1
JEE Main 2022 (Online) 25th July Evening Shift
Numerical
+4
-1

Let $$A=\left[\begin{array}{lll} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right], a, b \in \mathbb{R}$$. If for some

$$n \in \mathbb{N}, A^{n}=\left[\begin{array}{ccc} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{array}\right]$$ then $$n+a+b$$ is equal to ____________.

2
JEE Main 2022 (Online) 25th July Morning Shift
Numerical
+4
-1

Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and $$B=A-I$$. If $$\omega=\frac{\sqrt{3} i-1}{2}$$, then the number of elements in the $$\operatorname{set}\left\{n \in\{1,2, \ldots, 100\}: A^{n}+(\omega B)^{n}=A+B\right\}$$ is equal to ____________.

3
JEE Main 2022 (Online) 29th June Evening Shift
Numerical
+4
-1

Let $$M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right]$$, where $$\alpha$$ is a non-zero real number an $$N = \sum\limits_{k = 1}^{49} {{M^{2k}}}$$. If $$(I - {M^2})N = - 2I$$, then the positive integral value of $$\alpha$$ is ____________.

4
JEE Main 2022 (Online) 28th June Evening Shift
Numerical
+4
-1

If the system of linear equations
$$2x - 3y = \gamma + 5$$,
$$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R has infinitely many solutions then the value
of | 9$$\alpha$$ + 3$$\beta$$ + 5$$\gamma$$ | is equal to ____________.