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Consider the system of linear equations
\( -x + y + 2z = 0 \)
\( 3x - ay + 5z = 1 \)
\( 2x - 2y - az = 7 \)
Let \( S_{1} \) be the set of all \( a \in \mathbb{R} \) for which the system is inconsistent and \( S_{2} \) be the set of all \( a \in \mathbb{R} \) for which the system has infinitely many solutions. If \( n(S_{1}) \) and \( n(S_{2}) \) denote the number of elements in \( S_{1} \) and \( S_{2} \) respectively, then:

If the system of linear equations
\( 2x + y - z = 3 \)
\( x - y - z = \alpha \)
\( 3x + 3y + \beta z = 3 \)
has infinitely many solution, then \( \alpha + \beta - \alpha \beta \) is equal to __________.

Let \( \theta \in \left(0, \frac{\pi}{2}\right) \). If the system of linear equation.
\( (1 + \cos^2 \theta)x + \sin^2 \theta \, y + 4 \sin 3\theta \, z = 0 \)
\( \cos^2 \theta \, x + (1 + \sin^2 \theta) y + 4 \sin 3\theta \, z = 0 \)
\( \cos^2 \theta \, x + \sin^2 \theta \, y + (1 + 4 \sin 3\theta) z = 0 \)
Has a non-trivial solution, then the value of \( \theta \) is:

If \(\displaystyle P = \begin{bmatrix} 1 & 0 \\ 1/2 & 1 \end{bmatrix}\), then \( P^{50} \) is:

The number of elements in the set  where \( I \) is \( 2 \times 2 \) identity matrix, is _________.

The system of equations \( kx + y + z = 1 \), \( x + ky + z = k \) and \( x + y + zk = k^2 \) has no solution if \( k \) is equal to:

Let \( J_{n,m} = \int_{0}^{\tfrac{1}{2}} \frac{x^m}{x^n - 1} \, dx \), \( \forall \, n > m \) and \( n, m \in \mathbb{N} \). Consider a matrix \( A = [a_{ij}]_{3 \times 3} \) where Then \( \lvert \operatorname{adj}\, A^{-1} \rvert \) is:

, then value of \( \det(A^{4})+\det\!\big(A^{10}-(\operatorname{Adj}(2A))^{10}\big) \) is equal to ______.

Let \( \alpha, \beta, \gamma \) be the real roots of the equation, \( x^3 + ax^2 + bx + c = 0 \), \( (a,b,c \in \mathbb{R} \) and \( a,b \ne 0) \). If the system of equations (in \( u, v, w \)) given by \( \alpha u + \beta v + \gamma w = 0 \); \( \beta u + \gamma v + \alpha w = 0 \); \( \gamma u + \alpha v + \beta w = 0 \) has non-trivial solution, then value of \( \dfrac{\alpha^2}{b} \) is:

If the matrix  satisfies \( A\!\left(A^{3} + 3I\right) = 2I \), then the value of \( k \) is:

Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix, where
Let a function \( f:\mathbb{R}\to\mathbb{R} \) be defined as \( f(x)=\det(A) \). Then the sum of maximum and minimum value of \( f \) on \( \mathbb{R} \) is equal to:

where \( [t] \) denotes the greatest integer less than or equal to \( t \). If \( \det(A) = 192 \), then the set of values of \( x \) is the interval:

Let \( A=[a_{ij}] \) be a real matrix of order \( 3\times 3 \), such that \( a_{i1}+a_{i2}+a_{i3}=1 \), for \( i=1,2,3 \). Then, the sum of all the entries of the matrix \( A^{3} \) is equal to:

,\( x\in[0,\pi]. \) Then the maximum value of \( f(x) \) is equal to ______.

. If \( \operatorname{Tr}(A) \) denotes the sum of all diagonal elements of the matrix \( A \), then \( \operatorname{Tr}(A) - \operatorname{Tr}(B) \) has value equal to:

Let the system of linear equations
\( 4x + \lambda y + 2z = 0 \)
\( 2x - y + z = 0 \)
\( \mu x + 2y + 3z = 0, \lambda, \mu \in \mathbb{R} \).
Has a non-trivial solution. Then which of the following is true?

Let \( a, b, c, d \) be in arithmetic progression with common difference \( \lambda \). ,
then value of \( \lambda^{2} \) is equal to _______.

Define a relation \( R \) over a class of \( n \times n \) real matrices \( A \) and \( B \) as “\( ARB \) iff there exists a non-singular matrix \( P \) such that \( P A P^{-1} = B \)”. Then which of the following is true?

The solutions of the equation 

Let \( A \) be a \( 3\times 3 \) real matrix. If \( \det\!\big(2\,\mathrm{Adj}(2\,\mathrm{Adj}(\mathrm{Adj}(2A)))\big)=2^{41} \), then the value of \( \det(A^{2}) \) equals _________.

For real numbers \( \alpha \) and \( \beta \), consider the following system of linear equations:
\( x+y-z=2, \quad x+2y+\alpha z=1, \quad 2x-y+z=\beta \).
If the system has infinite solutions, then \( \alpha+\beta \) is equal to ________.

The number of distinct real roots of in the interval \( -\dfrac{\pi}{4} \le x \le \dfrac{\pi}{4} \) is:

Let \(A\) and \(B\) be \(3 \times 3\) real matrices such that \(A\) is symmetric matrix and \(B\) is skew-symmetric matrix. Then the system of linear equations \(\left(A^{2}B^{2}-B^{2}A^{2}\right)X=O\), where \(X\) is a \(3 \times 1\) column matrix of unknown variables and \(O\) is a \(3 \times 1\) null matrix, has:

\( i = \sqrt{-1} \), and \( Q = A^T B A \), then the inverse of the matrix \( A Q^{2021} A^T \) is equal to:

If the following system of linear equations
\( 2x+y+z=5 \)
\( x-y+z=3 \)
\( x+y+az=b \)
has no solution, then:

Let \( A=\{a_{ij}\} \) be a \(3\times3\) matrix, where

Then \( \det\!\big(3\,\operatorname{Adj}(2A^{-1})\big) \) is equal to ________.

The values of \( a \) and \( b \) for which the system of equations
\( 2x+3y+6z=8 \)
\( x+2y+az=5 \)
\( 3x+5y+9z=b \)
has no solution, are: