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UPSC NDA Math Model Paper 4
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© examsnet.com
Question : 13
Total: 120
If sin x + a cos x = b, then |a sin x - cos x| is:
√
a
2
+
b
2
+
1
√
a
2
−
b
2
+
1
√
a
2
+
b
2
−
1
None of the above.
Validate
Solution:
sin
x
+
a
cos
x
=
b
⇒
(
sin
x
+
a
cos
x
)
2
=
b
2
⇒
sin
2
x
+
a
2
cos
2
x
+
2
a
sin
x
cos
x
=
b
2
⇒
(
1
−
cos
2
x
)
+
a
2
cos
2
x
+
2
a
sin
x
cos
x
=
b
2
.
.
.
[
Using
sin
2
x
=
1
−
cos
2
x
]
⇒
(
a
2
−
1
)
cos
2
x
+
2
a
sin
x
cos
x
=
b
2
−
1
⇒
2
a
sin
x
cos
x
=
b
2
−
1
+
(
1
−
a
2
)
cos
2
x
........(1)
Let
k
=
|
a
sin
x
−
cos
x
|
⇒
k
2
=
(
a
sin
x
−
cos
x
)
2
⇒
k
2
=
a
2
sin
2
x
+
cos
2
x
−
2
a
sin
x
cos
x
⇒
k
2
=
a
2
(
1
−
cos
2
x
)
+
cos
2
x
−
2
a
sin
x
cos
x
.
.
.
[
Using
sin
2
x
=
1
−
cos
2
x
]
⇒
k
2
=
a
2
+
(
1
−
a
2
)
cos
2
x
−
2
a
sin
x
cos
x
⇒
k
2
=
a
2
+
(
1
−
a
2
)
cos
2
x
−
[
b
2
−
1
+
(
1
−
a
2
)
cos
2
x
]
... [Using equation (1)]
⇒
k
2
=
a
2
−
b
2
+
1
⇒
k
=
√
a
2
−
b
2
+
1
∴
|
a
sin
x
−
cos
x
|
=
√
a
2
−
b
2
+
1
© examsnet.com
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