1
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

For the system of linear equations $$\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta$$, which one of the following statements is NOT correct?

A
It has infinitely many solutions if $$\alpha=1$$ and $$\beta=1$$
B
It has infinitely many solutions if $$\alpha=2$$ and $$\beta=-1$$
C
$$x+y+z=\frac{3}{4}$$ if $$\alpha=2$$ and $$\beta=1$$
D
It has no solution if $$\alpha=-2$$ and $$\beta=1$$
2
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

If $$A = {1 \over 2}\left[ {\matrix{ 1 & {\sqrt 3 } \cr { - \sqrt 3 } & 1 \cr } } \right]$$, then :

A
$$\mathrm{A^{30}-A^{25}=2I}$$
B
$$\mathrm{A^{30}+A^{25}-A=I}$$
C
$$\mathrm{A^{30}=A^{25}}$$
D
$$\mathrm{A^{30}+A^{25}+A=I}$$
3
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1

Let $$S$$ denote the set of all real values of $$\lambda$$ such that the system of equations

$$\lambda x+y+z=1$$

$$x+\lambda y+z=1$$

$$x+y+\lambda z=1$$

is inconsistent, then $$\sum_\limits{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right)$$ is equal to

A
12
B
2
C
4
D
6
4
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

For the system of linear equations

$$x+y+z=6$$

$$\alpha x+\beta y+7 z=3$$

$$x+2 y+3 z=14$$

which of the following is NOT true ?

A
If $$\alpha=\beta=7$$, then the system has no solution
B
For every point $$(\alpha, \beta) \neq(7,7)$$ on the line $$x-2 y+7=0$$, the system has infinitely many solutions
C
There is a unique point $$(\alpha, \beta)$$ on the line $$x+2 y+18=0$$ for which the system has infinitely many solutions
D
If $$\alpha=\beta$$ and $$\alpha \neq 7$$, then the system has a unique solution
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