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:

A balloon is moving up in air vertically above a point \(A\) on the ground. When it is at a height \(h_1\), a girl standing at a distance \(d\) (point \(B\)) from \(A\) (See figure) sees it at an angle \(45^\circ\) with respect to the vertical. When the balloon climbs up a further height \(h_2\), it is seen at an angle \(60^\circ\) with respect to the vertical if the girl moves further by a distance \(2.464 d\) (point \(C\)). Then the height \(h_2\) is: (given \(\tan 30^{\circ}=0.5774\)):

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? 

In an octagon \(\text{ABCDEFGH}\) of equal side, what is the sum of \(\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{AD}}+\overrightarrow{\mathrm{AE}}+\overrightarrow{\mathrm{AF}}+\overrightarrow{\mathrm{AG}}+\overrightarrow{\mathrm{AH}}\) if \(\overrightarrow{\mathrm{AO}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}\)

  
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. \(180^\circ\)