If p and q are the non-zero roots of the equation x² + px + q = 0, then how many possible values can q have?

If $\rm \Delta = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}$ then what is $\rm \begin{vmatrix} 3d + 5g & 4a + 7g & 6g\\ 3e + 5h & 4b + 7h & 6h\\ 3f + 5i & 4c + 7i & 6i \end{vmatrix}$ equal to?

$\frac{1}{b+c}, \frac{1}{c+a},\frac{1}{a+b}$ are in HP, then which of the following is/are correct?

1. a, b, c are in AP

2. (b + c)², (c + a)², (a + b)² are in GP. Select the correct answer using the code given below.

$\rm A=\begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}$ where a ∈ ℕ, then is A¹⁰⁰ - A⁵⁰ - 2A²⁵ equal to?

where I is the identity matrix.