Consider the statements : P : There exists some x IR such that f(x) + 2x = 2(1+x2) Q : There exists some x IR such that 2f(x) +1 = 2x(1+x) Then (A) both P and Q are true (B) P is true and Q is false (C) P is false and Q is true (D) both P and Q are false.

Options

A. both P and Q are true
B. P is true and Q is false
C. P is false and Q is true
D. both P and Q are false

Explanation

To solve the given problem, we need to analyze the two statements P and Q based on the function \( f(x) \).

**Given:**
- Statement P: There exists some \( x \in \mathbb{R} \) such that \( f(x) + 2x = 2(1+x^2) \)
- Statement Q: There exists some \( x \in \mathbb{R} \) such that \( 2f(x) + 1 = 2x(1+x) \)

We are also provided with the function:
\[ f(x) = (1 - x^2) \sin^2 x + x^2 \]

### Step 1: Analyze Statement P

We need to check if there exists some \( x \) such that:
\[ f(x) + 2x = 2(1 + x^2) \]

Substituting \( f(x) \):
\[ (1 - x^2) \sin^2 x + x^2 + 2x = 2(1 + x^2) \]
This simplifies to:
\[ (1 - x^2) \sin^2 x + x^2 + 2x = 2 + 2x^2 \]
Rearranging gives:
\[ (1 - x^2) \sin^2 x = x^2 - 2x + 2 \]
This can be rewritten as:
\[ (1 - x^2) \sin^2 x = (x - 1)^2 + 1 \]

### Step 2: Analyze the Equation

We can analyze the left side:
\[ (1 - x^2) \sin^2 x \]
The term \( 1 - x^2 \) is non-positive for \( |x| \geq 1 \) and non-negative for \( |x| < 1 \). The term \( \sin^2 x \) is always non-negative.

Now, for the right side:
\[ (x - 1)^2 + 1 \geq 1 \]
This means the left side must also be greater than or equal to 1 for the equality to hold.

### Step 3: Evaluate the Possibility

For \( |x| < 1 \), \( 1 - x^2 > 0 \) and \( \sin^2 x \) can take values between 0 and 1. Thus, the left side can be zero or positive, but we need to check if it can equal the right side, which is always greater than or equal to 1.

For \( |x| \geq 1 \), \( 1 - x^2 \leq 0 \), thus making the left side non-positive, which cannot equal the positive right side.

### Conclusion for Statement P:
Since we cannot find any \( x \) such that the equation holds true, **Statement P is false**.

### Step 4: Analyze Statement Q

Now we check Statement Q:
\[ 2f(x) + 1 = 2x(1+x) \]

Substituting \( f(x) \):
\[ 2((1 - x^2) \sin^2 x + x^2) + 1 = 2x(1 + x) \]
This simplifies to:
\[ 2(1 - x^2) \sin^2 x + 2x^2 + 1 = 2x + 2x^2 \]
Rearranging gives:
\[ 2(1 - x^2) \sin^2 x + 1 = 2x \]

### Step 5: Evaluate the Equation

This can be simplified to:
\[ 2(1 - x^2) \sin^2 x = 2x - 1 \]
The left side can be zero or negative depending on \( x \), while the right side can take values depending on \( x \).

### Conclusion for Statement Q:
It is possible to find values of \( x \) such that both sides can equal, thus **Statement Q is true**.

### Final Answer:
- Statement P is false.
- Statement Q is true.

Thus, the correct option is **(C) P is false and Q is true**.