Statement-1: If a,b,c are distinct real number and `omega( ne 1)` is a cube root of unity, then `|(a+bomega+comega^(2))/(aomega^(2)+b+comega)|=1` Statement-2: For any non-zero complex number `z,|z / bar z)|=1

Options

A. Statement-1 is True, Statement-2 is True: Statement-2 is a correct exp,anation for statement-1.
B. Statement-1 is true, statement -2 is true, Statement-2 is not a correct explanation for statement-1.
C. Statement-1 is True, statement-2 is false,
D. statement-1 is False, Statement-2 is true.

Answer

To solve the given problem, we need to analyze both statements and show that Statement 1 is true and Statement 2 is also true.

### Step-by-Step Solution:

**Step 1: Understanding the cube roots of unity**

The cube roots of unity are given by:
- \( \omega = e^{2\pi i / 3} \)
- \( \omega^2 = e^{-2\pi i / 3} \)
- \( \omega^3 = 1 \)

We know that \( 1 + \omega + \omega^2 = 0 \).

**Step 2: Define the expression**

Let:
\[ z = a + b\omega + c\omega^2 \]
We need to show that:
\[ \left| \frac{z}{a\omega^2 + b + c\omega} \right| = 1 \]

**Step 3: Multiply both sides by \( \omega^2 \)**

Multiply both the numerator and the denominator by \( \omega^2 \):
\[ z \omega^2 = (a + b\omega + c\omega^2) \omega^2 = a\omega^2 + b\omega^3 + c\omega^4 \]
Since \( \omega^3 = 1 \) and \( \omega^4 = \omega \), we can rewrite this as:
\[ z \omega^2 = a\omega^2 + b + c\omega \]

**Step 4: Substitute in the original expression**

Now, substituting back into our expression:
\[ \left| \frac{z}{a\omega^2 + b + c\omega} \right| = \left| \frac{z \omega^2}{z} \right| = \left| \frac{z \omega^2}{z} \right| = \left| \omega^2 \right| \]

**Step 5: Calculate the modulus**

Since \( |\omega| = 1 \):
\[ |\omega^2| = 1 \]
Thus:
\[ \left| \frac{z}{a\omega^2 + b + c\omega} \right| = 1 \]

**Conclusion for Statement 1:**
Statement 1 is true.

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**Step 6: Analyze Statement 2**

Statement 2 states:
\[ \left| \frac{z}{\bar{z}} \right| = 1 \]
for any non-zero complex number \( z \).

**Step 7: Understanding the modulus of a complex number and its conjugate**

For any complex number \( z = x + iy \):
\[ \bar{z} = x - iy \]
The modulus of \( z \) is:
\[ |z| = \sqrt{x^2 + y^2} \]
And the modulus of \( \bar{z} \) is also:
\[ |\bar{z}| = \sqrt{x^2 + y^2} \]

**Step 8: Calculate the modulus of the ratio**

Thus:
\[ \left| \frac{z}{\bar{z}} \right| = \frac{|z|}{|\bar{z}|} = \frac{|z|}{|z|} = 1 \]

**Conclusion for Statement 2:**
Statement 2 is also true.

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### Final Conclusion:
Both Statement 1 and Statement 2 are true.