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UPSC 2016 CDS I Math Paper
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© examsnet.com
Question : 9
Total: 100
If the polynomial
x
6
+
p
x
5
+
q
x
4
−
x
2
−
x
−
3
is divisible by
(
x
4
−
1
)
,
then the value of
p
2
+
q
2
is
[2016 CDS-I]
1
9
10
13
Validate
Solution:
Let
p
(
x
)
=
x
6
+
p
x
5
+
q
x
4
−
x
2
−
x
−
3
x
4
−
1
=
(
x
2
+
1
)
(
x
+
1
)
(
x
−
1
)
If p(x) is divisible by
x
4
−
1
, then p(a) = 0 and
p
(
−
1
)
=
0
.
p
(
1
)
=
1
+
p
+
q
−
1
−
1
−
3
=
p
+
q
−
4
=
0
⇒
p
+
q
=
4
......(1)
p
(
−
1
)
=
1
−
p
+
q
−
1
+
1
−
3
−
p
+
q
−
2
=
0
⇒
−
p
+
q
=
2
....(2)
Adding (1) and (2), we have
2
q
=
6
⇒
q
=
3
∴
p
=
1
So,
p
2
+
q
2
=
1
+
9
=
10
© examsnet.com
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