Matrices and Determinants

If the system of linear equations :
$$x+y+z=6,\qquad x+2y+5z=10,\qquad 2x+3y+\lambda z=\mu$$
has infinitely many solutions, then the value of $\lambda+\mu$ equals :

Let $A=\begin{bmatrix}\alpha&1&2\\2&3&0\\0&4&5\end{bmatrix}$ and $B=\begin{bmatrix}1&0&0\\0&-5\alpha&0\\0&4\alpha&-2\alpha\end{bmatrix}+\operatorname{adj}(A)$. If $\det(B)=66$, then $\det(\operatorname{adj}(A))$ equals :

Consider the matrix

$$P = \begin{pmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{pmatrix}.$$

Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3\times3$ invertible matrices $Q$ with integer entries, such that

$$Q^{-1} = Q^T \quad\text{and}\quad PQ = QP,$$

is

Let $\vec w = \hat i + \hat j - 2\hat k$, and $\vec u$ and $\vec v$ be two vectors, such that $\vec u \times \vec v = \vec w$ and $\vec v \times \vec w = \vec u$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that

$$\vec u = \alpha\hat i + \beta\hat j + \gamma\hat k,\quad -t\alpha + \beta + \gamma = 0,\quad \alpha - t\beta + \gamma = 0,\quad\text{and}\quad \alpha + \beta - t\gamma = 0.$$

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let $I = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$ and $P = \begin{pmatrix} 2 & 0\\ 0 & 3 \end{pmatrix}$. Let $Q = \begin{pmatrix} x & y\\ z & 4 \end{pmatrix}$ for some non-zero real numbers $x$, $y$, and $z$, for which there is a $2\times2$ matrix $R$ with all entries being non-zero real numbers, such that $QR = RP$.

Then which of the following statements is (are) TRUE?

Consider the vectors

$$\vec x = \hat i + 2\hat j + 3\hat k,\qquad \vec y = 2\hat i + 3\hat j + \hat k,\qquad\text{and}\qquad \vec z = 3\hat i + \hat j + 2\hat k.$$

For two distinct positive real numbers $\alpha$ and $\beta$, define

$$\vec X = \alpha\vec x + \beta\vec y - \vec z,\qquad \vec Y = \alpha\vec y + \beta\vec z - \vec x,\qquad\text{and}\qquad \vec Z = \alpha\vec z + \beta\vec x - \vec y.$$

If the vectors $\vec X$, $\vec Y$, and $\vec Z$ lie in a plane, then the value of $\alpha + \beta - 3$ is _____.

Let $\mathbb{R}^2$ denote $\mathbb{R}\times\mathbb{R}$. Let

$$S = \left\{(a,b,c) : a,b,c \in \mathbb{R} \text{ and } ax^2 + 2bxy + cy^2 > 0 \text{ for all } (x,y) \in \mathbb{R}^2 - \{(0,0)\}\right\}.$$

Then which of the following statements is (are) TRUE?

Let $S = \left\{A = \begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0,1\} \text{ and } |A| \in \{-1, 1\}\right\}$, where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is _____.

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2 + x - 1 = 0$. Consider the set $T = \{1, \alpha, \beta\}$. For a $3\times 3$ matrix $M = (a_{ij})_{3\times 3}$, define $R_i = a_{i1} + a_{i2} + a_{i3}$ and $C_j = a_{1j} + a_{2j} + a_{3j}$ for $i = 1, 2, 3$ and $j = 1, 2, 3$.

Match each entry in List-I to the correct entry in List-II.

The correct option is

Let $\alpha$, $\beta$ and $\gamma$ be real numbers. Consider the following system of linear equations

$$\begin{aligned} x + 2y + z &= 7\\ x + \alpha z &= 11\\ 2x - 3y + \beta z &= \gamma \end{aligned}$$

Match each entry in List-I to the correct entries in List-II.

The correct option is:

Let $M = (a_{ij})$, $i, j \in \{1,2,3\}$, be the $3\times3$ matrix such that $a_{ij} = 1$ if $j+1$ is divisible by $i$, otherwise $a_{ij} = 0$. Then which of the following statements is(are) true?

