Question

Let $k \in \mathbb{R}$. Which of the following statements is/are correct for the roots of the quadratic equation \[ x^2 + 2(k + 1)x + 9k - 5 = 0 \] ?

• A. If $k \leq 1$, then the roots are real and positive
• B. If $2 \leq k \leq 4$, then the roots are complex
• C. If $4<k<6$, then the roots are real and opposite in sign
• D. If $k \geq 6$, then the roots are real and negative

Hint:

Always analyze quadratic roots using discriminant sign for nature and coefficient signs for positivity or negativity.

Answer 1

The Correct Option is B, D

Step 1: Compute discriminant.
\[ D = [2(k+1)]^2 - 4(1)(9k - 5) = 4(k^2 + 2k + 1 - 9k + 5) = 4(k^2 - 7k + 6) = 4(k - 1)(k - 6). \]
Step 2: Identify nature of roots.
- If $D>0$, roots are real: $k<1$ or $k>6$. - If $D = 0$, repeated roots: $k = 1$ or $k = 6$. - If $D<0$, complex roots: $1<k<6$.
Step 3: Sign of roots.
Sum of roots $= -2(k+1)$; product of roots $= 9k - 5$. - For $k \le 1$: both positive if sum $>0$ and product $>0$ ⇒ false (sum negative). - For $2 \le k \le 4$: complex ⇒ (B) correct. - For $4<k<6$: real ($D>0$) and product $9k-5>0$, sum $-2(k+1)<0$ ⇒ opposite signs ⇒ (C) correct. - For $k \ge 6$: both negative (sum and product positive/negative check).
Step 4: Conclusion.
Correct statements: (B) and (C).