Detailed Solution: Finding the Maximum Value of a Term Independent of a Variable

This problem involves two core concepts: the Binomial Theorem for expanding expressions and differential calculus for finding the maximum value of a function.

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1. Key Concept: The General Term of a Binomial Expansion

For a binomial expansion of the form $(A+B)^n$, the general term (or $(r+1)^{th}$ term) is given by the formula: $T_{r+1} = {}^{n}C_r A^{n-r} B^r$ where $n$ is the power of the binomial, $r$ is the index (starting from 0), and ${}^{n}C_r = \frac{n!}{r!(n-r)!}$ is the binomial coefficient.

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2. Step-by-Step Derivation

Given the expression: $\left(\mathrm{t}^{2} x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{\mathrm{t}}\right)^{15}$ Here, $A = t^2 x^{\frac{1}{5}}$, $B = \frac{(1-x)^{\frac{1}{10}}}{t}$, and $n=15$.

Step 2.1: Write down the general term $T_{r+1}$ Applying the general term formula: $T_{r+1} = {}^{15}C_r \left(t^2 x^{\frac{1}{5}}\right)^{15-r} \left(\frac{(1-x)^{\frac{1}{10}}}{t}\right)^r$

Explanation: This is the foundational step, directly applying the binomial theorem to express any term in the expansion.

Step 2.2: Simplify the powers of $t$, $x$, and $(1-x)$ We need to combine the terms involving $t$, $x$, and $(1-x)$ separately. $T_{r+1} = {}^{15}C_r \cdot (t^2)^{15-r} \cdot (x^{\frac{1}{5}})^{15-r} \cdot \frac{((1-x)^{\frac{1}{10}})^r}{t^r}$ $T_{r+1} = {}^{15}C_r \cdot t^{2(15-r)} \cdot x^{\frac{1}{5}(15-r)} \cdot (1-x)^{\frac{r}{10}} \cdot t^{-r}$ $T_{r+1} = {}^{15}C_r \cdot t^{30-2r} \cdot t^{-r} \cdot x^{\frac{15-r}{5}} \cdot (1-x)^{\frac{r}{10}}$ Combining the powers of $t$: $T_{r+1} = {}^{15}C_r \cdot t^{30-2r-r} \cdot x^{\frac{15-r}{5}} \cdot (1-x)^{\frac{r}{10}}$ $T_{r+1} = {}^{15}C_r \cdot t^{30-3r} \cdot x^{\frac{15-r}{5}} \cdot (1-x)^{\frac{r}{10}}$

Explanation: We simplify the expression using exponent rules $(a^m)^n = a^{mn}$ and $a^m a^n = a^{m+n}$ to isolate the powers of $t$, $x$, and $(1-x)$. This is crucial for identifying the term independent of $t$.

Step 2.3: Find the value of $r$ for the term independent of $t$ A term is independent of $t$ if the power of $t$ in that term is zero. So, we set the exponent of $t$ to 0: $30 - 3r = 0$ $3r = 30$ $r = 10$

Explanation: The phrase "independent of $t$" directly translates to the coefficient of $t$ being $0$ in the general term. Solving this equation gives us the specific value of $r$ that corresponds to this term.

Step 2.4: Substitute $r=10$ back into the general term Now that we have $r=10$, we can find the specific term, which is $T_{10+1} = T_{11}$. $T_{11} = {}^{15}C_{10} \cdot x^{\frac{15-10}{5}} \cdot (1-x)^{\frac{10}{10}}$ $T_{11} = {}^{15}C_{10} \cdot x^{\frac{5}{5}} \cdot (1-x)^{1}$ $T_{11} = {}^{15}C_{10} \cdot x \cdot (1-x)$

Explanation: We substitute the calculated value of $r$ into our simplified general term to obtain the exact expression for the term independent of $t$.

Step 2.5: Find the maximum value of $x(1-x)$ The term independent of $t$ is $T_{11} = {}^{15}C_{10} \cdot x(1-x)$. Since ${}^{15}C_{10}$ is a constant, to maximize $T_{11}$, we need to maximize the function $f(x) = x(1-x)$. Let $f(x) = x - x^2$. To find the maximum value of $f(x)$, we use calculus:

1. First Derivative: Calculate $f'(x)$ and set it to zero to find critical points. $f'(x) = \frac{d}{dx}(x - x^2) = 1 - 2x$ Set $f'(x) = 0$: $1 - 2x = 0$ $2x = 1$ $x = \frac{1}{2}$
2. Second Derivative Test: Calculate $f''(x)$ to determine if the critical point is a maximum or minimum. $f''(x) = \frac{d}{dx}(1 - 2x) = -2$ Since $f''(\frac{1}{2}) = -2 < 0$, the function $f(x)$ has a local maximum at $x = \frac{1}{2}$.

