JEE Main 2025 · 2 April · evening · Single correct · +4 / −1

If the system of equations
2 x+λ y+3 z=5 ; 3 x+2 y-z=7 ; 4 x+5 y+μ z=9
has infinitely many solutions, then (λ²+μ²) is equal to :

  1. (a) 30
  2. (b) 26
  3. (c) 22
  4. (d) 18
Answer and solution

Answer (b) 26

Solution

Maths sir · 4 steps

Infinitely many solutions means the third equation is just a mix of the first two. Find that mix.

  1. Why combine. Infinitely many solutions means the three equations are not independent. So the third equation is a times the first plus b times the second, with the constants on the right matching too.
  2. Find the mix. Use the parts with no unknowns: the x coefficients and the constants. That gives 2a + 3b = 4 and 5a + 7b = 9, which solve to a = -1 and b = 2.

    (4x+5y+μ z = 9) = -1 × (2x+λ y+3z = 5) + 2 × (3x+2y-z = 7)

  3. Match y and z. Coefficient of y: -λ + 4 = 5, so λ = -1. Coefficient of z: -3 - 2 = μ, so μ = -5. As a check, with these values Δ = 7μ + 35 = 0.
  4. So. λ² + μ² = 1 + 25 = 26, which is option (b).

The idea to keep. For infinitely many solutions in a 3 by 3 system, write one equation as a combination of the other two. Find the multipliers from the rows with no unknowns, then read off the parameters.

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

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Asked about 190 times in JEE Main since 2020. This question is from 2025.