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UPSC NDA Math Model Paper 9
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© examsnet.com
Question : 13
Total: 120
If, a and b are real numbers such that
a
2
+
b
2
=
1
, then value of
x
in equation
1
+
i
x
1
−
i
x
=
(
a
−
i
b
)
Purly imaginary
Purely real
Complex number
None of the above
Validate
Solution:
Here,
1
+
i
x
1
−
i
x
=
(
a
−
i
b
)
1
Now, using componendo and dividendo property
⇒
1
+
i
x
+
1
−
i
x
1
+
i
x
−
1
+
i
x
=
(
a
−
i
b
+
1
)
a
−
i
b
−
1
⇒
2
2
i
x
=
a
−
i
b
+
1
a
−
i
b
−
1
⇒
i
x
=
(
a
−
1
)
−
i
b
(
a
+
1
)
−
i
b
⇒
i
x
=
(
a
−
1
)
−
i
b
(
a
+
1
)
−
i
b
×
(
a
−
1
)
+
i
b
(
a
+
1
)
+
i
b
⇒
i
x
=
(
a
2
−
1
+
b
2
)
+
[
(
a
−
1
)
b
−
(
a
+
1
)
b
]
i
(
a
+
1
)
2
+
b
2
⇒
i
x
=
0
−
2
b
i
(
a
+
1
)
2
+
b
2
=
−
2
b
i
(
a
+
1
)
2
+
b
2
.
.
.
.
.
.
(
∵
a
2
+
b
2
=
1
)
⇒
x
=
−
2
b
(
a
+
1
)
2
+
b
2
So, the value of x in the given equation is purely real .
Hence, option (2) is correct.
© examsnet.com
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