1
JEE Main 2021 (Online) 16th March Evening Shift
Numerical
+4
-1
Let $$A = \left[ {\matrix{ {{a_1}} \cr {{a_2}} \cr } } \right]$$ and $$B = \left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr } } \right]$$ be two 2 $$\times$$ 1 matrices with real entries such that A = XB, where

$$X = {1 \over {\sqrt 3 }}\left[ {\matrix{ 1 & { - 1} \cr 1 & k \cr } } \right]$$, and k$$\in$$R.

If $$a_1^2$$ + $$a_2^2$$ = $${2 \over 3}$$(b$$_1^2$$ + b$$_2^2$$) and (k2 + 1) b$$_2^2$$ $$\ne$$ $$-$$2b1b2, then the value of k is __________.
2
JEE Main 2021 (Online) 16th March Morning Shift
Numerical
+4
-1
Let $$P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {60} & {14} \cr } } \right]$$ and

$$A = \left[ {\matrix{ 2 & 7 & {{\omega ^2}} \cr { - 1} & { - \omega } & 1 \cr 0 & { - \omega } & { - \omega + 1} \cr } } \right]$$ where

$$\omega = {{ - 1 + i\sqrt 3 } \over 2}$$, and I3 be the identity matrix of order 3. If the
determinant of the matrix (P$$-$$1AP$$-$$I3)2 is $$\alpha$$$$\omega$$2, then the value of $$\alpha$$ is equal to ______________.
3
JEE Main 2021 (Online) 16th March Morning Shift
Numerical
+4
-1
The total number of 3 $$\times$$ 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is equal to _____________.
4
JEE Main 2021 (Online) 26th February Evening Shift
Numerical
+4
-1
If the matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \cr } } \right]$$ satisfies the equation

$${A^{20}} + \alpha {A^{19}} + \beta A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & 0 \cr 0 & 0 & 1 \cr } } \right]$$ for some real numbers $$\alpha$$ and $$\beta$$, then $$\beta$$ $$-$$ $$\alpha$$ is equal to ___________.
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