Question

The ratio of the coefficient of x¹⁵ to the term independent of x in `x^2 + 2^15/x` is ______.

Options

• 12:32

• 1:32

• 32:12

• 32:1

Answer 1

The ratio of the coefficient of x¹⁵ to the term independent of x in `x^2 + 2^15/x` is 1:32.

Explanation:

Let T_r+1 be the general term of `x^2 + 2^15/x`

So, T_r+1 = `""^15"C"_r  (x^2)^(15 - r)  2^r/x`

= `""^15"C"_r (2)^r  x^(30 - 3r)`  ....(1)

Now, for the coefficient of term containing x¹⁵

30 – 3r = 15

i.e., r = 5

Therefore, ¹⁵C₅ (2)5 is the coefficient of x¹⁵   ....(From (1))

To find the term independent of x

Put 30 – 3r = 0

Thus ¹⁵C₁₀ 2¹⁰ is the term independent of x ....(From (1))

Now the ratio is `(""^15"C"_5  2^5)/(""^15"C"_10  2^10) = 1/2^5 = 1/32`