1
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1

If the system of equations

$$x+y+a z=b$$

$$2 x+5 y+2 z=6$$

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

has infinitely many solutions, then $$2 a+3 b$$ is equal to :

A
28
B
25
C
20
D
23
2
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1

Let $$\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{2 \times 2}$$, where $$\mathrm{a}_{\mathrm{ij}} \neq 0$$ for all $$\mathrm{i}, \mathrm{j}$$ and $$\mathrm{A}^{2}=\mathrm{I}$$. Let a be the sum of all diagonal elements of $$\mathrm{A}$$ and $$\mathrm{b}=|\mathrm{A}|$$. Then $$3 a^{2}+4 b^{2}$$ is equal to :

A
4
B
3
C
14
D
7
3
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 $$\beta=1$$
4
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
$$\mathrm{A^{30}+A^{25}+A=I}$$
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