1
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 Let A be a 3 $$\times$$ 3 real matrix such that

$$A\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 1 \cr } } \right);A\left( {\matrix{ 1 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ { - 1} \cr 0 \cr 1 \cr } } \right)$$ and $$A\left( {\matrix{ 0 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 2 \cr } } \right)$$.

If $$X = {({x_1},{x_2},{x_3})^T}$$ and I is an identity matrix of order 3, then the system $$(A - 2I)X = \left( {\matrix{ 4 \cr 1 \cr 1 \cr } } \right)$$ has :

A
no solution
B
infinitely many solutions
C
unique solution
D
exactly two solutions
2
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 Let $$A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$$. If M and N are two matrices given by $$M = \sum\limits_{k = 1}^{10} {{A^{2k}}}$$ and $$N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}}$$ then MN2 is :

A
a non-identity symmetric matrix
B
a skew-symmetric matrix
C
neither symmetric nor skew-symmetric matrix
D
an identity matrix
3
JEE Main 2022 (Online) 24th June Evening Shift
+4
-1 Let the system of linear equations

x + y + $$\alpha$$z = 2

3x + y + z = 4

x + 2z = 1

have a unique solution (x$$^ *$$, y$$^ *$$, z$$^ *$$). If ($$\alpha$$, x$$^ *$$), (y$$^ *$$, $$\alpha$$) and (x$$^ *$$, $$-$$y$$^ *$$) are collinear points, then the sum of absolute values of all possible values of $$\alpha$$ is

A
4
B
3
C
2
D
1
4
JEE Main 2022 (Online) 24th June Morning Shift
+4
-1 The number of values of $$\alpha$$ for which the system of equations :

x + y + z = $$\alpha$$

$$\alpha$$x + 2$$\alpha$$y + 3z = $$-$$1

x + 3$$\alpha$$y + 5z = 4

is inconsistent, is

A
0
B
1
C
2
D
3
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