Q. No.1. \(\vec{A}|\Delta \theta|\)
2. \(\mathrm{|\vec{B}| \Delta \theta-|\vec{A}|}\)
3. \(|\vec{\mathrm{A}}|\left(1-\frac{\Delta \theta^2}{2}\right)\)
4. \(0\)
1. \(\sqrt{\frac{9}{12}}\)
2. \(\sqrt{\frac{20}{9}}\)
3. \(\sqrt{\frac{5}{9}}\)
4. \(\sqrt{\frac{10}{9}}\)
Let \(\left|\vec{A_1}\right|=3,\left|\vec{A_2}\right|=5\) and \(\left|\vec{A_1}+\vec{A_2}\right|=5.\) The value of \(\left(\vec{2 A_1}+3 \vec{A_2}\right) \cdot\left(3 \vec{A_1}-2 \vec{A_2}\right) \) is:
| 1. | \(-112.5 \) | 2. | \(-106.5 \) |
| 3. | \(-118.5 \) | 4. | \(-99.5 \) |
A balloon is rising vertically above a fixed point \(A\) on the ground. A girl at point \(B,\) located \(d\) metres from \(A,\) observes the balloon at a height \(h_1\) and sees it at an angle of \(45^\circ\) with respect to the vertical. After the balloon ascends an additional height \(h_2,\) the girl walks \(2.464d\) metres further away from point \(A\) to reach point \(C.\) From this new position, she observes the balloon at an angle of \(60^\circ\) with the vertical. If \(\tan 30^{\circ}=0.5774,\) then what is the value of \(h_2\) in terms of \(d \text{?}\)
1. \(d\)
2. \(0.732d \)
3. \(1.464 d \)
4. \(0.464 d \)
A particle moving in the xy plane experiences a velocity-dependent force \(\vec F= k\left(v_y\hat i +v_x \hat j\right)\) where \(v_x\) and \(v_y\) are the \(x\) and \(y\) components of its velocity \(\vec{v}\). If \(\vec{a}\) is the acceleration of the particle, then which of the following statements is true for the particle?
| 1. | Quantity \(\vec{v}.\vec{a}\) is constant in time. |
| 2. | Kinetic energy of particle is constant in time. |
| 3. | Quantity \(\vec{v}\times\vec{a}\) is constant in time. |
| 4. | \(\vec{F}\) arises due to a magnetic field. |
In a regular octagon \({ABCDEFGH},\) all sides are equal in length. The position vector of point \(A\) with respect to the center \(O\) of the octagon is given by: \(\overrightarrow{{AO}}=2 \hat{{i}}+3 \hat{{j}}-4 \hat{{k}}.\)
What is the value of the vector sum: \(\overrightarrow{{AB}}+\overrightarrow{{AC}}+\overrightarrow{{AD}}+\overrightarrow{{AE}}+\overrightarrow{{AF}}+\overrightarrow{{AG}}+\overrightarrow{{AH}} ~\text{?}\)
1. \( -16 \hat{i}-24 \hat{j}+32 \hat{k} \)
2. \( 16 \hat{i}+24 \hat{j}-32 \hat{k} \)
3. \( 16 \hat{i}+24 \hat{j}+32 \hat{k} \)
4. \(16 \hat{i}-24 \hat{j}+32 \hat{k} \)
If \(\vec{P} \times \vec{Q}=\vec{Q} \times \vec{P},\) the angle between \(\vec{P}\) and \(\vec{Q}\) is \(\theta\) \((0^\circ<\theta<360^\circ),\) then the value of \(\theta\) will be:
1. \(30^\circ\)
2. \(60^\circ\)
3. \(90^\circ\)
4. \(180^\circ\)
\(\vec A\) is a vector quantity such that \(| \vec A|\) = non-zero constant. Which of the following expressions is true for \(\vec A\)?
1. \(\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{A}}=0 \)
2. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}<0 \)
3. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}=0 \)
4. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}>0\)
1. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|= |\hat{\mathrm{A}}-\hat{\mathrm{B}} \mid \tan \frac{\theta}{2} \)
2. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \tan \frac{\theta}{2} \)
3. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|=|\hat{\mathrm{A}}-\hat{\mathrm{B}}| \cos \frac{\theta}{2} \)
4. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \cos \frac{\theta}{2}\)
1. \(\sin ^{-1}\left(\frac{3}{5}\right) \)
2. \(\sin ^{-1}\left(\frac{1}{3}\right) \)
3. \(\cos ^{-1}\left(\frac{3}{5}\right) \)
4. \(\cos ^{-1}\left(\frac{1}{3}\right)\)
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