If `(2x-5y)^3-(2x+5y)^3=y[Ax^2+By^2]`, then what is the value of `(2A-B)`?

Options

A. (A).25
B. (B).40
C. (C).15
D. (D).10

Answer

To solve the equation \((2x - 5y)^3 - (2x + 5y)^3 = y[Ax^2 + By^2]\), we can use the formula for the difference of cubes:

\[ a^3 - b^3 = (a - b)(a^2 + b^2 + ab) \]

Here, let \(a = 2x - 5y\) and \(b = 2x + 5y\).

### Step 1: Calculate \(a - b\)

\[ a - b = (2x - 5y) - (2x + 5y) = -10y \]

### Step 2: Calculate \(a^2 + b^2 + ab\)

First, we need to find \(a^2\), \(b^2\), and \(ab\):

- \(a^2 = (2x - 5y)^2 = 4x^2 - 20xy + 25y^2\)
- \(b^2 = (2x + 5y)^2 = 4x^2 + 20xy + 25y^2\)
- \(ab = (2x - 5y)(2x + 5y) = 4x^2 - 25y^2\)

Now, we can combine these:

\[ a^2 + b^2 + ab = (4x^2 - 20xy + 25y^2) + (4x^2 + 20xy + 25y^2) + (4x^2 - 25y^2) \]

Combining like terms:

\[ = 4x^2 + 4x^2 + 4x^2 + (-20xy + 20xy) + (25y^2 + 25y^2 - 25y^2) \]
\[ = 12x^2 + 0xy + 25y^2 = 12x^2 + 25y^2 \]

### Step 3: Substitute back into the equation

Now substituting back into the difference of cubes formula:

\[ (2x - 5y)^3 - (2x + 5y)^3 = (-10y)(12x^2 + 25y^2) \]

This gives us:

\[ -10y(12x^2 + 25y^2) = y[Ax^2 + By^2] \]

### Step 4: Equate coefficients

From the equation, we can equate coefficients:

\[ -10y(12x^2 + 25y^2) = y[Ax^2 + By^2] \]

This implies:

\[ Ax^2 + By^2 = -120x^2 - 250y^2 \]

Thus, we have:

\[ A = -120, \quad B = -250 \]

### Step 5: Calculate \(2A - B\)

Now we need to find \(2A - B\):

\[ 2A - B = 2(-120) - (-250) = -240 + 250 = 10 \]

### Final Answer

The value of \(2A - B\) is:

\[ \boxed{10} \]