1. Key Concept: Binomial Theorem and General Term

The Binomial Theorem provides a formula for expanding expressions of the form $(A+B)^n$. For such an expansion, the general term, often denoted as $(T_{r+1})$, which represents the $(r+1)^{th}$ term, is given by:

$T_{r+1} = {^nC_r} A^{n-r} B^r$

Here, ${^nC_r}$ is the binomial coefficient, calculated as ${^nC_r} = \frac{n!}{r!(n-r)!}$.

A term is considered "independent of $x$" if it does not contain the variable $x$. Mathematically, this means the exponent of $x$ in that particular term must be zero, since $x^0 = 1$.

2. Deriving the General Term for the Given Expression

The given binomial expression is $\left(\sqrt{\mathrm{a}} x^2+\frac{1}{2 x^3}\right)^{10}$. Comparing this with $(A+B)^n$, we identify:

• $n = 10$ (the power of the binomial)
• $A = \sqrt{a} x^2$ (the first term)
• $B = \frac{1}{2 x^3}$ (the second term)

Now, we substitute these into the general term formula:

$T_{r+1} = {^{10}C_r} \left(\sqrt{a} x^2\right)^{10-r} \left(\frac{1}{2 x^3}\right)^r$

To find the term independent of $x$, we need to isolate all parts containing $x$ and combine their exponents. Let's simplify the expression by applying the power rules $(uv)^m = u^m v^m$ and $\left(\frac{1}{u}\right)^m = u^{-m}$:

$T_{r+1} = {^{10}C_r} \left((\sqrt{a})^{10-r} (x^2)^{10-r}\right) \left(\left(\frac{1}{2}\right)^r \left(\frac{1}{x^3}\right)^r\right)$ $T_{r+1} = {^{10}C_r} (a^{1/2})^{10-r} x^{2(10-r)} \left(\frac{1}{2}\right)^r x^{-3r}$

Next, we combine the terms involving $x$ using the rule $x^m \cdot x^k = x^{m+k}$:

$T_{r+1} = {^{10}C_r} a^{(10-r)/2} \left(\frac{1}{2}\right)^r x^{(20-2r) - 3r}$ $T_{r+1} = {^{10}C_r} a^{(10-r)/2} \left(\frac{1}{2}\right)^r x^{20-5r}$

This is the general term for the expansion, where the power of $x$ is $(20-5r)$.

3. Finding the Value of r for the Term Independent of x

For the term to be independent of $x$, its exponent must be zero. Therefore, we set the power of $x$ from our general term to zero:

$20 - 5r = 0$

Now, we solve for $r$:

$5r = 20$ $r = \frac{20}{5}$ $r = 4$

Tip: In binomial expansions, $r$ must always be a non-negative integer (an integer between 0 and $n$, inclusive). Our calculated value $r=4$ is valid, as $0 \le 4 \le 10$. If $r$ were a negative number or a fraction, it would indicate that there is no term independent of $x$ in the expansion.

4. Calculating the Term Independent of x

Now that we have found $r=4$, we substitute this value back into the general term expression (without the $x$ part, as its exponent will be 0):

$T_{4+1} = T_5 = {^{10}C_4} a^{(10-4)/2} \left(\frac{1}{2}\right)^4$ $T_5 = {^{10}C_4} a^{6/2} \left(\frac{1}{16}\right)$ $T_5 = {^{10}C_4} a^3 \left(\frac{1}{16}\right)$

Next, we calculate the binomial coefficient ${^{10}C_4}$:

${^{10}C_4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}$ ${^{10}C_4} = \frac{10 \times 9 \times 8 \times 7}{24} = 10 \times 3 \times 7 = 210$

Substitute this value back into the expression for $T_5$:

$T_5 = 210 \cdot a^3 \cdot \frac{1}{16}$

We are given that the value of this term independent of $x$ is 105:

$210 \cdot a^3 \cdot \frac{1}{16} = 105$

5. Solving for a and a^2

Now, we need to solve this equation for $a$:

$210 a^3 = 105 \times 16$

Divide both sides by 210:

$a^3 = \frac{105 \times 16}{210}$

Notice that 210 is exactly twice 105 ($210 = 2 \times 105$). So, we can simplify the fraction:

$a^3 = \frac{1}{2} \times 16$ $a^3 = 8$

To find $a$, we take the cube root of both sides:

$a = \sqrt[3]{8}$ $a = 2$

Finally, the question asks for the value of $a^2$:

$a^2 = 2^2$ $a^2 = 4$

Common Mistake: Be meticulous with arithmetic calculations, especially when dealing with binomial coefficients, powers, and fractions. A small error can lead to an incorrect final answer. It's often helpful to simplify fractions before multiplying large numbers.

6. Summary and Key Takeaway

This problem is a classic application of the Binomial Theorem. The key steps involve:

1. Writing down the general term $(T_{r+1})$ of the binomial expansion.
2. Collecting all terms involving $x$ and determining the total exponent of $x$.
3. Setting this exponent to zero to find the value of $r$ that corresponds to the term independent of $x$.
4. Substituting the obtained value of $r$ back into the general term (excluding the $x$ part) to calculate its numerical value.
5. Equating this numerical value to the given constant to solve for the unknown variable, in this case, $a$.

Understanding how to manipulate exponents and binomial coefficients is crucial for solving such problems efficiently and accurately.