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KEAM 2013 Math Paper
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© examsnet.com
Question : 26
Total: 120
Let
t
n
denote the
n
t
h
term in a binomial expansion. If
t
6
t
5
in the expansion of
(
a
+
b
)
n
+
4
and
t
5
t
4
in the expansion of
(
a
+
b
)
n
are equal, then
n
is
9
11
13
15
17
Validate
Solution:
t
6
and
t
5
in the expansion of
(
a
+
b
)
n
+
4
is
t
5
=
t
4
+
1
=
n
+
4
C
4
a
n
+
4
−
4
b
4
=
n
+
4
C
4
a
n
⋅
b
4
and
t
6
=
t
5
+
1
=
n
+
4
C
5
a
n
+
4
−
5
b
5
=
n
+
4
C
5
a
n
−
1
b
5
∴
t
6
t
5
=
n
+
4
C
5
⋅
a
n
−
1
⋅
b
5
n
+
4
C
4
⋅
a
n
⋅
b
4
=
n
+
4
C
5
n
+
4
C
4
(
b
a
)
=
n
5
(
b
a
)
......(i)
Now,
t
5
and
t
4
in the expansion of
(
a
+
b
)
n
is
t
5
=
t
4
+
1
=
n
C
4
⋅
a
n
−
4
⋅
b
4
and
t
4
=
t
3
+
1
=
n
C
3
⋅
a
n
−
3
⋅
b
3
∴
t
5
t
4
=
n
C
4
⋅
a
n
−
4
⋅
b
4
n
C
3
⋅
a
n
−
3
⋅
b
3
=
n
−
3
4
⋅
(
b
a
)
.....(ii)
On equating Eqs. (i) and (ii), we get
n
5
(
b
a
)
=
n
−
3
4
(
b
a
)
⇒
4
n
=
5
n
−
15
⇒
n
=
15
© examsnet.com
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