1
JEE Main 2021 (Online) 26th August Morning Shift
+4
-1
Let $$\theta \in \left( {0,{\pi \over 2}} \right)$$. If the system of linear equations

$$(1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\theta z = 0$$

$${\cos ^2}\theta x + (1 + {\sin ^2}\theta )y + 4\sin 3\,\theta z = 0$$

$${\cos ^2}\theta x + {\sin ^2}\theta y + (1 + 4\sin 3\,\theta )z = 0$$

has a non-trivial solution, then the value of $$\theta$$ is :
A
$${{4\pi } \over 9}$$
B
$${{7\pi } \over {18}}$$
C
$${\pi \over {18}}$$
D
$${{5\pi } \over {18}}$$
2
JEE Main 2021 (Online) 26th August Morning Shift
+4
-1
If $$A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr } } \right)$$, $$B = \left( {\matrix{ 1 & 0 \cr i & 1 \cr } } \right)$$, $$i = \sqrt { - 1}$$, and Q = ATBA, then the inverse of the matrix A Q2021 AT is equal to :
A
$$\left( {\matrix{ {{1 \over {\sqrt 5 }}} & { - 2021} \cr {2021} & {{1 \over {\sqrt 5 }}} \cr } } \right)$$
B
$$\left( {\matrix{ 1 & 0 \cr { - 2021i} & 1 \cr } } \right)$$
C
$$\left( {\matrix{ 1 & 0 \cr {2021i} & 1 \cr } } \right)$$
D
$$\left( {\matrix{ 1 & { - 2021i} \cr 0 & 1 \cr } } \right)$$
3
JEE Main 2021 (Online) 27th July Evening Shift
+4
-1
Let A and B be two 3 $$\times$$ 3 real matrices such that (A2 $$-$$ B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3, then the value of the determinant of the matrix A3 + B3 is equal to :
A
2
B
4
C
1
D
0
4
JEE Main 2021 (Online) 27th July Morning Shift
+4
-1
Let $$A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]$$. If A$$-$$1 = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$\in$$ R, I is a 2 $$\times$$ 2 identity matrix then 4($$\alpha$$ $$-$$ $$\beta$$) is equal to :
A
5
B
$${8 \over 3}$$
C
2
D
4
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