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

Let $$P$$ be a square matrix such that $$P^{2}=I-P$$. For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$P^{\alpha}+P^{\beta}=\gamma I-29 P$$ and $$P^{\alpha}-P^{\beta}=\delta I-13 P$$, then $$\alpha+\beta+\gamma-\delta$$ is equal to :

A
18
B
22
C
24
D
40
2
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1

For the system of equations

$$x+y+z=6$$

$$x+2 y+\alpha z=10$$

$$x+3 y+5 z=\beta$$, which one of the following is NOT true?

A
System has a unique solution for $$\alpha=3,\beta\ne14$$.
B
System has infinitely many solutions for $$\alpha=3, \beta=14$$.
C
System has no solution for $$\alpha=3, \beta=24$$.
D
System has a unique solution for $$\alpha=-3, \beta=14$$.
3
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
4
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
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