Sum Of Coefficients
If every variable in the polynomial expansion of (3x - 4y)^{12} is set equal to one, the resulting sum of all the coefficients of the expansion equals which value?
Select the correct option:
Solution
1
The sum of all coefficients of any polynomial expression is obtained by substituting one for every variable, a quick evaluation trick central to JEE Advanced binomial problems. For (3x - 4y)^{12}, setting x = 1 and y = 1 gives (3·1 - 4·1)^{12} = (3 - 4)^{12} = (-1)^{12}. Since the exponent 12 is even, (-1)^{12} = 1, so the sum of all coefficients is exactly 1. Option -1 would arise only for an odd exponent, which is not the case here. Option 7^{12} mistakenly adds the magnitudes 3 and 4 instead of subtracting, ignoring the sign in the binomial. Option 0 incorrectly assumes the positive and negative coefficients cancel completely, which they do not. The key insight is that substituting unity collapses the entire structure into a single power of the numerical base. Plausibility check: since the base 3 - 4 = -1 has magnitude one, any power of it must have magnitude one, and the even exponent fixes the sign as positive, fully consistent with the answer 1.
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About This Question
- Subject
- mathematics
- Chapter
- binomial theorem and its simple applications
- Topic
- sum of coefficients
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
1
The sum of all coefficients of any polynomial expression is obtained by substituting one for every variable, a quick evaluation trick central to JEE Advanced binomial problems. For (3x - 4y)^{12}, setting x = 1 and y = 1 gives (3·1 - 4·1)^{12} = (3 - 4)^{12} = (-1)^{12}. Since the exponent 12 is even, (-1)^{12} = 1, so the sum of all coefficients is exactly 1. Option -1 would arise only for an odd exponent, which is not the case here. Option 7^{12} mistakenly adds the magnitudes 3 and 4 instead of subtracting, ignoring the sign in the binomial. Option 0 incorrectly assumes the positive and negative coefficients cancel completely, which they do not. The key insight is that substituting unity collapses the entire structure into a single power of the numerical base. Plausibility check: since the base 3 - 4 = -1 has magnitude one, any power of it must have magnitude one, and the even exponent fixes the sign as positive, fully consistent with the answer 1.
This easy difficulty mathematics question is from the chapter binomial theorem and its simple applications, covering the topic of sum of coefficients. It appeared in the 2025 exam.
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