Question

If the quadratic expression \[ (\lambda+2)x^2 - 3\lambda x + 4\lambda = 0, \qquad \lambda \ne -2 \] has two positive roots, then the number of possible integral values of \(\lambda\) is:

• A. \(2\)
• B. \(4\)
• C. \(1\)
• D. \(3\)

Answer 1

The Correct Option is A

Concept: For a quadratic equation \[ ax^2 + bx + c = 0 \] to have two positive roots: \[ \Delta \ge 0, \qquad \frac{-b}{a} > 0, \qquad \frac{c}{a} > 0 \] where \[ \Delta = b^2 - 4ac \] Step 1: Identify coefficients. \[ a = (\lambda + 2), \quad b = -3\lambda, \quad c = 4\lambda \] Step 2: Condition for positive product of roots. \[ \frac{c}{a} > 0 \] \[ \frac{4\lambda}{\lambda+2} > 0 \] This gives \[ \lambda > 0 \quad \text{or} \quad \lambda < -2 \] Step 3: Condition for positive sum of roots. \[ \frac{-b}{a} > 0 \] \[ \frac{3\lambda}{\lambda+2} > 0 \] Again, \[ \lambda > 0 \quad \text{or} \quad \lambda < -2 \] Step 4: Discriminant condition. \[ \Delta = (-3\lambda)^2 - 4(\lambda+2)(4\lambda) \] \[ \Delta = 9\lambda^2 -16\lambda(\lambda+2) \] \[ \Delta = 9\lambda^2 -16\lambda^2 -32\lambda \] \[ \Delta = -7\lambda^2 -32\lambda \] \[ \Delta \ge 0 \] \[ -7\lambda^2 -32\lambda \ge 0 \] \[ 7\lambda^2 + 32\lambda \le 0 \] \[ \lambda(7\lambda+32) \le 0 \] \[ -\frac{32}{7} \le \lambda \le 0 \] Step 5: Combine conditions. From previous condition: \[ \lambda > 0 \quad \text{or} \quad \lambda < -2 \] Intersecting with \[ -\frac{32}{7} \le \lambda \le 0 \] gives \[ -\frac{32}{7} \le \lambda < -2 \] Step 6: Find integer values. \[ -\frac{32}{7} \approx -4.57 \] Thus, \[ \lambda = -4, -3 \] Total number of integral values: \[ 2 \]