Statement-1 (Assertion) and Statement-2 (Reason)
Each of the these examples also has four laternative choices ,
only one of which is the correct answer. You have to select the correct choice as given below .
Number of distincet terms in the
sum of expansion ` (1 + ax)^(10)+ (1-ax)^(10) ` is 22.

Options

A. Statement-1 is ture ,Statement-2 is treu, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is ture ,Statement-2 is treu, Statement-2 is not a correct explanation for Statement-1
C. Statement-1 is true ,Statement-2 is false
D. Statement-1 is true ,Statement-2 is ture

Explanation

To solve the problem of finding the number of distinct terms in the expansion of \( (1 + ax)^{10} + (1 - ax)^{10} \), we can follow these steps:

### Step 1: Expand the expressions
We start by expanding both expressions using the Binomial Theorem. According to the Binomial Theorem, the expansion of \( (1 + ax)^n \) is given by:

\[ (1 + ax)^n = \sum_{k=0}^{n} \binom{n}{k} (ax)^k = \sum_{k=0}^{n} \binom{n}{k} a^k x^k \]

Thus, we can write:

\[ (1 + ax)^{10} = \sum_{k=0}^{10} \binom{10}{k} a^k x^k \]

\[ (1 - ax)^{10} = \sum_{k=0}^{10} \binom{10}{k} (-a)^k x^k \]

### Step 2: Combine the expansions
Now, we combine the expansions:

\[ (1 + ax)^{10} + (1 - ax)^{10} = \sum_{k=0}^{10} \binom{10}{k} a^k x^k + \sum_{k=0}^{10} \binom{10}{k} (-a)^k x^k \]

### Step 3: Simplify the combined expression
When we add these two expansions, we notice that terms with odd powers of \( ax \) will cancel out, while terms with even powers will double:

\[ = 2 \sum_{k=0, k \text{ even}}^{10} \binom{10}{k} a^k x^k \]

The even powers of \( k \) are \( 0, 2, 4, 6, 8, 10 \).

### Step 4: Count the distinct terms
The distinct terms in the combined expansion correspond to the even powers of \( x \) from \( 0 \) to \( 10 \). The even integers in this range are:

- \( x^0 \)
- \( x^2 \)
- \( x^4 \)
- \( x^6 \)
- \( x^8 \)
- \( x^{10} \)

This gives us a total of 6 distinct terms.

### Step 5: Conclusion
Thus, the number of distinct terms in the expansion of \( (1 + ax)^{10} + (1 - ax)^{10} \) is 6, not 22. Therefore, the assertion that the number of distinct terms is 22 is false.

### Final Answer
The correct choice is that Statement-1 is false and Statement-2 is true.

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