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NIMCET 2019 Question Paper with solutions
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© examsnet.com
Question : 9
Total: 120
Let f(x) be a polynomial satisfying, f(0) = 2, f'(0) = 3 and f''(x) = f(x). Then f(4) is equal to
5
(
e
8
−
1
)
2
e
4
(
5
e
8
−
1
)
2
e
4
2
e
4
(
5
e
8
−
1
)
2
e
4
5
(
e
8
−
1
)
Validate
Solution:
Given,
f
(
0
)
=
2
,
f
′
(
0
)
=
3
and
f
′
′
(
x
)
=
f
(
x
)
⇒
d
2
y
d
x
2
−
y
=
0
Auxiliary equation
=
m
2
−
1
=
0
m
=
±
1
Solution of given differential equation is
y
=
a
e
x
+
b
e
−
x
⇒
f
(
x
)
=
a
e
x
+
b
e
−
x
f
(
0
)
=
a
+
b
=
2
....(i)
f
′
(
x
)
=
a
e
x
−
b
e
−
x
f
′
(
0
)
=
a
−
b
=
3
...(ii)
Solving Eqs. (i) and (ii), we get
a
=
5
2
b
=
=
−
1
2
∴
f
(
x
)
=
1
2
(
5
e
x
−
e
−
x
)
f
(
4
)
=
1
2
(
5
e
4
−
e
−
4
)
f
(
4
)
=
1
2
(
5
e
8
−
1
e
4
)
f
(
4
)
=
5
e
8
−
1
2
e
4
© examsnet.com
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