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KVPY SX SB Exam 2013 Question Paper
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© examsnet.com
Question : 1
Total: 120
Part-I
MATHEMATICS
The sum of non-real roots of the polynomial equation
x
3
+
3
x
2
+
3
x
+
3
=
0
equals 0
lies between 0 and 1
lies between -1 and 0
has absolute value bigger than 1
Validate
Solution:
f
(
x
)
=
x
3
+
3
x
2
+
3
x
+
3
=
0
f
′
(
x
)
=
3
x
2
+
6
x
+
3
=
3
(
x
+
1
)
2
≥
0
f(x) is Increasing function
Now
f
(
−
3
)
=
−
6
<
0
f
(
−
2
)
=
1
>
0
∴ real root
γ
lies between - 3 and -2
α
+
β
+
γ
=
−
3
;
−
3
<
γ
<
−
2
α
+
β
−
3
<
α
+
β
+
γ
<
α
+
β
−
2
α
+
β
−
3
<
−
3
<
α
+
β
−
2
−
1
<
α
+
β
<
0
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