Maharashtra State BoardHSC Science (General) 11th
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Answer the following: If the coefficient of x2 and x3 in the expansion of (3 + kx)9 are equal, find k - Mathematics and Statistics

Sum

Answer the following:

If the coefficient of x2 and x3 in the expansion of (3 + kx)9 are equal, find k

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Solution

Here a = 3, b = kx, n = 9

We have, tr+1 = nCr, an–r .br 

= 9Cr (3)9–r (kx)r

= 9Cr (3)9–r . kr(x) 

For coefficient of x2, r = 2

For coefficient of x3, r = 3

But, coefficient of x2 = coefficient of x3

9C2 (3)7 . k2 = 9C3. 36 .k3

∴ `(9!)/(2!7!) (3)^7"k"^2 = (9!)/(3!6!) (3)^6 . "k"^3`

∴ `(9 xx 8 xx 7!)/(2 xx 1 xx 7!) (3)^7"k"^2 = (9 xx 8 xx 7 xx 6!)/(3 xx 2 xx 1 xx 6!) (3)^6*"k"^3`

∴ 36 × 37 × k2 = 84 × 36 × k3

∴ `36/84 xx 3` = k

∴ k = `9/7`

Concept: General Term in Expansion of (a + b)n
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 4 Methods of Induction and Binomial Theorem
Miscellaneous Exercise 4 | Q II. (14) | Page 86
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