STATEMENT-1 : If ` a^(x) = b^(y) = c^(z) , ` where x,y,z are unequal positive numbers and a, b,c are in G.P. , then
` x^(3) + z^(3) gt 2y^(3)` and
STATEMENT-2 : If a, b,c are in H,P, ` a^(3) + c^(3) ge 2b^(3)` , where a, b, c are positive real numbers .

Options

A. Statemant-1 is True , Statement-2 is True, Statement -2 is a correct explanation for Statement-1
B. Statemant-1 is True , Statement-2 is True, Statement -2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Stetement-2 is False.
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze the two statements provided and prove their validity step by step.

### Step 1: Understanding the Given Conditions
We are given that \( a^x = b^y = c^z = k \) for some positive \( k \), where \( x, y, z \) are unequal positive numbers and \( a, b, c \) are in geometric progression (G.P.).

### Step 2: Expressing \( a, b, c \) in Terms of \( k \)
From the equality \( a^x = b^y = c^z = k \), we can express \( a, b, c \) as:
- \( a = k^{1/x} \)
- \( b = k^{1/y} \)
- \( c = k^{1/z} \)

### Step 3: Using the G.P. Condition
Since \( a, b, c \) are in G.P., we have the condition:
\[ b^2 = ac \]
Substituting the expressions for \( a, b, c \):
\[ (k^{1/y})^2 = (k^{1/x})(k^{1/z}) \]
This simplifies to:
\[ k^{2/y} = k^{1/x + 1/z} \]

### Step 4: Equating Exponents
Since the bases are the same, we can equate the exponents:
\[ \frac{2}{y} = \frac{1}{x} + \frac{1}{z} \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ \frac{2}{y} - \frac{1}{x} = \frac{1}{z} \]
This can be rewritten as:
\[ \frac{1}{y} - \frac{1}{x} = \frac{1}{z} - \frac{1}{y} \]
This shows that \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in arithmetic progression (A.P.).

### Step 6: Concluding About \( x, y, z \)
Since \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in A.P., it follows that \( x, y, z \) are in harmonic progression (H.P.).

### Step 7: Applying the H.P. Condition
For numbers in H.P., we have the property that:
\[ x^3 + z^3 \geq 2y^3 \]
This confirms that the first statement is true.

### Step 8: Analyzing the Second Statement
The second statement claims that if \( a, b, c \) are in H.P., then:
\[ a^3 + c^3 \geq 2b^3 \]
This is also a known property of numbers in H.P. and is true.

### Conclusion
Both statements are true, and the second statement provides a correct explanation for the first statement.

### Final Answer
Both Statement 1 and Statement 2 are true, and Statement 2 is the correct explanation for Statement 1.

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