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CDS Model Paper 1 Mathematics
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© examsnet.com
Question : 68
Total: 100
If
m
and
n
are the roots of the equation
a
x
2
+
b
x
+
c
=
0
, then the equation whose roots are
(
m
2
+
1
)
m
and
(
n
2
+
1
)
n
is
a
c
x
2
+
(
a
b
+
b
c
)
x
+
b
2
+
(
a
−
c
)
2
=
0
a
c
x
2
+
(
a
b
−
b
c
)
x
+
b
2
+
(
a
−
c
)
2
=
0
a
c
x
2
−
(
a
b
−
b
c
)
x
+
b
2
−
(
a
−
c
)
2
=
0
a
c
x
2
−
(
a
b
+
b
c
)
x
+
b
2
−
(
a
−
c
)
2
=
0
Validate
Solution:
For the given equation
a
x
2
+
b
x
+
c
=
0
,
m
and
n
are the roots
∴ Sum of roots
=
m
+
n
=
−
b
a
and Product of roots
=
m
n
=
c
a
Sum of roots \& of required equation
=
m
2
+
1
m
and
n
2
+
1
n
=
m
2
+
1
m
+
n
2
+
1
n
=
m
2
n
+
n
+
m
n
2
+
m
m
n
=
m
n
(
m
+
n
)
+
(
m
+
n
)
m
n
=
(
m
+
n
)
(
m
n
+
1
)
m
n
=
−
b
a
(
c
a
+
1
)
c
a
=
−
b
(
a
+
c
)
a
c
Put
m
+
n
=
−
b
a
and
m
n
=
c
a
∴ Product of roots
=
m
2
+
1
m
×
n
2
+
1
n
=
(
m
2
+
1
)
(
n
2
+
1
)
m
n
=
m
2
n
2
+
n
2
+
m
2
+
1
m
n
=
(
m
n
)
2
+
(
m
+
n
)
2
−
2
m
n
+
1
m
n
=
(
c
a
)
2
+
(
−
b
a
)
2
−
2
(
c
a
)
+
1
c
a
=
c
2
+
b
2
−
2
a
c
+
a
2
a
c
=
b
2
+
(
a
−
c
)
2
a
c
We know that, quadratic equation is of the form
x
2
−
(Sum of roots)
x
+
Product of roots
=
0
⇒
x
2
−
(
−
b
(
a
+
c
)
a
c
)
x
+
(
b
2
+
(
a
−
c
)
2
a
c
)
=
0
⇒
a
c
x
2
+
b
(
a
+
c
)
x
+
b
2
+
(
a
−
c
)
2
=
0
⇒
a
c
x
2
+
(
a
b
+
b
c
)
x
+
b
2
+
(
a
−
c
)
2
=
0
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