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UPSC NDA 2 2022 Mathematics Solved Paper

Question Numbers: 22-24

Consider the binomial expansion of (p + qx)9 :

Question : 22 of 120

Marks: +1, -0

What is the value of q if the coefficients of x3 and x6 are equal ?

Solution:

Concept:

In binomial expansion

(a+b)^n

, the general term is

T_{r+1} = {}^nC_r a^{n-r} b^r

Explanation:

For

(p+qx)^9

, the term containing

x^r

{}^9C_r p^{9-r} (qx)^r = {}^9C_r p^{9-r} q^r x^r

The coefficient of

x^3

occurs when

r=3

, giving

{}^9C_3 p^{6} q^3 = 84 p^6 q^3

The coefficient of

x^6

occurs when

r=6

, giving

{}^9C_6 p^{3} q^6 = 84 p^3 q^6

Set these coefficients equal:

84 p^6 q^3 = 84 p^3 q^6

Divide both sides by

84 p^3 q^3

(assuming

p>0

q>0

p^3 = q^3

Therefore,

p = q

Answer:

q = p

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