Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where
\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]
How much charge is there on a hole? Draw the circuit symbol of p-n junction diode.
Find the minimum value of ( z = x + 3y ) under the following constraints:
• x + y ≤ 8• 3x + 5y ≥ 15• x ≥ 0, y ≥ 0
A charge \( Q \) is distributed over two concentric hollow spheres of radii \( r \) and \( R \) (\( R>r \)) such that their surface charge densities are equal. Find the electric potential at their common center.
Find out the angle of refraction in a medium of refractive index \(\sqrt{3}\), when the angle of incidence is \(60^\circ\).
There are 10 black and 5 white balls in a bag. Two balls are taken out, one after another, and the first ball is not placed back before the second is taken out. Assume that the drawing of each ball from the bag is equally likely. What is the probability that both the balls drawn are black?