Consider the matrices : A=[2, -5; 3, m], B=[20; m] and X=[x; y]. Let the set of all m, for which the system of equations A X=B has a negative solution (i.e., x<0 and y<0), be the interval (a, b). Then 8 ∫ aᵇ|A| d m is equal to _________.

Solution

Maths sir · 4 steps

Write x and y by Cramer's rule, see that they share the denominator |A|, and read off their signs.

1. Three determinants. The system is 2x - 5y = 20 and 3x + my = m. So |A| = 2m + 15, Dₓ = 20m + 5m = 25m and D_y = 2m - 60.

   x = 25m/(2m+15), y = (2m-60)/(2m+15)

2. Both negative. x < 0 needs 25m and 2m+15 of opposite signs, so -15/2 < m < 0. y < 0 needs -15/2 < m < 30. Both hold together on (-15/2, 0), so a = -15/2 and b = 0.
3. Integrate. ∫_(-15/2)⁰ (2m+15) dm = [m² + 15m]_(-15/2)⁰ = 0 - (225/4 - 225/2) = 225/4.
4. So. 8 × 225/4 = 450.

The idea to keep. For a sign condition on a 2 by 2 system, use Cramer's rule: x and y share the denominator |A|, so each sign condition reduces to a simple inequality in the parameter.

Answer from the NTA final answer key. Solution by PrepWiser.