion of the tip of the second hand of a clock. In one minute (assuming \(R\) to be the length of the second hand), its:
| 1. | displacement is \(2\pi R\) |
| 2. | distance covered is \(2R\) |
| 3. | displacement is zero. |
| 4. | distance covered is zero. |
A person reaches a point directly opposite on the other bank of a flowing river while swimming at a speed of \(5~\text{m/s}\)at an angle of \(120^\circ\) with the flow. The speed of the flow must be:
1. \(2.5~\text{m/s}\)
2. \(3~\text{m/s}\)
3. \(4~\text{m/s}\)
4. \(1.5~\text{m/s}\)
When a particle is projected at some angle to the horizontal, it has a range \(R\) and time of flight \(t_1\). If the same particle is projected with the same speed at some other angle to have the same range, its time of flight is \(t_2\), then:
1. \(t_{1} + t_{2} = \frac{2 R}{g}\)
2. \(t_{1} - t_{2} = \frac{R}{g}\)
3. \(t_{1} t_{2} = \frac{2 R}{g}\)
4. \(t_{1} t_{2} = \frac{R}{g}\)
A car is moving at a speed of \(40\) m/s on a circular track of radius \(400\) m. This speed is increasing at the rate of \(3\) m/s2. The acceleration of the car is:
1. \(4\) m/s2
2. \(7\) m/s2
3. \(5\) m/s2
4. \(3\) m/s2
In a two-dimensional motion, instantaneous speed \(v_0\) is a positive constant. Then, which of the following is necessarily true?
| 1. | the average velocity is not zero at any time. |
| 2. | average acceleration must always vanish. |
| 3. | displacements in equal time intervals are equal. |
| 4. | equal path lengths are traversed in equal intervals. |
A cat is situated at point \(A\) (\(0,3,4\)) and a rat is situated at point \(B\) (\(5,3,-8\)). The cat is free to move but the rat is always at rest. The minimum distance travelled by the cat to catch the rat is:
1. \(5\) unit
2. \(12\) unit
3. \(13\) unit
4. \(17\) unit
If \(\left| \vec{A}\right|\) = \(2\) and \(\left| \vec{B}\right|\) = \(4,\) then match the relations in column-I with the angle \(\theta\) between \(\vec{A}\) and \(\vec{B}\) in column-II.
| Column-I | Column-II |
| (A) \(\left| \vec{A}\times \vec{B}\right|\) \(=0\) | (p) \(\theta=30^\circ\) |
| (B)\(\left| \vec{A}\times \vec{B}\right|\)\(=8\) | (q) \(\theta=45^\circ\) |
| (C) \(\left| \vec{A}\times \vec{B}\right|\) \(=4\) | (r) \(\theta=90^\circ\) |
| (D) \(\left| \vec{A}\times \vec{B}\right|\) \(=4\sqrt2\) | (s) \(\theta=0^\circ\) |
| 1. | A(s), B(r), C(q), D(p) |
| 2. | A(s), B(p), C(r), D(q) |
| 3. | A(s), B(p), C(q), D(r) |
| 4. | A(s), B(r), C(p), D(q) |
1. \(45^{\circ} \)
2. \(90^{\circ} \)
3. \(-45^{\circ} \)
4. \(180^{\circ}\)
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