If A denotes the sum of all the coefficients in the expansion of (1-3 x+10 x²)ⁿ and B denotes the sum of all the coefficients in the expansion of (1+x²)ⁿ, then :

Solution

Maths sir · 4 steps

Put x = 1 in both expressions, and the two sums come out as powers of 8 and 2.

1. Why x = 1. In any expansion, each term is a coefficient times a power of x. At x = 1 every power of x becomes 1, so only the coefficients are left, and they add up.
2. Find A. Put x = 1 in (1 - 3x + 10x²)ⁿ: A = (1 - 3 + 10)ⁿ = 8ⁿ.
3. Find B. Put x = 1 in (1 + x²)ⁿ: B = (1 + 1)ⁿ = 2ⁿ.

   A = 8ⁿ = (2³)ⁿ = (2ⁿ)³ = B³

4. So. A = B³, which is option (d).

The idea to keep. Sum of all coefficients means put x = 1. Then compare the two answers by writing both as powers of the same base.

Answer from the NTA final answer key. Solution by PrepWiser.