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UPSC 2014 CDS II Math Paper
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© examsnet.com
Question : 35
Total: 100
Consider the following statements :
1.
(
a
−
b
−
c
)
is one of the factors of
3
a
b
c
+
b
3
+
c
3
−
a
3
2.
(
b
+
c
−
1
)
is one of the factors of
3
b
c
+
b
3
+
c
3
−
1
Which of the above statements is/are correct?
[2014 CDS-II]
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Validate
Solution:
We know,
x
3
+
y
3
+
z
3
−
3
x
y
z
=
(
x
+
y
+
z
)
(
x
2
+
y
2
+
z
2
−
x
y
−
y
z
−
z
x
)
3
a
b
c
+
b
3
+
c
3
−
a
3
=
−
[
a
3
+
(
−
b
)
3
+
(
−
c
)
3
−
3
×
a
×
(
−
b
)
×
(
−
c
)
]
=
−
[
a
+
(
−
b
)
+
(
−
c
)
]
[
a
2
+
(
−
b
)
2
+
(
−
c
)
2
−
a
×
(
−
b
)
−
(
−
b
)
×
(
−
c
)
−
(
−
c
)
×
a
]
=
−
(
a
−
b
−
c
)
(
a
2
+
b
2
+
c
2
+
a
b
−
b
c
+
c
a
)
Hence, statement 1 is correct.
3
b
c
+
b
3
+
c
3
−
1
=
b
3
+
c
3
+
(
−
1
)
3
−
3
×
b
×
c
×
(
−
1
)
=
[
b
+
c
+
(
−
1
)
]
[
b
2
+
c
2
+
(
−
1
)
2
−
b
×
c
]
−
c
×
(
−
1
)
−
(
−
1
)
×
b
=
(
b
+
c
−
1
)
(
b
2
+
c
2
+
1
−
b
c
+
c
+
b
)
Hence, statement 2 is correct.
© examsnet.com
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