Let q be the maximum integral value of p in 0 , 10 for which the roots of the equation x 2 - p x + 5 4 p = 0 are rational. Then the area of the region x , y : 0 ≤ y ≤ ( x - q ) 2 , 0 ≤ x ≤ q is

Step-by-step solution

x 2 - p x + 5 p 4 = 0 For rational roots, D must be a perfect square. D = p 2 - 5 p = p p - 5 Now put values in reverse order as q is the maximum value of p At p = 10 , D = 50 → not a perfect square At p = 9 , D = 36 → perfect square So, the maximum integral value of p is 9 ∴   q = 9 0 ≤ y ≤ x - 9 2 Area of shaded region = ∫ 0 9 x - 9 2 d x = x - 9 3 3 0 9 = 243   unit 2