Let α and β be the distinct roots of the equation x²+x-1=0. Consider the set T=\{1, α, β\}. For a 3 × 3 matrix M= (a_{i j}) 3 × 3) , define R_i=a_{i 1}+a_{i 2}+a_β and C_j=a₁ j}+a₂ j}+a₃ j} for i=1,2,3 and j=1,2,3 Match each entry in List-I to the correct entry in List-II. List-I List-II (P) The number of matrices M= (a_{i j}) 3 × 3 with all entries in T such that R_i=C_j=0 for all i, j is (1) 1 (Q) The number of symmetric matrices M= (a_{i j}) 3 × 3 with all entries in T such that C_j=0 for all j is (2) 12 (R) Let M= (a_{i j}) ₃ × 3} be a skew symmetric matrix such that a_{ij} T for ij . Then the number of elements in the set \{ {pmatrix} x \ y \ z {pmatrix} : x, y · z R, M{pmatrix} x \ y \ z {pmatrix} = {pmatrix} a₁2} \ 0 \ -a₂3} {pmatrix} \} is (3) Infinite (S) Let M= (a_{i j}) ₃ × 3} be a matrix with all entries in T such that R_i=0 for all i . Then the absolute value of the determinant of M is (4) 6 (5) 0 The correct option is [JEE Adv 2024 P1]

Question

Let α and β be the distinct roots of the equation x²+x-1=0. Consider the set T=\{1, α, β\}. For a 3 × 3 matrix M= (a_{i j}) 3 × 3) , define R_i=a_{i 1}+a_{i 2}+a_β and C_j=a₁ j}+a₂ j}+a₃ j} for i=1,2,3 and j=1,2,3 Match each entry in List-I to the correct entry in List-II.    List-I    List-II  (P)  The number of matrices M= (a_{i j}) 3 × 3 with all entries in T such that R_i=C_j=0 for all i, j is  (1)  1  (Q)  The number of symmetric matrices M= (a_{i j}) 3 × 3 with all entries in T such that C_j=0 for all j is  (2)  12  (R)  Let M= (a_{i j}) ₃ × 3} be a skew symmetric matrix such that a_{ij} T for i>j . Then the number of elements in the set \{ {pmatrix} x \ y \ z {pmatrix} : x, y · z R, M{pmatrix} x \ y \ z {pmatrix} = {pmatrix} a₁2} \ 0 \ -a₂3} {pmatrix} \} is  (3)  Infinite  (S)  Let M= (a_{i j}) ₃ × 3} be a matrix with all entries in T such that R_i=0 for all i . Then the absolute value of the determinant of M is  (4)  6      (5)  0 The correct option is [JEE Adv 2024 P1]

Examsnet · Accepted Answer

Correct Answer is : (P) ⟶⟶ (2) (Q) ⟶⟶ (4) (R) ⟶⟶ (3) (S) ⟶⟶ (5)

	List-I		List-II
(P)	The number of matrices $M= (a_{i j}) 3 \times 3$ with all entries in $T$ such that $R_i=C_j=0$ for all $i, j$ is	(1)	1
(Q)	The number of symmetric matrices $M= (a_{i j}) 3 \times 3$ with all entries in $T$ such that $C_j=0$ for all $j$ is	(2)	12
(R)	Let $M= (a_{i j}) _{3 \times 3}$ be a skew symmetric matrix such that $a_{ij} \in T$ for $i>j$ . Then the number of elements in the set $\{ \begin{pmatrix} x \\ y \\ z \end{pmatrix} : x, y \cdot z \in R, M\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_{12} \\ 0 \\ -a_{23} \end{pmatrix} \}$ is	(3)	Infinite
(S)	Let $M= (a_{i j}) _{3 \times 3}$ be a matrix with all entries in $T$ such that $R_i=0$ for all $i$ . Then the absolute value of the determinant of $M$ is	(4)	6
		(5)	0

JEE Advanced 2024 Paper 1