If ` x- y = 11 , 1 // x - 1//y = 11//24 , ` then find ` x^(3) - y ^(3) `

Options

A. 429
B. 485
C. 529
D. 539

Answer

To solve the problem step by step, we will use the given equations to find the value of \( x^3 - y^3 \).

### Step 1: Write down the equations
We are given:
1. \( x - y = 11 \) (Equation 1)
2. \( \frac{1}{x} - \frac{1}{y} = \frac{11}{24} \) (Equation 2)

### Step 2: Simplify Equation 2
From Equation 2, we can rewrite it using a common denominator:
\[ \frac{y - x}{xy} = \frac{11}{24} \]
Since \( y - x = -(x - y) \), we can substitute:
\[ \frac{-(x - y)}{xy} = \frac{11}{24} \]
Substituting \( x - y = 11 \):
\[ \frac{-11}{xy} = \frac{11}{24} \]

### Step 3: Cross-multiply to solve for \( xy \)
Cross-multiplying gives:
\[ -11 \cdot 24 = 11 \cdot xy \]
\[ -264 = 11xy \]
Dividing both sides by 11:
\[ xy = -24 \]

### Step 4: Use the identity for \( x^3 - y^3 \)
We can use the identity:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
We already know \( x - y = 11 \) and \( xy = -24 \).

### Step 5: Find \( x^2 + xy + y^2 \)
We can express \( x^2 + y^2 \) using the identity:
\[ x^2 + y^2 = (x - y)^2 + 2xy \]
Substituting the known values:
\[ x^2 + y^2 = (11)^2 + 2(-24) = 121 - 48 = 73 \]
Now, substituting back to find \( x^2 + xy + y^2 \):
\[ x^2 + xy + y^2 = x^2 + y^2 + xy = 73 - 24 = 49 \]

### Step 6: Calculate \( x^3 - y^3 \)
Now substituting back into the identity:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) = 11 \cdot 49 = 539 \]

### Final Answer
Thus, the value of \( x^3 - y^3 \) is \( 539 \).