Which of the following is not true ?

Options

A. The number whose conjugate is `1 /(1-i) is 1/(1+i) `
B. if `sinx + icos 2x and cos x -isin2x ` are conjugate to each other then number of values of x is zero
C. if x+ 1+iy and 2 + 3i are conjugate of each other then the value of x + y is -2.
D. `2 + I gt 3 + i`

Answer

To solve the problem of identifying which statement is not true among the given options, let's analyze each statement step by step.

### Step 1: Analyze the First Statement
The first statement claims that the number whose conjugate is \( \frac{1}{1 - i} \) is \( \frac{1}{1 + i} \).

1. Let \( z = \frac{1}{1 + i} \).
2. To find the conjugate \( z^* \), we first rationalize \( z \):
\[ z = \frac{1}{1 + i} \cdot \frac{1 - i}{1 - i} = \frac{1 - i}{1^2 + 1^2} = \frac{1 - i}{2} = \frac{1}{2} - \frac{1}{2}i \]
3. The conjugate \( z^* = \frac{1}{2} + \frac{1}{2}i \).
4. Now, let's find \( z^* \) directly from \( \frac{1}{1 - i} \):
\[ z^* = \frac{1}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{1 + i}{1^2 + 1^2} = \frac{1 + i}{2} = \frac{1}{2} + \frac{1}{2}i \]
5. Since both calculations yield the same result, the first statement is **true**.

### Step 2: Analyze the Second Statement
The second statement claims that if \( \sin x + i \cos 2x \) and \( \cos x - i \sin 2x \) are conjugates, then the number of values of \( x \) is zero.

1. Let \( z = \sin x + i \cos 2x \).
2. The conjugate \( z^* = \sin x - i \cos 2x \).
3. Set \( z^* = \cos x - i \sin 2x \):
\[ \sin x - i \cos 2x = \cos x - i \sin 2x \]
4. Equate the real and imaginary parts:
- Real: \( \sin x = \cos x \)
- Imaginary: \( -\cos 2x = -\sin 2x \) or \( \cos 2x = \sin 2x \)
5. The equations \( \sin x = \cos x \) gives \( x = \frac{\pi}{4} + n\pi \) and \( \cos 2x = \sin 2x \) gives \( 2x = \frac{\pi}{4} + n\pi \).
6. There are solutions for \( x \), hence the statement is **false**.

### Step 3: Analyze the Third Statement
The third statement claims that if \( x + 1 + i y \) and \( 2 + 3i \) are conjugates, then \( x + y = -2 \).

1. Let \( z = x + 1 + i y \).
2. The conjugate \( z^* = x + 1 - i y \).
3. Set \( z^* = 2 + 3i \):
\[ x + 1 - i y = 2 + 3i \]
4. Equate the real and imaginary parts:
- Real: \( x + 1 = 2 \) → \( x = 1 \)
- Imaginary: \( -y = 3 \) → \( y = -3 \)
5. Thus, \( x + y = 1 - 3 = -2 \), confirming the statement is **true**.

### Step 4: Analyze the Fourth Statement
The fourth statement claims that \( 2 + i \) is greater than \( 3 + i \).

1. For complex numbers \( z_1 = 2 + i \) and \( z_2 = 3 + i \):
- Real part of \( z_1 \) is \( 2 \) and of \( z_2 \) is \( 3 \).
- Imaginary part of both is \( 1 \).
2. For \( z_1 > z_2 \), both the real part and imaginary part must satisfy:
- \( 2 > 3 \) (false)
- \( 1 > 1 \) (false)
3. Since neither condition holds, the statement is **false**.

### Conclusion
The only statement that is not true is the fourth statement.