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IPMAT Indore 2022 (SA) PYQs

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IPMAT Indore 2022

Modern Math
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Binomial Theorem

Easy

The sum of the coefficients of all the terms in the expansion of (5x−9)4(5 x-9)^{4} is __________.

Entered answer:

✅ Correct Answer: 256
To find the sum of coefficients in any polynomial expansion, substitute x=1x = 1.
=(5x−9)4=(5x - 9)^4 \newline =(5(1)−9)4=(5(1) - 9)^4 \newline =(5−9)4=(5 - 9)^4 \newline =(−4)4=(-4)^4 \newline =256=\boxed{256}
How does this work?
When we expand (5x−9)4(5x - 9)^4, we get: ax4+bx3+cx2+dx+eax^4 + bx^3 + cx^2 + dx + e
To find a+b+c+d+ea + b + c + d + e, substitute x=1x = 1:
a(1)4+b(1)3+c(1)2+d(1)+e=a+b+c+d+ea(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = a + b + c + d + e
This method works for any polynomial and is much faster than full expansion.
∴\therefore The sum of coefficients =256= 256
You do not have to expand it entirely:

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IPMAT Indore 2023

If three consecutive coefficients in the expansion of (x+y)n(x+y)^n are in the ratio 1:9:631:9:63, then the value of nn is

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