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UPSC 2017 CDS II Math Paper
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© examsnet.com
Question : 34
Total: 100
What is
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
(
x
−
y
)
3
+
(
y
−
z
)
3
+
(
z
−
x
)
3
equal to ?
[2017 CDS-II]
−
1
3
1
3
3
-3
Validate
Solution:
If
a
+
b
+
c
=
0
, then
a
3
+
b
3
+
c
3
=
3
a
b
c
∴
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
(
x
−
y
)
3
+
(
y
−
z
)
3
+
(
z
−
x
)
3
=
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
3
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
[
(
x
−
y
)
+
(
y
−
z
)
+
(
z
−
x
)
=
0
]
=
1
3
Alternative Method:
Take x = 1,y = 2, z = 3
We have
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
(
x
−
y
)
3
+
(
y
−
z
)
3
+
(
z
−
x
)
3
=
(
1
−
2
)
(
2
−
3
)
(
3
−
1
)
(
1
−
2
)
3
+
(
2
−
3
)
3
+
(
3
−
1
)
3
=
(
−
1
)
×
(
−
1
)
×
2
(
−
1
)
3
+
(
−
1
)
3
+
2
3
=
2
−
1
−
1
+
8
=
2
6
=
1
3
© examsnet.com
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