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JEE Advance | PYQ

Let \(\displaystyle I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \text{ and } P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}. \text{ Let } Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix}\)for some non-zero real numbers \( x, y \) and \( z \), for which there is a \( 2 \times 2 \) matrix \( R \) with all entries being non-zero real numbers, such that \( Q R = R P \). 
Then which of the following statements is (are) TRUE?

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?

For any \( 3 \times 3 \) matrix \( M \), let \( |M| \) denote the determinant of \( M \). Let 

If \( Q \) is a nonsingular matrix of order \( 3 \times 3 \), then which of the following statements is (are) TRUE?

Let \( p, q, r \) be nonzero real numbers that are, respectively, the \( 10^{\text{th}} \), \( 100^{\text{th}} \) and \( 1000^{\text{th}} \) terms of a harmonic progression. Consider the system of linear equations
\( x + y + z = 1 \Rightarrow 10x + 100y + 1000z = 0 \Rightarrow q r x + p r y + p q z = 0 \)

The correct option is:

Let \( R \) denote the set of all real numbers and let \( i = \sqrt{-1} \), Consider the matrices . Let \( a, b, c, d \) be real numbers such that .
Let $H = \{x + iy : \quad x,y \in \mathbb{R} \text{ and } y > 0\}$
Then which of the following statements is (are) TRUE?

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

Then the number of invertible matrices in \( R \) is:

Let \( \alpha, \beta \) and \( \gamma \) be real numbers such that the system of linear equations
\( x + 2y + 3z = \alpha \)
\( 4x + 5y + 6z = \beta \)
\( 7x + 8y + 9z = \gamma - 1 \)
is consistent. Let \( |M| \) represent the determinant of the matrix 

Let \( P \) be the plane containing all those \( (\alpha, \beta, \gamma) \) for which the above system of linear equations is consistent, and \( D \) be the square of the distance of the point \( (0, 1, 0) \) from the plane \( P \).
The value of \( D \) is _______.

For \( 3 \times 3 \) matrix \( M \), let \( |M| \) denote the determinant of \( M \). Let \( I \) be the \( 3 \times 3 \) identity matrix. Let \( E \) and \( F \) be two \( 3 \times 3 \) matrices such that \( (I - EF) \) is invertible. If \( G = (I - EF)^{-1} \), then which of the following statements is(are) TRUE?

Which one of the following matrices can be obtained by performing elementary row transformations on the \( 3\times 3 \) identity matrix?

Consider the matrix . Let \( p, q, r, s, a, b, c \) and \( d \) be integers such that .
Then which of the following statements is (are) TRUE?

Let \( \alpha, \beta \) and \( \gamma \) be real numbers such that the system of linear equations
\( x + 2y + 3z = \alpha \)
\( 4x + 5y + 6z = \beta \)
\( 7x + 8y + 9z = \gamma - 1 \)
is consistent. Let \( |M| \) represent the determinant of the matrix 

Let \( P \) be the plane containing all those \( (\alpha, \beta, \gamma) \) for which the above system of linear equations is consistent, and \( D \) be the square of the distance of the point \( (0, 1, 0) \) from the plane \( P \).
The value of \( |M| \) 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:

Consider the matrix \(\displaystyle P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}\).
Let the transpose of a matrix \( X \) be denoted by \( X^T \). Then the number of \( 3\times 3 \) invertible matrices \( Q \) with integer entries, such that \( Q^{-1}=Q^T \) and \( PQ=QP \) is: