1
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 ___________.
2
JEE Main 2021 (Online) 25th February Morning Slot
Numerical
+4
-1
If $$A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over 2}} \right)} & 0 \cr } } \right]$$ and
$$({I_2} + A){({I_2} - A)^{ - 1}} = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]$$, then $$13({a^2} + {b^2})$$ is equal to
3
JEE Main 2021 (Online) 25th February Morning Slot
Numerical
+4
-1
Let $$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]$$, where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If $${A^2} = {I_3}$$, then the value of $${x^3} + {y^3} + {z^3}$$ is ____________.
4
JEE Main 2021 (Online) 25th February Morning Slot
Numerical
+4
-1
If the system of equations

kx + y + 2z = 1

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

$$-$$2x $$-$$2y $$-$$4z = 3

has infinitely many solutions, then k is equal to __________.
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