Consider the following statements
I. The term containing `x^(2)` does not exist in the given expansion.
II. The sum of the coefficients of all the terms in the given expansion is `2^(15)`.
Which of the above statement (s) is `//` are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the two statements regarding the expansion of the expression \( (x^2 + \frac{1}{x})^{15} \).

### Step 1: Determine the General Term
The general term \( T_{r+1} \) in the binomial expansion of \( (x^2 + \frac{1}{x})^{15} \) can be expressed as:
\[ T_{r+1} = \binom{15}{r} (x^2)^{15-r} \left(\frac{1}{x}\right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{15}{r} x^{2(15-r)} x^{-r} = \binom{15}{r} x^{30 - 3r} \]

### Step 2: Check for the Existence of the Term \( x^2 \)
To find if the term containing \( x^2 \) exists, we set the exponent equal to 2:
\[ 30 - 3r = 2 \]
Solving for \( r \):
\[ 30 - 2 = 3r \implies 28 = 3r \implies r = \frac{28}{3} \]
Since \( r \) must be a non-negative integer, \( \frac{28}{3} \) is not valid. Therefore, there is **no term** containing \( x^2 \) in the expansion.

### Step 3: Verify the First Statement
Since we found that \( r \) cannot be a fraction, the first statement is **correct**: "The term containing \( x^2 \) does not exist in the given expansion."

### Step 4: Sum of the Coefficients
To find the sum of the coefficients of all terms in the expansion, we substitute \( x = 1 \):
\[ (1^2 + \frac{1}{1})^{15} = (1 + 1)^{15} = 2^{15} \]
This confirms that the sum of the coefficients is indeed \( 2^{15} \).

### Step 5: Verify the Second Statement
Since we calculated the sum of the coefficients to be \( 2^{15} \), the second statement is also **correct**: "The sum of the coefficients of all the terms in the given expansion is \( 2^{15} \)."

### Conclusion
Both statements are correct. Therefore, the answer is that both statements I and II are correct.

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