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if a cos 2 sin 2 sin 2 cos 2 and a at i where i is
Question:
If
A
=
[
cos
2
θ
−
sin
2
θ
sin
2
θ
cos
2
θ
]
and
A
+
A
T
=
I
Where
I
is the unit matrix of
2
×
2
and
A
T
is the transpose of
A
, then the value of
θ
is equal to:
MHT CET
Updated On:
Jun 23, 2024
(A)
3
π
2
(B)
π
(C)
π
3
(D)
π
6
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The Correct Option is
D
Solution and Explanation
Explanation:
Given,
A
=
[
cos
2
θ
−
sin
2
θ
sin
2
θ
cos
2
θ
]
A
+
A
T
=
1
The transpose of matrix
A
is given by,
A
T
=
[
cos
2
θ
sin
2
θ
−
sin
2
θ
cos
2
θ
]
As,
A
+
A
T
=
I
[
cos
2
θ
−
sin
2
θ
sin
2
θ
cos
2
θ
]
+
[
cos
2
θ
sin
2
θ
−
sin
2
θ
cos
2
θ
]
=
I
⇒
[
2
cos
2
θ
0
0
2
cos
2
θ
]
=
[
1
0
0
1
]
As we know that,If two matrices
A
and
B
are equal then their corresponding elements are also equal.
∴
2
cos
2
θ
=
1
⇒
cos
2
θ
=
1
2
⇒
cos
2
θ
=
cos
60
∘
⇒
2
θ
=
π
3
[
∵
cos
60
∘
=
π
3
]
∴
θ
=
π
6
So, the value of
θ
is
π
6
.Hence, the correct option is (D).
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