If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of ( 2x³ + 3/x )¹⁰ is 5¹⁰ - β . 3⁹, then β is equal to ____________.

Solution

Maths sir · 4 steps

Every power here is even, so take the full sum 5¹⁰ and subtract only the negative-power terms.

1. General term. Tᵣ₊₁ = C(10, r)(2x³)¹⁰⁻ʳ(3/x)ʳ = C(10, r) 2¹⁰⁻ʳ 3ʳ x³⁰⁻⁴ʳ. The power 30-4r is always even, so every term has an even power of x.
2. Which are positive. 30-4r > 0 means r ≤ 7. The terms with r = 8, 9, 10 have powers -2, -6, -10. Putting x = 1 gives the sum of all coefficients as (2+3)¹⁰ = 5¹⁰, so the required sum is 5¹⁰ minus the coefficients for r = 8, 9, 10.

   C(10, 8)2² 3⁸ + C(10, 9)2· 3⁹ + 3¹⁰ = 60· 3⁹ + 20· 3⁹ + 3· 3⁹ = 83· 3⁹

3. Check the pieces. C(10, 8)× 4 × 3⁸ = 180 × 3⁸ = 60 × 3⁹, C(10, 9)× 2 × 3⁹ = 20 × 3⁹, and 3¹⁰ = 3 × 3⁹.
4. So. The sum is 5¹⁰ - 83 · 3⁹, so β = 83.

The common slip: 80. Leaving out the r = 10 term 3¹⁰ = 3 · 3⁹ gives 80, not 83.

The idea to keep. When you need the sum of coefficients over a set of powers, put x = 1 for the full total. Then subtract the few terms you do not want, instead of adding up the many you do.

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