The image contains a mathematics exam paper for class 11, focusing on higher mathematics. Here's an interpretation and breakdown of the problems presented:

Question 1

Matrices $A$ and $D$ are provided:

• $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 3 \\ 3 & 1 & 1 \end{bmatrix}$
• $D = \begin{bmatrix} 3 + x & 4 & x \\ 4 & x + 3 & 3 \\ x & 3 & 4 + x \end{bmatrix}$

Sub-questions:

1. (A) If $x = 5$, find the value of $2A^2$.
2. (B) If $x = 0$, find the inverse of $A$, i.e., $A^{-1}$.
3. (C) If $x = 2$ and $D = 0$, solve the equation.

Question 2

Another set of matrices $A$, $B$, and $C$ is provided:

• $A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 4 \\ 0 & 5 & 6 \end{bmatrix}$, with $\begin{bmatrix} u \\ v \\ w \end{bmatrix} = 0$, given $u = 0$, $v = 0$, $w = 0$.
• $B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $C = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$.

Sub-questions:

1. (A) Find the transpose of the matrix product $(BC)^T$.
2. (B) Prove that $u = v = w$.
3. (C) Find the inverse of $A$, i.e., $A^{-1}$.

Would you like me to solve a specific question or provide a step-by-step solution for all sub-parts? Let me know!

Relative Questions:

1. What does matrix inversion mean, and how is it applied?
2. How do you compute the square of a matrix like $A^2$?
3. What is the significance of transpose $(BC)^T$ in a matrix?
4. Can a matrix $D$ equaling zero always provide solvable equations?
5. How do you prove equality $u = v = w$ in matrix equations?

Tip:

When solving matrix-related questions, ensure all conditions like invertibility or determinant values are verified to avoid undefined results.