Let A and B be matrices of order 3 × 3. If \( |A| = \frac{1}{2\sqrt{2}} \) and \( |B| = \frac{1}{729} \), then what is the value of \( |2B[adj(3A)]| \)?
A man has 7 relatives (4 women and 3 men). His wife also has 7 relatives (3 women and 4 men). In how many ways can they invite 3 women and 3 men so that 3 of them are man’s relatives and 3 of them are his wife’s relatives?
A triangle PQR is such that 3 points lie on the side PQ, 4 points on QR and 5 points on RP respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?
If \( -\sqrt{2} \) and \( \sqrt{3} \) are roots of the equation \( a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + x^{4} = 0 \) where \( a_{0}, a_{1}, a_{2}, a_{3} \) are integers, then which one of the following is correct?
Let \( z_{1} \) and \( z_{2} \) be two complex numbers such that \( \left| \frac{z_{1} + z_{2}}{z_{1} - z_{2}} \right| = 1 \), then what is \( Re\left(\frac{z_{1}}{z_{2}}\right) + 1 \) equal to?
