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Step-by-Step Solution
Step 1: Write the Binomial Expression
The given expression is
(2x^3 + \frac{3}{x})^{10} .
We want to find the sum of coefficients of all positive even powers of x in its expansion.
Step 2: Determine the General Term
The general term T_{r+1} in the expansion of
(a + b)^n
is given by
\binom{n}{r} \, a^{n-r} \, b^r.
Here,
a = 2 x^3, \; b = \frac{3}{x}, \; n = 10.
So the general term of
(2x^3 + \frac{3}{x})^{10}
is
T_{r+1} = \binom{10}{r} \, (2x^3)^{\,10-r} \, \left(\frac{3}{x}\right)^{r}.
Step 3: Simplify the General Term
Simplifying,
T_{r+1}
= \binom{10}{r} \, 2^{\,10-r} \, x^{3(10-r)} \, 3^r \, x^{-r}
= \binom{10}{r} \, 2^{\,10-r} \, 3^r \, x^{30 - 3r - r}
= \binom{10}{r} \, 2^{\,10-r} \, 3^r \, x^{30 - 4r}.
Step 4: Find the Condition for Positive Even Powers of x
We require the exponent of x to be both positive and even. That is:
30 - 4r \text{ is positive and even.}
Evenness: 30 - 4r is always even (since 30 is even and 4r is always even).
Positivity: 30 - 4r > 0 \implies 4r < 30 \implies r < 7.5.
Hence, the values of r that satisfy are r = 0, 1, 2, 3, 4, 5, 6, 7.
(Beyond r = 7, the power of x would become non-positive.)
Step 5: Sum of Coefficients for These Terms
The coefficient of x^{30-4r} in the general term is
\binom{10}{r} 2^{\,10-r} 3^r.
Therefore, the sum of the coefficients of all positive even powers of x is
S = \sum_{r=0}^{7} \binom{10}{r} \, 2^{\,10-r} \, 3^r.
Step 6: Relate to the Full Binomial Sum and Subtract Unwanted Terms
The sum of coefficients in the full expansion of
(2 + 3)^{10}
is
(5)^{10} = 5^{10}.
That sum corresponds to
\sum_{r=0}^{10} \binom{10}{r} 2^{\,10-r} 3^r = 5^{10}.
We only need terms for r = 0 to 7 . Hence, we must subtract the terms corresponding to r = 8, 9, 10 from 5^{10} .
Those terms are:
\binom{10}{8} 2^{\,10-8} 3^8, \quad
\binom{10}{9} 2^{\,10-9} 3^9, \quad
\binom{10}{10} 2^{\,10-10} 3^{10}.
Numerically,
\binom{10}{8} = 45, \quad 2^2 = 4, \\
\binom{10}{9} = 10, \quad 2^1 = 2, \\
\binom{10}{10} = 1, \quad 2^0 = 1.
Hence, the sum of these three unwanted terms is:
45 \times 4 \times 3^8 + 10 \times 2 \times 3^9 + 1 \times 1 \times 3^{10}.
Step 7: Final Computation
Therefore,
S = 5^{10} - \left[ 45 \cdot 4 \cdot 3^8 + 10 \cdot 2 \cdot 3^9 + 3^{10} \right].
After factoring out 3^9 where convenient, one obtains:
S = 5^{10} - \left( 60 \times 3^9 + 20 \times 3^9 + 3 \times 3^9 \right)
= 5^{10} - 83 \times 3^9.
Step 8: Identify the Parameter \beta
We are given that the sum of coefficients is
5^{10} - \beta \, 3^9.
Comparing this with the result
5^{10} - 83 \times 3^9,
we see that
\beta = 83.
Answer:
\beta = 83.
