If the sum of the coefficients of all the positive even powers of x in the binomial expansion of \(\left(2 x^{3}+\frac{3}{x}\right)^{10}\) is \(5^{10}-\beta \cdot 3^{9}\), then \(\beta\) is equal to

JEE Main · 2022 · 25 Jun 2022 · Shift 2 · Binomial Theorem

If the sum of the coefficients of all the positive even powers of x in the binomial expansion of

(2 x^{3} + \frac{3}{x})^{10}

5^{10} - β \cdot 3^{9}

, then

β

is equal to

Answer: 83

The general term of

(2 x^{3} + \frac{3}{x})^{10}

(r 10) (2 x^{3})^{10 - r} (\frac{3}{x})^{r} = (r 10) 2^{10 - r} 3^{r} x^{30 - 4 r}

, so the exponent of

x

30 - 4 r

, which is even for every

r

. Putting

x = 1

sums all the coefficients to

(2 + 3)^{10} = 5^{10}

. The exponent

30 - 4 r

is positive exactly for

r = 0

through

7

, so the sum over the positive even powers is

5^{10}

reduced by the contributions of

r = 8, 9, 10

, where the powers become negative. Those three terms are

(8 10) 2^{2} 3^{8} + (9 10) 2 \cdot 3^{9} + (10 10) 3^{10} = 180 \cdot 3^{8} + 20 \cdot 3^{9} + 3^{10}

, and rewriting each as a multiple of

3^{9}

gives

60 \cdot 3^{9} + 20 \cdot 3^{9} + 3 \cdot 3^{9} = 83 \cdot 3^{9}

. Hence the required sum is

5^{10} - 83 \cdot 3^{9}

, so

β = 83

JEE Main · 2022 · 25 Jun 2022 · Shift 2 · Integer answer