Let $R = \left\{\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix} : a, b, c, d \in \{0, 3, 5, 7, 11, 13, 17, 19\}\right\}$. Then the number of invertible matrices in $R$ is _____.

Let $|M|$ denote the determinant of a square matrix $M$. Let $g : \left[0, \dfrac\pi2\right] \to \mathbb{R}$ be the function defined by
$$g(\theta) = \sqrt{f(\theta) - 1} + \sqrt{f\left(\frac\pi2 - \theta\right) - 1}$$
where
$$f(\theta) = \frac12\begin{vmatrix}1 & \sin\theta & 1\\ -\sin\theta & 1 & \sin\theta\\ -1 & -\sin\theta & 1\end{vmatrix} + \begin{vmatrix}\sin\pi & \cos\left(\theta + \frac\pi4\right) & \tan\left(\theta - \frac\pi4\right)\\ \sin\left(\theta - \frac\pi4\right) & -\cos\frac\pi2 & \log_e\left(\frac4\pi\right)\\ \cot\left(\theta + \frac\pi4\right) & \log_e\left(\frac\pi4\right) & \tan\pi\end{vmatrix}.$$
Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2) = 2 - \sqrt2$. Then, which of the following is/are TRUE?

Let $\beta$ be a real number. Consider the matrix
$$A = \begin{pmatrix}\beta & 0 & 1\\ 2 & 1 & -2\\ 3 & 1 & -2\end{pmatrix}.$$
If $A^7 - (\beta-1)A^6 - \beta A^5$ is a singular matrix, then the value of $9\beta$ is _____.

Let $\hat i, \hat j$ and $\hat k$ be the unit vectors along the three positive coordinate axes. Let
$$\vec a = 3\hat i + \hat j - \hat k,\qquad \vec b = \hat i + b_2\hat j + b_3\hat k,\ b_2, b_3 \in \mathbb{R},\qquad \vec c = c_1\hat i + c_2\hat j + c_3\hat k,\ c_1, c_2, c_3 \in \mathbb{R}$$
be three vectors such that $b_2b_3 > 0$, $\vec a\cdot\vec b = 0$ and
$$\begin{pmatrix}0 & -c_3 & c_2\\ c_3 & 0 & -c_1\\ -c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix}1\\ b_2\\ b_3\end{pmatrix} = \begin{pmatrix}3 - c_1\\ 1 - c_2\\ -1 - c_3\end{pmatrix}.$$
Then, which of the following is/are TRUE?

If $M = \begin{pmatrix}\frac52 & \frac32\\ -\frac32 & -\frac12\end{pmatrix}$, then which of the following matrices is equal to $M^{2022}$?

The number of values of $\alpha$ for which the system of equations :
$$x+y+z=\alpha$$
$$\alpha x+2\alpha y+3z=-1$$
$$x+3\alpha y+5z=4$$
is inconsistent, is

Let $S=\{\sqrt n:1\le n\le50$ and $n$ is odd$\}$.

Let $a\in S$ and $A=\begin{bmatrix}1&0&a\\-1&1&0\\-a&0&1\end{bmatrix}$.

If $\displaystyle\sum_{a\in S}\det(\operatorname{adj}A)=100\lambda$, then $\lambda$ is equal to :

Let the system of linear equations
$$x+y+\alpha z=2$$
$$3x+y+z=4$$
$$x+2z=1$$
have a unique solution $(x^*,y^*,z^*)$. If $(\alpha,x^*)$, $(y^*,\alpha)$ and $(x^*,-y^*)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

Let $S=\left\{\begin{pmatrix}-1&a\\0&b\end{pmatrix};a,b\in\{1,2,3,\dots,100\}\right\}$ and let $T_n=\left\{A\in S:A^{n(n+1)}=I\right\}$. Then the number of elements in $\displaystyle\bigcap_{n=1}^{100}T_n$ is ______.