lf `(1 +x)^(10) = a_0 +a_1x + a_2x^2++a_(10)x^(10)`, then value of `(a_0-a_2+a_4-a_6+a_8-a_(10))^2+(a_1-a_3+a_5-a_7+a_9)^2` is

Options

A. `2^(9)`
B. `3^(9)`
C. `2^(10)`
D. `3^(10)`

Answer

To solve the problem, we need to find the value of \((a_0 - a_2 + a_4 - a_6 + a_8 - a_{10})^2 + (a_1 - a_3 + a_5 - a_7 + a_9)^2\) where \( (1 + x)^{10} = a_0 + a_1x + a_2x^2 + \ldots + a_{10}x^{10} \).

### Step 1: Identify the coefficients
From the binomial theorem, we know that:
\[ (1 + x)^{10} = \sum_{k=0}^{10} \binom{10}{k} x^k \]
Thus, the coefficients are:
- \( a_k = \binom{10}{k} \)

### Step 2: Calculate \( a_0, a_2, a_4, a_6, a_8, a_{10} \)
Using the binomial coefficients:
- \( a_0 = \binom{10}{0} = 1 \)
- \( a_2 = \binom{10}{2} = 45 \)
- \( a_4 = \binom{10}{4} = 210 \)
- \( a_6 = \binom{10}{6} = 210 \)
- \( a_8 = \binom{10}{8} = 45 \)
- \( a_{10} = \binom{10}{10} = 1 \)

### Step 3: Calculate \( a_0 - a_2 + a_4 - a_6 + a_8 - a_{10} \)
Now we compute:
\[ a_0 - a_2 + a_4 - a_6 + a_8 - a_{10} = 1 - 45 + 210 - 210 + 45 - 1 \]
Calculating step-by-step:
- \( 1 - 45 = -44 \)
- \( -44 + 210 = 166 \)
- \( 166 - 210 = -44 \)
- \( -44 + 45 = 1 \)
- \( 1 - 1 = 0 \)

So, \( a_0 - a_2 + a_4 - a_6 + a_8 - a_{10} = 0 \).

### Step 4: Calculate \( a_1, a_3, a_5, a_7, a_9 \)
Next, we calculate:
- \( a_1 = \binom{10}{1} = 10 \)
- \( a_3 = \binom{10}{3} = 120 \)
- \( a_5 = \binom{10}{5} = 252 \)
- \( a_7 = \binom{10}{7} = 120 \)
- \( a_9 = \binom{10}{9} = 10 \)

### Step 5: Calculate \( a_1 - a_3 + a_5 - a_7 + a_9 \)
Now we compute:
\[ a_1 - a_3 + a_5 - a_7 + a_9 = 10 - 120 + 252 - 120 + 10 \]
Calculating step-by-step:
- \( 10 - 120 = -110 \)
- \( -110 + 252 = 142 \)
- \( 142 - 120 = 22 \)
- \( 22 + 10 = 32 \)

So, \( a_1 - a_3 + a_5 - a_7 + a_9 = 32 \).

### Step 6: Calculate the final expression
Now we need to compute:
\[ (a_0 - a_2 + a_4 - a_6 + a_8 - a_{10})^2 + (a_1 - a_3 + a_5 - a_7 + a_9)^2 \]
Substituting the values we found:
\[ 0^2 + 32^2 = 0 + 1024 = 1024 \]

### Final Answer
Thus, the value is:
\[ \boxed{1024} \]