Multiple choice

In a mathematical competition, the scores obtained by a student in two subjects are given by the quadratic equation ax2 + bx + c = 0, where roots of the given equation represent the scores in two subjects. The maximum possible score in each subject is 100. It is known that the student's score in one subject is at least 40, and the product of the student's scores in the two subjects is 1,600. Find the maximum possible value of b + c.

  1. 1,500

  2. 1,511

  3. 1,518

  4. 1,520
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the roots be r1 and r2. r1*r2 = 1600. r1 >= 40. Since r1*r2 = 1600, r2 = 1600/r1. Since r1 <= 100, r2 >= 16. The equation is (x-r1)(x-r2) = x^2 - (r1+r2)x + r1*r2 = 0. So b = -(r1+r2) and c = 1600. We want to maximize b+c = 1600 - (r1+r2). To maximize this, we minimize r1+r2. Given r1*r2 = 1600 and r1 >= 40, the minimum sum occurs at the boundary r1=40, r2=40. Sum = 80. b+c = 1600 - 80 = 1520.

Keep practicing — related questions