Let $z = \cos \theta + i \sin \theta$. Then the value of $\sum_{m=1}^{15} \operatorname{Im}(z^{2m-1})$ at $\theta = 2^{\circ}$ is

If $x_n = \cos \left(\frac{\pi}{4^n}\right) + i \sin \left(\frac{\pi}{4^n}\right)$,then the product $x_1 x_2 x_3 \ldots \infty$ is equal to

$A$ conducting rod with resistance $r$ per unit length is moving inside a vertical magnetic field $\vec B$ at constant speed $v$ on two horizontal parallel ideal conductor rails. The ends of the rails are connected to a resistor $R$. The separation between the rails is $d$. The rod maintains a tilted angle $\theta$ to the rails. Find the external force required to keep the rod moving.

Orlon is a polymer of

For the stationary wave $y = 4 \sin \left( \frac{\pi x}{15} \right) \cos (96 \pi t)$,the distance between a node and the next antinode is

If $|\vec{A} \times \vec{B}| = \sqrt{3} \vec{A} \cdot \vec{B}$,then the value of $|\vec{A} + \vec{B}|$ is