Let the system of equations x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+λ z=μ have infinite number of solutions. Then λ+2 μ is equal to :

Solution

Maths sir · 5 steps

Infinite solutions means the third equation is just a mix of the first two, so find that mix.

1. Find lambda. For infinitely many solutions the coefficient determinant must be zero. The determinant D below must vanish.

   D = |1, 2, 3; 2, 3, 1; 4, 3, λ| = 1(3λ - 3) - 2(2λ - 4) + 3(6 - 12) = -λ - 13

2. So lambda is. -λ - 13 = 0 gives λ = -13.
3. Find the mix. Write equation 3 as a times equation 1 plus b times equation 2. Matching the x and y coefficients gives a + 2b = 4 and 2a + 3b = 3, so b = 5 and a = -6. Check on z: 3(-6) + 5 = -13 = λ, as it should be.
4. Find mu. The right-hand sides must combine in the same way: μ = -6(5) + 5(9) = -30 + 45 = 15. Any other μ would make equation 3 contradict the first two, and then there would be no solution at all.
5. So. λ + 2μ = -13 + 30 = 17, which is option (b).

The idea to keep. For infinitely many solutions, D = 0 alone is not enough. The constants must also fit, so the dependent equation has to be the same combination of the others on both sides.

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