CT 24: Quadratic Equaiton – IV

1.It is given that the roots of the equation x² – 2x – log₂ K = 0 are real. For this, the minimum value of K is

2. What is the value of k for which the sum of the squares of the roots of 2x² - 2(k - 2) x - (k + 1) = 0 is minimum?

3. It is given that the roots of the equation x² – 4x – log₃ P = 0 are real. For this, the minimum value of P is

4. If the roots of the quadratic equation (log₅ k)x² − 2x + 1 = 0 are real and equal, then the maximum value of k is

5. α is the maximum value of 1 − 2x − 5x² and β is the minimum value of x² − 2x + r. If 5αx² + βx + 6 > 0 for all real values x, then the interval in which r lies is

6. The minimum value of the expression 2x2 + 3x + 1 is

7. What is the maximum value of expression x²/(2x + 3)?

8. At what value of x, 40 - x² + 5x will be maximum and the minimum value?

9. If α and β are the roots of the quadratic equation 2x² + 6x + k = 0, where k < 0, then what is the maximum value of (α/β + β/α)?

10. Find the minimum value of (6x2 - 15x + 8).