$A$ is a $3 \times 3$ matrix satisfying $A^3-5A^2+7A+I=0$. If $A^5-6A^4+12A^3-6A^2+2A+2I=lA+mI$, then $l+m=$
- A$5$
- B-$1$
- C$4$
- D$2$
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If matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$,then which of the following statements is incorrect?
For $M=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and for any $n \in N$,the matrix $M^{n+1}-M^n=$
$A$ and $B$ are two non-singular square matrices of order $3 \times 3$ such that $AB = A$ and $|A + B| \neq 0$. Then:
If $A, B$ are two non-singular matrices of order $3$ and $|B|=k$, where $k$ is a positive integer, then match the items of List-$I$ with the items of List-$II$.
List-$I$ List-$II$ $A$. $|k^{-1} A^{-1}|$ $I$. $BA^k + A^kB$ $B$. $|\text{Adj}(A^{-1})|$ $II$. $\frac{B\text{Adj}(B)}{|B|}$ $C$. $BAB^{-1} = I \Rightarrow BA^kB^{-1} =$ $III$. $\frac{1}{|B|^3|A|}$ $D$. $\text{Adj}(\text{Adj}(A^{-1})) =$ $IV$. $\frac{1}{|A|}(A^{-1})$ $V$. $\frac{1}{|A|^2}$
| List-$I$ | List-$II$ |
| $A$. $|k^{-1} A^{-1}|$ | $I$. $BA^k + A^kB$ |
| $B$. $|\text{Adj}(A^{-1})|$ | $II$. $\frac{B\text{Adj}(B)}{|B|}$ |
| $C$. $BAB^{-1} = I \Rightarrow BA^kB^{-1} =$ | $III$. $\frac{1}{|B|^3|A|}$ |
| $D$. $\text{Adj}(\text{Adj}(A^{-1})) =$ | $IV$. $\frac{1}{|A|}(A^{-1})$ |
| $V$. $\frac{1}{|A|^2}$ |
Let $a = \min \{x^2 + 2x + 3, x \in R\}$ and $b = \lim_{x \to 0} \frac{\sin x \cos x}{e^x - e^{-x}}$. Then the value of $\sum_{r=0}^n a^r b^{n-r}$ is
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