If $A_i = \begin{bmatrix} a^i & b^i \\ b^i & a^i \end{bmatrix}$ and if $|a| < 1, |b| < 1$,then $\sum_{i=1}^{\infty} \det(A_i)$ is equal to

If $A = \begin{bmatrix} -8 & 5 \\ 2 & 4 \end{bmatrix}$ satisfies the equation $x^2 + 4x - p = 0$, then $p$ is equal to

If the function $f:[a, b] \rightarrow \left[-\frac{\sqrt{3}}{4}, \frac{1}{2}\right]$ defined by $f(x) = \left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\sin x & 1 \\ 1+\cos x & 1 & 1 \end{array} \right|$ is one-one and onto, then:

$A$ and $B$ are two $3 \times 3$ non-singular matrices such that $\operatorname{adj} A = |A| B$. If $\operatorname{tr}(X)$ denotes the trace of a square matrix $X$ and $C = \begin{bmatrix} 4 & 4 & 7 \\ 3 & -2 & 5 \\ -2 & 3 & 6 \end{bmatrix}$, then $\sum_{k=1}^{\infty} \operatorname{tr}\left(\frac{1}{3^k}(A B)^k C\right)$ is equal to

Suppose $a_1, a_2, \dots$ are real numbers,with $a_1 \neq 0$. If $a_1, a_2, a_3, \dots$ are in $A.P.$,then:

Let $M = \begin{bmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{bmatrix}$ and $\operatorname{adj} M = \begin{bmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{bmatrix}$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?
$(1)$ $a+b=3$
$(2)$ $\operatorname{det}(\operatorname{adj} M^2) = 81$
$(3)$ $(\operatorname{adj} M)^{-1} + \operatorname{adj} M^{-1} = -M$
$(4)$ If $M \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$,then $\alpha - \beta + \gamma = 3$