\( 1 \)
Step 1: Understanding the Given Condition
We are given three complex numbers \( Z_1, Z_2, Z_3 \) with unit modulus, i.e., \[ |Z_1| = |Z_2| = |Z_3| = 1. \] Also, we are given the equation: \[ |Z_1 - Z_2|^2 + |Z_1 - Z_3|^2 = 4. \]
Step 2: Expanding the Modulus Expressions
Expanding the squared modulus terms: \[ (Z_1 - Z_2)(\overline{Z_1 - Z_2}) + (Z_1 - Z_3)(\overline{Z_1 - Z_3}) = 4. \] Using properties of conjugates, \[ |Z_1|^2 + |Z_2|^2 - Z_1 \overline{Z_2} - \overline{Z_1} Z_2 + |Z_1|^2 + |Z_3|^2 - Z_1 \overline{Z_3} - \overline{Z_1} Z_3 = 4. \] Since \( |Z_1|^2 = |Z_2|^2 = |Z_3|^2 = 1 \), we simplify: \[ 1 + 1 - Z_1 \overline{Z_2} - \overline{Z_1} Z_2 + 1 + 1 - Z_1 \overline{Z_3} - \overline{Z_1} Z_3 = 4. \]
Step 3: Evaluating the Summation
Rearranging terms: \[ 4 - (Z_1 \overline{Z_2} + \overline{Z_1} Z_2 + Z_1 \overline{Z_3} + \overline{Z_1} Z_3) = 4. \] Thus, \[ Z_1 \overline{Z_2} + \overline{Z_1} Z_2 + Z_1 \overline{Z_3} + \overline{Z_1} Z_3 = 0. \]
If \( x \) and \( y \) are two positive real numbers such that \( x + iy = \frac{13\sqrt{5} + 12i}{(2 - 3i)(3 + 2i)} \), then \( 13y - 26x = \):
\[ \text{The domain of the real-valued function } f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) \text{ is} \]
Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \).
\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]