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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)

Subtopic:  Vector Product |
 86%
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The angle between \(\mathrm{A}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\mathrm{B}=\hat{\mathbf{i}}-\hat{\mathbf{j}}\) is:
1. \(45^{\circ} \)
2. \(90^{\circ} \)
3. \(-45^{\circ} \)
4. \(180^{\circ}\)
Subtopic:  Scalar Product |
 78%
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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.
Subtopic:  Acceleration |
 57%
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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

Subtopic:  Circular Motion |
 76%
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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. \(t_{1} t_{2} = \frac{R}{g}\)

Subtopic:  Projectile Motion |
 74%
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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. \(1.5~\text{m/s}\)

Subtopic:  Relative Motion |
 84%
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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.
Subtopic:  Position & Displacement |
 87%
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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. \(A = B = C\)

Subtopic:  Position & Displacement |
 85%
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A stone tied to the end of a \(1\) m long string is whirled in a horizontal circle at a constant speed. If the stone makes \(22\) revolutions in \(44\) seconds, what is the magnitude and direction of acceleration of the stone?

1. \(\pi^2 ~\text{ms}^{-2} \) and direction along the tangent to the circle.
2. \(\pi^2 ~\text{ms}^{-2} \)  and direction along the radius towards the centre.
3. \(\frac{\pi^2}{4}~\text{ms}^{-2} \) and direction along the radius towards the centre.
4. \(\pi^2~\text{ms}^{-2} \) and direction along the radius away from the centre.

Subtopic:  Circular Motion |
 78%
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AIPMT - 2005
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Two boys are standing at the ends \(A\) and \(B\) of the ground where \(AB =a.\) The boy at \(B\) starts running in a direction perpendicular to \(AB\) with velocity \(v_1.\) The boy at \(A\) starts running simultaneously with velocity \(v\) and catches the other boy in a time \(t,\) where \(t\) is:

1. \(\frac{a}{\sqrt{v^2+v^2_1}}\) 2. \(\frac{a}{\sqrt{v^2-v^2_1}}\)
3. \(\frac{a}{v-v_1}\) 4. \(\frac{a}{v+v_1}\)
Subtopic:  Relative Motion |
 63%
From NCERT
AIPMT - 2005
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