1
JEE Main 2023 (Online) 8th April Morning Shift
+4
-1 Let $$P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$$ and $$Q=P A P^{T}$$. If $$P^{T} Q^{2007} P=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$$, then $$2 a+b-3 c-4 d$$ equal to :

A
2004
B
2006
C
2007
D
2005
2
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
3
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 $$\alpha=-3, \beta=14$$.
4
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
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