Q. No.1. \(25\) m
2. \(50\) m
3. \(100\) m
4. \(200\) m
A particle starting from the point \((1,2)\) moves in a straight line in the XY-plane. Its coordinates at a later time are \((2,3).\) The path of the particle makes with \(x\)-axis an angle of:
| 1. | \(30^\circ\) | 2. | \(45^\circ\) |
| 3. | \(60^\circ\) | 4. | data is insufficient |
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) |
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. |
The \(x\) and \(y\) coordinates of the particle at any time are \(x = 5t-2t^2\) and \(y=10t\) respectively, where \(x\) and \(y\) are in metres and \(t\) is in seconds. The acceleration of the particle at \(t=2\) s is:
1. \(0\) m/s2
2. \(5\) m/s2
3. \(-4\) m/s2
4. \(-8\) m/s2
The position of a particle at time \(t\) is given by, \(x=3t^3\), \(y=2t^2+8t\), and \(z=6t-5\). The initial velocity of the particle is:
| 1. | \(20\) unit | 2. | \(10\) unit |
| 3. | \(5\) unit | 4. | \(13\) unit |
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
| 1. | \(\dfrac{{v}^{2}}{r}\) | 2. | \(a\) |
| 3. | \(\sqrt{{a}^{2}{+}{\left({\dfrac{{v}^{2}}{r}}\right)}^{2}}\) | 4. | \(\sqrt{a+\dfrac{v^{2}}{r}}\) |
| 1. | radial acceleration \(a_{r}=0;\) tangential acceleration \(a_{t}\neq 0.\) |
| 2. | radial acceleration \(a_{r}=0;\) tangential acceleration \(a_{t}=0.\) |
| 3. | radial acceleration \(a_{r}\neq 0;\) tangential acceleration \(a_{t}\neq 0.\) |
| 4. | radial acceleration \(a_{r}\neq 0;\) tangential acceleration \(a_{t}=0\) |
1. \(4m\pi \nu^2R\)
2. \(4\pi^2 \nu R\)
3. \(4\pi^2 \nu^2R\)
4. \(4\pi^2 \nu^2R^2\)
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