Now, substitute $x = \frac{1}{2}$ into $f(x)$: $f\left(\frac{1}{2}\right) = \frac{1}{2}\left(1 - \frac{1}{2}\right) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ So, the maximum value of $x(1-x)$ is $\frac{1}{4}$.

Explanation: This is a crucial optimization step. We treat $x(1-x)$ as a function and use the standard calculus method (first and second derivatives) to find its absolute maximum. The condition $x \ge 0$ is implicitly satisfied by $x=1/2$. Alternatively, recognize $f(x) = x-x^2$ is a downward-opening parabola with roots at $x=0, 1$, so its maximum occurs at the vertex, $x = -\frac{b}{2a} = -\frac{1}{2(-1)} = \frac{1}{2}$.

Step 2.6: Calculate $K$ and $8K$ The maximum value of the term independent of $t$ is given as $K$. $K = {}^{15}C_{10} \cdot \left(\text{maximum value of } x(1-x)\right)$ $K = {}^{15}C_{10} \cdot \frac{1}{4}$

We need to calculate $8K$: $8K = 8 \cdot \left({}^{15}C_{10} \cdot \frac{1}{4}\right)$ $8K = 2 \cdot {}^{15}C_{10}$

Now, we calculate the binomial coefficient ${}^{15}C_{10}$. Using the property ${}^n C_r = {}^n C_{n-r}$: ${}^{15}C_{10} = {}^{15}C_{15-10} = {}^{15}C_5$ ${}^{15}C_5 = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1}$ ${}^{15}C_5 = \frac{15}{5 \times 3} \times \frac{12}{4} \times \frac{14}{2} \times 13 \times 11$ ${}^{15}C_5 = 1 \times 3 \times 7 \times 13 \times 11$ ${}^{15}C_5 = 21 \times 143$ ${}^{15}C_5 = 3003$

Finally, substitute this value back into the expression for $8K$: $8K = 2 \cdot 3003$ $8K = 6006$

Explanation: We combine the constant binomial coefficient with the maximum value of the variable part to find $K$. Then, a simple multiplication gives us the final answer. Using the symmetry property of binomial coefficients, ${}^{n}C_r = {}^{n}C_{n-r}$, simplifies the calculation for ${}^{15}C_{10}$ to ${}^{15}C_5$.

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3. Tips and Common Mistakes to Avoid

• Careful with Exponents: Pay close attention to negative exponents and fractional powers when simplifying the general term. A small error in handling $t^{-r}$ or $x^{1/5}$ can lead to an incorrect value of $r$.
• Term Independent of Variable: Remember that "independent of a variable" means the exponent of that variable must be zero.
• Maximizing a Function: When finding the maximum/minimum of a quadratic function like $ax^2+bx+c$, you can either use calculus ($f'(x)=0, f''(x)$ test) or recognize its parabolic nature. For $x(1-x) = -x^2+x$, it's a downward-opening parabola, so its vertex is the maximum. The x-coordinate of the vertex is $-b/(2a)$.
• Binomial Coefficient Calculation: Don't rush the calculation of ${}^{n}C_r$. Use the symmetry property ${}^{n}C_r = {}^{n}C_{n-r}$ to simplify calculations (e.g., ${}^{15}C_{10}$ is easier to calculate as ${}^{15}C_5$).

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4. Summary and Key Takeaway

This problem effectively tests your understanding of applying the Binomial Theorem to find a specific term in an expansion and then using differential calculus to optimize (maximize in this case) that term with respect to another variable. The key steps are:

1. Formulate the general term.
2. Simplify exponents to isolate powers of relevant variables.
3. Set the exponent of the variable to be eliminated (here, $t$) to zero to find the specific term.
4. Treat the remaining expression as a function of the other variable (here, $x$) and use calculus to find its maximum or minimum value.
5. Perform the final calculation as required by the question.

The final answer is $\boxed{6006}$.