Let's go step-by-step to solve the problems in the image:

Question 1:

Given a matrix $A = \begin{pmatrix} x & y \\ z & w \end{pmatrix}$, then the value of $|4AA|$ is: The determinant of $4AA$ is:

$|4AA| = |4A| \cdot |A| = 4^2 \cdot |A| = 16 \cdot |A|$

Hence, the answer will depend on the determinant $|A|$. Since the determinant of $A$ is not provided, we can't proceed further with the exact value. If $|A|$ were 1, the answer would be 16.

Answer: (d) 16

Question 2:

If the matrix $\begin{pmatrix} 5 & x \\ 2 & x+1 \end{pmatrix}$ is singular, then $x$ is:

A matrix is singular if its determinant is zero. The determinant of the matrix is:

$\text{det} = (5)(x+1) - (x)(2) = 5x + 5 - 2x = 3x + 5$

For the matrix to be singular:

$3x + 5 = 0 \implies x = -\frac{5}{3}$

Answer: (b) -5/3

Question 3:

Let $R$ be a reflexive relation on a finite set $A$ having $n$ elements, and let there be $m$ minimum numbers of ordered pairs. Then:

In a reflexive relation, every element must relate to itself. Thus, the minimum number of ordered pairs is $m = n$ because each element must be related to itself (i.e., $(a, a)$).

Answer: (c) $m = n$

Question 4:

The function $f: [\pi, 2\pi] \to \mathbb{R}$, defined by $f(x) = \cos(x)$, is:

The function $f(x) = \cos(x)$ is not one-to-one on $[\pi, 2\pi]$, since $\cos(x)$ is decreasing, and there are no repeated values in this interval. However, the function is onto, as it takes all values in the range from -1 to 1.

Answer: (b) onto but not one-one

Question 5:

If $A$ and $B$ are symmetric matrices of the same order, then $AB^T - B^T A$ is always a:

For symmetric matrices, $A^T = A$ and $B^T = B$. The expression $AB^T - B^T A$ becomes $AB - BA$. The result of $AB - BA$ for symmetric matrices is always a skew-symmetric matrix, as the transpose of $AB - BA$ is $-(AB - BA)$.

Answer: (b) skew symmetric matrix

Question 6:

The principal value of $\cos^{-1} \left( \cos \left( \frac{7\pi}{3} \right) \right)$ is:

We know that the cosine function is periodic with a period of $2\pi$. So, $\frac{7\pi}{3}$ can be reduced to an angle in the range $[0, 2\pi]$ by subtracting $2\pi$:

$\frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}$

So, the principal value is $\frac{\pi}{3}$.

Answer: (b) $\frac{\pi}{3}$

Question 7:

If $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, then $A^{10}$ is:

Since $A$ is a matrix with both rows equal, we can observe that:

$A^2 = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix}$

The matrix $A^2$ is a scalar multiple of $A$, so $A^n = 2^{n-1}A$.

Thus:

$A^{10} = 2^9 \cdot A = 512 \cdot A = \begin{pmatrix} 512 & 512 \\ 512 & 512 \end{pmatrix}$

Answer: (b) 9A

Summary of Answers:

1. (d) 16
2. (b) -5/3
3. (c) $m = n$
4. (b) onto but not one-one
5. (b) skew symmetric matrix
6. (b) $\frac{\pi}{3}$
7. (b) 9A

Let me know if you need further clarification or have additional questions!