1
JEE Advanced 2024 Paper 1 Online
+3
-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 $M=\left(a_{i j}\right)_{3 \times 3}$, define $R_i=a_{i 1}+a_{i 2}+a_{i 3}$ and $C_j=a_{1 j}+a_{2 j}+a_{3 j}$ 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 $M = (a_{ij})_{3x3}$ 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 $M = (a_{ij})_{3x3}$ with all entries in $T$ such that $C_j = 0$ for all $j$, is (2) 12
(R) Let $M = (a_{ij})_{3x3}$ be a skew symmetric matrix such that $a_{ij} \in T$ for $i > j$.

Then the number of elements in the set

$\left\{ \begin{pmatrix} x \\ y \\ z \end{pmatrix} : x, y, z \in \mathbb{R}, M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_{12} \\ 0 \\ a_{13} \end{pmatrix} \right\}$ is
(3) infinite
(S) Let $M = (a_{ij})_{3x3}$ 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

The correct option is
A
(P) $\rightarrow$ (4) $\quad$ (Q) $\rightarrow(2) \quad(\mathrm{R}) \rightarrow(5) \quad$ (S) $\rightarrow$ (1)
B
$(\mathrm{P}) \rightarrow(2) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(5)$
C
$(\mathrm{P}) \rightarrow(2) \quad$ (Q) $\rightarrow(4) \quad(\mathrm{R}) \rightarrow(3) \quad$ (S) $\rightarrow$ (5)
D
(P) $\rightarrow$ (1) $\quad$ (Q) $\rightarrow$ (5) $\quad$ (R) $\rightarrow$ (3) $\quad$ (S) $\rightarrow$ (4)
2
JEE Advanced 2023 Paper 1 Online
+3
-1
Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations

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

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

List - I List - II
(P) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has (1) a unique solution
(Q) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has (2) no solution
(R) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$, then the system has (3) infinitely many solutions
(S) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma=28$, then the system has (4) $x=11, y=-2$ and $z=0$ as a solution
(5) $x=-15, y=4$ and $z=0$ as a solution

The correct option is:
A
$(P) \rightarrow(3) ~~ (Q) \rightarrow(2) ~~(R) \rightarrow(1)~~ (S) \rightarrow(4)$
B
$(P) \rightarrow(3) ~~(Q) \rightarrow(2) ~~(R) \rightarrow(5)~~ (S) \rightarrow(4)$
C
$(P) \rightarrow(2)~~ (Q) \rightarrow(1) ~~ (R) \rightarrow(4) ~~ (S) \rightarrow(5)$
D
$(P) \rightarrow(2) ~~ (Q) \rightarrow(1) ~~ (R) \rightarrow(1) ~~ (S) \rightarrow(3)$
3
JEE Advanced 2022 Paper 2 Online
+3
-1
If $M=\left(\begin{array}{rr}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right)$, then which of the

following matrices is equal to $M^{2022} ?$
A
$\left(\begin{array}{rr}3034 & 3033 \\ -3033 & -3032\end{array}\right)$
B
$\left(\begin{array}{ll}3034 & -3033 \\ 3033 & -3032\end{array}\right)$
C
$\left(\begin{array}{rr}3033 & 3032 \\ -3032 & -3031\end{array}\right)$
D
$\left(\begin{array}{rr}3032 & 3031 \\ -3031 & -3030\end{array}\right)$
4
JEE Advanced 2022 Paper 1 Online
+3
-1

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

$$\begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered}$$\$

List-I List-II
(I) If $$\frac{q}{r}=10$$, then the system of linear equations has (P) $$x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$$ as a solution
(II) If $$\frac{p}{r} \neq 100$$, then the system of linear equations has (Q) $$x=\frac{10}{9}, y=-\frac{1}{9}, z=0$$ as a solution
(III) If $$\frac{p}{q} \neq 10$$, then the system of linear equations has (R) infinitely many solutions
(IV) If $$\frac{p}{q}=10$$, then the system of linear equations has (S) no solution
(T) at least one solution

The correct option is:

A
(I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (R); (III) $$\rightarrow$$ (S); (IV) $$\rightarrow$$ (T)
B
(I) $$\rightarrow$$ (Q); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (S); (IV) $$\rightarrow$$ (R)
C
(I) $$\rightarrow(\mathrm{Q})$$; (II) $$\rightarrow$$ (R); (III) $$\rightarrow(\mathrm{P})$$; (IV) $$\rightarrow$$ (R)
D
(I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (P); (IV) $$\rightarrow$$ (T)
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