If A = x$^2$ (1 – 3y$^2$), B = x (xy$^2$ – 2x) and C = 3y (y – 4x$^2$y), then
A + B – C = px$^2$ + qx$^2$y$^2$ + ry$^2$.
What is the value of (5p + q – r) ?

Question

If A = x$^2$ (1 – 3y$^2$), B = x (xy$^2$ – 2x) and C = 3y (y – 4x$^2$y), then
A + B – C = px$^2$ + qx$^2$y$^2$ + ry$^2$. 
What is the value of (5p + q – r) ?

Accepted Answer

To solve the problem and verify why Option 4 is the correct answer, we need to evaluate the expression A + B - C and compare it to the form px$^2$ + qx$^2$y$^2$ + ry$^2$. Then, we will calculate the value of (5p + q - r).

Given:
- A = x$^2$ (1 – 3y$^2$)
- B = x (xy$^2$ – 2x)
- C = 3y (y – 4x$^2$y)

First, let's expand each expression:

1. **A = x$^2$ (1 – 3y$^2$):**
   $$
   A = x^2 - 3x^2y^2
   $$

2. **B = x (xy$^2$ – 2x):**
   $$
   B = x^2y^2 - 2x^2
   $$

3. **C = 3y (y – 4x$^2$y):**
   $$
   C = 3y^2 - 12x^2y^2
   $$

Now, calculate A + B - C:

$$
A + B - C = (x^2 - 3x^2y^2) + (x^2y^2 - 2x^2) - (3y^2 - 12x^2y^2)
$$

Combine like terms:

- The x$^2$ terms: $x^2 - 2x^2 = -x^2$
- The x$^2$y$^2$ terms: $-3x^2y^2 + x^2y^2 + 12x^2y^2 = 10x^2y^2$
- The y$^2$ terms: $-3y^2$

Thus, the expression simplifies to:

$$
A + B - C = -x^2 + 10x^2y^2 - 3y^2
$$

This matches the form px$^2$ + qx$^2$y$^2$ + ry$^2$, where:
- $p = -1$
- $q = 10$
- $r = -3$

Now, calculate (5p + q - r):

$$
5p + q - r = 5(-1) + 10 - (-3) = -5 + 10 + 3 = 8
$$

Therefore, the value of (5p + q - r) is 8, which corresponds to Option 4.

**Explanation of Incorrect Options:**

1. **Option 1 (12):** This would require a different combination of coefficients for p, q, and r, which does not match the derived values.

2. **Option 2 (10):** This value would result from a different calculation of (5p + q - r), which is not supported by the derived coefficients.

3. **Option 3 (18):** This would imply a significantly different set of coefficients, which is not consistent with the given expressions.

Thus, Option 4 is the correct answer, as it accurately reflects the calculated value of (5p + q - r).