1
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1

If A and B are two non-zero n $$\times$$ n matrices such that $$\mathrm{A^2+B=A^2B}$$, then :

A
$$\mathrm{A^2B=I}$$
B
$$\mathrm{A^2=I}$$ or $$\mathrm{B=I}$$
C
$$\mathrm{A^2B=BA^2}$$
D
$$\mathrm{AB=I}$$
2
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1
Out of Syllabus

Let $$\alpha$$ be a root of the equation $$(a - c){x^2} + (b - a)x + (c - b) = 0$$ where a, b, c are distinct real numbers such that the matrix $$\left[ {\matrix{ {{\alpha ^2}} & \alpha & 1 \cr 1 & 1 & 1 \cr a & b & c \cr } } \right]$$ is singular. Then, the value of $${{{{(a - c)}^2}} \over {(b - a)(c - b)}} + {{{{(b - a)}^2}} \over {(a - c)(c - b)}} + {{{{(c - b)}^2}} \over {(a - c)(b - a)}}$$ is

A
3
B
6
C
12
D
9
3
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$$ by a single elementary row operation ?

A
$$\left[\begin{array}{cc}0 & 1 \\ 1 & -1\end{array}\right]$$
B
$$\left[\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right]$$
C
$$\left[\begin{array}{rr}-1 & 2 \\ -2 & 7\end{array}\right]$$
D
$$\left[\begin{array}{ll}-1 & 2 \\ -1 & 3\end{array}\right]$$
4
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

If the system of equations

\begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned}

has infinitely many solutions, then $$\alpha+\beta$$ is equal to

A
8
B
36
C
44
D
48
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