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Byju's Answer
Standard XII
Mathematics
Properties Derived from Trigonometric Identities
The equation ...
Question
The equation
cos
2
x
+
a
sin
x
=
2
a
−
7
has a solution for
A
all
a
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B
a
>
6
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C
a
<
2
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D
a
∈
[
2
,
6
]
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Solution
The correct option is
B
a
∈
[
2
,
6
]
Given equation is
cos
2
x
+
a
sin
x
=
2
a
−
7
⇒
1
−
2
sin
2
x
+
a
sin
x
=
2
a
−
7
⇒
2
sin
2
x
−
a
sin
x
+
2
a
−
8
=
0
We have a formula for solving quadratic equation
a
x
2
+
b
x
+
c
=
0
is
x
=
−
b
±
√
b
2
−
4
a
c
2
a
⇒
sin
x
=
a
±
√
a
2
−
16
(
a
−
4
)
4
=
a
±
(
a
−
8
)
4
⇒
sin
x
=
a
−
4
2
or
sin
x
=
2
(not possible)
∴
sin
x
=
a
−
4
2
∵
−
1
≤
sin
x
≤
1
∴
−
1
≤
a
−
4
2
≤
1
⇒
2
≤
a
≤
6
⟹
a
∈
[
2
,
6
]
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Assertion :The equation
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