1
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 ____________.

2
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 ____________.

3
JEE Main 2022 (Online) 28th June Evening Shift
Numerical
+4
-1 Let $$A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$$ where $$i = \sqrt { - 1}$$. Then, the number of elements in the set { n $$\in$$ {1, 2, ......, 100} : An = A } is ____________.

4
JEE Main 2022 (Online) 27th June Evening Shift
Numerical
+4
-1 Let A be a matrix of order 2 $$\times$$ 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is ___________.

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