Given the three equations:

Equation I: $$3x^2y - \frac{5}{2}x - \frac{1}{3}y + 2n + 1 = 0$$

Equation II: $$-2x^2y + \frac{1}{A}x + \frac{4}{3}y - 10 = 0$$

Equation III: $$x^2y - Ax + y - P = 0$$

Note:
- Equation I + Equation II = Equation III
- $n$ is the smallest whole number, and $A$ and $P$ are integers.

If $y = 1.4$ in Equation III, then which of the following pairs is the root of the equation?

Question

Given the three equations:

Equation I: $$3x^2y - \frac{5}{2}x - \frac{1}{3}y + 2n + 1 = 0$$

Equation II: $$-2x^2y + \frac{1}{A}x + \frac{4}{3}y - 10 = 0$$

Equation III: $$x^2y - Ax + y - P = 0$$

Note:
- Equation I + Equation II = Equation III
- $n$ is the smallest whole number, and $A$ and $P$ are integers.

If $y = 1.4$ in Equation III, then which of the following pairs is the root of the equation?

Accepted Answer

**Solution:**

1. From the problem, we know that Equation I plus Equation II equals Equation III:

$$ (3x^2y - \frac{5}{2}x - \frac{1}{3}y + 2n + 1) + (-2x^2y + \frac{1}{A}x + \frac{4}{3}y - 10) = x^2y - Ax + y - P $$

2. Simplify the left side:

$$ 3x^2y - 2x^2y = x^2y $$

$$ -\frac{5}{2}x + \frac{1}{A}x = -Ax $$ (from Equation III)

$$ -\frac{1}{3}y + \frac{4}{3}y = y $$

$$ 2n + 1 - 10 = -P $$

3. From the above, equate coefficients:

- For $$x^2y$$: Left side is $$x^2y$$, right side is $$x^2y$$ — consistent.

- For $x$: $$-\frac{5}{2} + \frac{1}{A} = -A$$

- For $y$: $$-\frac{1}{3} + \frac{4}{3} = 1$$ — consistent.

- For constants: $$2n + 1 - 10 = -P \Rightarrow 2n - 9 = -P \Rightarrow P = 9 - 2n$$

4. Since $n$ is the smallest whole number ($$n \geq 0$$), try $n = 0$:

- Then $P = 9$.

5. Find $A$ from the $x$ coefficient equation:

$$ -\frac{5}{2} + \frac{1}{A} = -A $$

Multiply both sides by $A$:

$$ -\frac{5}{2}A + 1 = -A^2 $$

Rearranged:

$$ A^2 - \frac{5}{2}A + 1 = 0 $$

Multiply entire equation by 2 to clear fraction:

$$ 2A^2 - 5A + 2 = 0 $$

6. Solve quadratic for $A$:

$$ A = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm 3}{4} $$

So,

- $$A = \frac{5 + 3}{4} = 2$$

- $$A = \frac{5 - 3}{4} = \frac{1}{2}$$ (not integer)

Hence, $A = 2$.

7. Now Equation III becomes:

$$ x^2 y - A x + y - P = 0 $$

Substitute $y = 1.4$, $A = 2$, and $P = 9$:

$$ 1.4 x^2 - 2x + 1.4 - 9 = 0 $$

Simplify:

$$ 1.4 x^2 - 2x - 7.6 = 0 $$

8. Multiply entire equation by 5 to clear decimals:

$$ 7 x^2 - 10 x - 38 = 0 $$

9. Calculate discriminant $D$:

$$ D = (-10)^2 - 4 \times 7 \times (-38) = 100 + 1064 = 1164 $$

10. Roots are:

$$ x = \frac{10 \pm \sqrt{1164}}{14} $$

Since $$\sqrt{1164} \approx 34.12$$,

$$ x_1 = \frac{10 + 34.12}{14} \approx 3.15 $$

$$ x_2 = \frac{10 - 34.12}{14} \approx -1.72 $$

11. None of the options given (7,11), (-21,11), (8,9) match these roots.

**Answer:** (d) None of these

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