JEE Main 2025 · 24 January · morning · Single correct · +4 / −1

If the system of equations
2 x-y+z=4 ; 5 x+λ y+3 z=12 ; 100 x-47 y+μ z=212
has infinitely many solutions, then μ-2 λ is equal to

  1. (a) 56
  2. (b) 59
  3. (c) 57
  4. (d) 55
Answer and solution

Answer (c) 57

Solution

Maths sir · 4 steps

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

  1. Why a combination. For infinitely many solutions, the three equations can carry only two independent pieces of information. The first two are not proportional, since the x-ratio is 5/2 but the z-ratio is 3. So the third equation, constant included, must equal a times the first plus b times the second.

    (100, -47, μ, 212) = a (2, -1, 1, 4) + b (5, λ, 3, 12)

  2. Find a and b. Use the columns with no unknowns, x and the constant: 2a + 5b = 100 and 4a + 12b = 212. Doubling the first gives 4a + 10b = 200. Subtracting gives 2b = 12, so b = 6 and a = 35.
  3. Read off unknowns. The y-column gives -35 + 6λ = -47, so λ = -2. The z-column gives μ = 35 + 3 × 6 = 53.
  4. So. μ - 2λ = 53 - 2(-2) = 53 + 4 = 57, which is option (c).

The idea to keep. Infinitely many solutions means one whole equation, constant included, is a combination of the other two. Fix the multipliers from fully known columns, then read off the unknowns.

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.