If the quadratic equation $x^2 + 12x + k = 0$ has roots $m$ and $n$ such that $m - n = 2$, which of the following statements is/are correct?

(I) $m imes n$ is a multiple of 5

(II) $m imes n 36$

(III) $m + n 11$

(IV) $m 5$

Question

If the quadratic equation $x^2 + 12x + k = 0$ has roots $m$ and $n$ such that $m - n = 2$, which of the following statements is/are correct?

(I) $m 	imes n$ is a multiple of 5

(II) $m 	imes n > 36$

(III) $m + n > 11$

(IV) $m > 5$

Accepted Answer

**Solution:**

1. Given the quadratic equation:
   $$x^2 + 12x + k = 0$$
   with roots $m$ and $n$.

2. From the properties of quadratic equations:
   - Sum of roots: $$m + n = -rac{b}{a} = -rac{12}{1} = -12$$
   - Product of roots: $$m 	imes n = rac{c}{a} = k$$

3. Given that:
   $$m - n = 2$$

4. Use the identity:
   $$(m - n)^2 = (m + n)^2 - 4mn$$

Substitute values:
   $$2^2 = (-12)^2 - 4k$$

$$4 = 144 - 4k$$

5. Solve for $k$:
   $$4k = 144 - 4$$
   $$4k = 140$$
   $$k = 35$$

6. Now check each statement:

(I) $m 	imes n = k = 35$, which is a multiple of 5. **True**

(II) $m 	imes n = 35$, which is not greater than 36. **False**

(III) $m + n = -12$, which is not greater than 11. **False**

(IV) Find $m$:

From $m + n = -12$ and $m - n = 2$, add both equations:
   $$2m = -10 ightarrow m = -5$$

So, $m = -5$, which is not greater than 5. **False**

7. However, the question options suggest statement IV as "$m > 5$". Since $m = -5$, this is false.

**Answer:** Only statement (I) is correct, but since option (d) "Only I and IV" is marked correct, there seems to be a discrepancy. Rechecking step 6(iv):

- Adding the two equations:
  $$m + n = -12$$
  $$m - n = 2$$

- Adding:
  $$2m = -10 ightarrow m = -5$$

- So $m = -5$, which is not greater than 5.

Therefore, only statement (I) is true.

**Correction:** The correct option should be (a) Only I.

If the provided answer key marks (d) as correct, it is likely a mistake.

**Final conclusion:**
- Statement (I) is true.
- Statements (II), (III), and (IV) are false.

**Answer:** Only I

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