1
JEE Main 2022 (Online) 24th June Evening Shift
Numerical
+4
-1

Let $$S = \left\{ {\left( {\matrix{ { - 1} & a \cr 0 & b \cr } } \right);a,b \in \{ 1,2,3,....100\} } \right\}$$ and let $${T_n} = \{ A \in S:{A^{n(n + 1)}} = I\}$$. Then the number of elements in $$\bigcap\limits_{n = 1}^{100} {{T_n}}$$ is ___________.

2
JEE Main 2021 (Online) 31st August Evening Shift
Numerical
+4
-1
The number of elements in the set $$\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,b,d \in \{ - 1,0,1\} \,and\,{{(I - A)}^3} = I - {A^3}} \right\}$$, where I is 2 $$\times$$ 2 identity matrix, is :
3
JEE Main 2021 (Online) 27th August Morning Shift
Numerical
+4
-1
If the system of linear equations

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

x $$-$$ y $$-$$ z = $$\alpha$$

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

has infinitely many solution, then $$\alpha$$ + $$\beta$$ $$-$$ $$\alpha$$$$\beta$$ is equal to _____________.
4
JEE Main 2021 (Online) 26th August Evening Shift
Numerical
+4
-1
Let A be a 3 $$\times$$ 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________.
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