The relation "congruence modulo $m$" is

$A$ $Ge$ specimen is doped with $Al$. The concentration of acceptor atoms is $\sim 10^{21} \text{ atoms}/m^3$. Given that the intrinsic concentration of electron-hole pairs is $\sim 10^{19} /m^3$,the concentration of electrons in the specimen is:

The equations of the circle which pass through the origin and make intercepts of lengths $4$ and $8$ on the $x$ and $y$-axes respectively are

If $A = \{2, 4, 5\}$ and $B = \{7, 8, 9\}$,then $n(A \times B)$ is equal to:

$A$ body of mass $2\,kg$ makes an elastic collision with another body at rest and continues to move in the original direction with one-fourth of its original speed. The mass of the second body which collides with the first body is .......... $kg$.

The area (in square units) of the triangle formed by the points with polar coordinates $(1, 0)$,$(2, \frac{\pi}{3})$,and $(3, \frac{2\pi}{3})$ is