$A$ Zener diode,having a breakdown voltage equal to $15 \, V$,is used in a voltage regulator circuit as shown in the figure. The current through the diode is.....$mA$.

For positive numbers $x, y, \text{ and } z$,the numerical value of the determinant $\left| \begin{array}{ccc} 1 & \log_x y & \log_x z \\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \end{array} \right|$ is

In the following circuit,find $I_1$ and $I_2$.

If the pair of lines represented by $3 x^2 - 5 x y + P y^2 = 0$ and $6 x^2 - x y - 5 y^2 = 0$ have one line in common,then the sum of all possible values of $P$ is

The product of the lengths of the perpendiculars drawn from the point $(-1, 5)$ to the pair of lines $2x^2 - xy - 3y^2 + 6x + y + 4 = 0$ is

$\sin 120^{\circ} \cos 150^{\circ} - \cos 240^{\circ} \sin 330^{\circ}$ is equal to :