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MCQ

Fill in the Blanks

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

#### Options

12:32

1:32

32:12

32:1

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#### Solution

The ratio of the coefficient of x^{15} 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 x^{15}

30 – 3r = 15

i.e., r = 5

Therefore, ^{15}C_{5} (2)5 is the coefficient of x^{15 } ....(From (1))

To find the term independent of x

Put 30 – 3r = 0

Thus ^{15}C_{10} 2^{10} 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`

Concept: General and Middle Terms

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