Matrices & Determinants · JEE Advanced previous year questions

2024 2 questions

1. Q102024 · 26 May 2024 · Paper 1· Numerical answer

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

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   Answer: 16

2. Q142024 · 26 May 2024 · Paper 1

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

   1. A.(P) $\to$ (4) \quad (Q) $\to$ (2) \quad (R) $\to$ (5) \quad (S) $\to$ (1)
   2. B.(P) $\to$ (2) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (5)
   3. C.(P) $\to$ (2) \quad (Q) $\to$ (4) \quad (R) $\to$ (3) \quad (S) $\to$ (5)
   4. D.(P) $\to$ (1) \quad (Q) $\to$ (5) \quad (R) $\to$ (3) \quad (S) $\to$ (4)
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   Answer: C