Given the following three equations:

$$ ext{I: } 3x^2y - \frac{5}{2}x - \frac{1}{3}y + 2^n + 1 = 0$$
$$ ext{II: } -2x^2y + \frac{1}{A}x + \frac{4}{3}y - 10 = 0$$
$$ ext{III: } x^2y - Ax + y - P = 0$$

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

Which of the following statements is/are true?

1. $A$ is a prime number
2. $A$ is a factor of $P$
3. Product of $A$ and 15 equals 32

Choose the correct option:

Question

Given the following three equations:

$$	ext{I: } 3x^2y - \frac{5}{2}x - \frac{1}{3}y + 2^n + 1 = 0$$
$$	ext{II: } -2x^2y + \frac{1}{A}x + \frac{4}{3}y - 10 = 0$$
$$	ext{III: } x^2y - Ax + y - P = 0$$

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

Which of the following statements is/are true?

1. $A$ is a prime number
2. $A$ is a factor of $P$
3. Product of $A$ and 15 equals 32

Choose the correct option:

Accepted Answer

**Solution:**

1. According to the problem, Equation I plus Equation II equals Equation III:

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

2. Combine like terms on the left side:

- For $x^2y$: $3x^2y - 2x^2y = x^2y$
   - For $x$: $-\frac{5}{2}x + \frac{1}{A}x = \left(-\frac{5}{2} + \frac{1}{A}ight)x$
   - For $y$: $-\frac{1}{3}y + \frac{4}{3}y = y$
   - Constants: $2^n + 1 - 10 = 2^n - 9$

3. So the left side becomes:

$$ x^2y + \left(-\frac{5}{2} + \frac{1}{A}ight)x + y + (2^n - 9) $$

4. This must equal the right side:

$$ x^2y - Ax + y - P $$

5. Equate the coefficients of like terms:

- Coefficient of $x^2y$: Left = Right, so no new info.
   - Coefficient of $x$: $$ -\frac{5}{2} + \frac{1}{A} = -A $$
   - Coefficient of $y$: Both sides have $y$, so equal.
   - Constants: $$ 2^n - 9 = -P $$

6. From the $x$ coefficient equation:

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

Multiply both sides by $A$ to clear denominator:

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

Rearranged:

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

Multiply entire equation by 2:

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

Or:

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

7. Solve quadratic equation for $A$:

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

So,

- $A = \frac{5 + 3}{4} = 2$
   - $A = \frac{5 - 3}{4} = \frac{2}{4} = 0.5$ (not an integer)

Since $A$ is an integer, $A = 2$.

8. Now find $P$ using constants:

$$ 2^n - 9 = -P \ P = 9 - 2^n $$

9. $n$ is the smallest whole number (starting from 0):

- For $n=0$: $P = 9 - 1 = 8$
   - For $n=1$: $P = 9 - 2 = 7$

Since $P$ must be integer, both are valid, but smallest $n$ is 0, so $P=8$.

10. Check statements:

- Statement 1: $A=2$ is a prime number — True.
    - Statement 2: $A=2$ is a factor of $P=8$ — True.
    - Statement 3: Product of $A$ and 15 equals 32?
      $$ 2 	imes 15 = 30 
eq 32 $$
      So, False.

**Answer:** Only statements 1 and 2 are true, so option (d).

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