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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
Solution
The ratio of the coefficient of x15 to the term independent of x in `x^2 + 2^15/x` is 1:32.
Explanation:
Let Tr+1 be the general term of `x^2 + 2^15/x`
So, Tr+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`
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