UPSEE Solved Model Paper 6

We can solve this problem using the definition of
Binomial theorem for a rational index.
Binomial Theorem for a rational Index
If n is a nonzero rational number and –1 < x < 1 then
(1+x)n = 1+

n

1!

x+

n(n−1)

2!

x2 +

n(n−1)(n−2)

3!

x3 + ... +

n(n−1)(n−2)...(n−r+1)

r!

xr + ... to ∞
In the above expansion, the first term must be unity. In the expansion of (a+x)n , where n is either a negative integer or a fraction, we proceed as follows
(a+x)n = [a(1+

x

a

)]n = an(1+

x

a

)n
= an[1+n

x

a

+

n(n−1)

2!

(

x

a

)2+...]
By using this definition we can find the power of x is valid
We have

1

√6−3x

=

1

√6(√1− 3x 6 )

=

1

√6(√1− x 2 )

The expansion in powers of x is valid if
|

x

2

| < 1 i.e., - 2 < x < 2
(value inside the square root should be positive)
Therefore
The expansion of

1

√6−3x

in powers of x is valid if
- 2 < x < 2