IIT JEE Advanced 2007 Paper 1

We have

lim

t→x

t2f(x)−x2f(t)

t−x

= 1
⇒

lim

t→x

2tf(x)−x2f′(t)

1−0

= 1
⇒ 2x f (x) - x2 f ' (x) = 1
⇒

x2f′(x)−2xf(x)

x4

= −

1

x4

⇒ d =

(f(x))

x2

= −

1

x4

(1)
On integrating both sides of this equation, we get

f(x)

x2

=

1

3x3

+c
Now, since
f (1) = 1 ⇒ c =

2

3

we get
f (x) =

1

3x

+

2

3

x2