JEE Main 2021 · 20 July · evening · Single correct · +4 / −1
For the natural numbers m, n, if (1 - y)ᵐ(1 + y)ⁿ = 1 + a₁y + a₂y² + .... + aₘ₊ₙyᵐ⁺ⁿ and a₁ = a₂ = 10, then the value of (m + n) is equal to :
- (a) 88
- (b) 64
- (c) 100
- (d) 80
Answer and solution
Answer (d) 80
Solution
Maths sir · 4 steps
Read off the y and y² coefficients of the product: two equations, and m+n drops out neatly.
- Coefficient of y. From (1-y)ᵐ the y term is -my, and from (1+y)ⁿ it is ny. So a₁ = n - m = 10.
- Coefficient of y squared. A y² term can come three ways: y² from the first bracket, y² from the second, or y from each. So a₂ = C(m, 2) + C(n, 2) - mn.
a₂ = (m² - m + n² - n)/2 - mn = ((n-m)² - (m+n))/2
- Put in numbers. With n - m = 10 and a₂ = 10: (100 - (m+n))/2 = 10, so 100 - (m+n) = 20. As a check, n = 45 and m = 35 are natural numbers.
- So. m + n = 80, which is option (d).
The idea to keep. For a product of two binomials, the y² coefficient collects every way the powers can add to 2. Rewrite it in terms of (n-m) and (m+n), and you often never need m and n separately.
Answer from the NTA final answer key. Solution by PrepWiser.