1
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 ____________.

2
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 ____________.

3
JEE Main 2022 (Online) 27th June Morning Shift
Numerical
+4
-1
Out of Syllabus

The positive value of the determinant of the matrix A, whose

Adj(Adj(A)) = $$\left( {\matrix{ {14} & {28} & { - 14} \cr { - 14} & {14} & {28} \cr {28} & { - 14} & {14} \cr } } \right)$$, is _____________.

4
JEE Main 2022 (Online) 26th June Evening Shift
Numerical
+4
-1

Let $$X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right],\,Y = \alpha I + \beta X + \gamma {X^2}$$ and $$Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma ){X^2}$$, $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R. If $${Y^{ - 1}} = \left[ {\matrix{ {{1 \over 5}} & {{{ - 2} \over 5}} & {{1 \over 5}} \cr 0 & {{1 \over 5}} & {{{ - 2} \over 5}} \cr 0 & 0 & {{1 \over 5}} \cr } } \right]$$, then ($$\alpha$$ $$-$$ $$\beta$$ + $$\gamma$$)2 is equal to ____________.