JEE Main 2023 · 10 April · evening · Numerical answer · +4 / −0

Let S be the set of values of λ, for which the system of equations
6 λ x-3 y+3 z=4 λ²,
2 x+6 λ y+4 z=1,
3 x+2 y+3 λ z=λ has no solution. Then 12 Σi ∈ S|λ| is equal to ___________.

Answer and solution

Answer 24

Solution

Maths sir · 3 steps

Find where D = 0, then check each root: a nonzero Dₓ, D_y or D_z means no solution at all.

  1. Set D to zero. A unique solution exists when D ≠ 0, so a system with no solution needs D = 0. Expanding along the first row gives D = 108λ³ - 84λ - 24 = 12(9λ³ - 7λ - 2).

    9λ³ - 7λ - 2 = (λ - 1)(3λ + 1)(3λ + 2) = 0 ⇒ λ = 1, -1/3, -2/3

  2. Test each root. With D = 0 the system has no solution if at least one of Dₓ, D_y, D_z is nonzero. For λ = -2/3 the y and z columns are negatives of each other, so Dₓ = 0 and you must check D_y instead.
    λDeterminant checkedValueVerdict
    1Dₓ25no solution
    -1/3Dₓ7/3no solution
    -2/3D_y115/9no solution
  3. So. All three roots are in S. Then Σ |λ| = 1 + 1/3 + 2/3 = 2, and 12 × 2 = 24.

The common slip: 0. Adding the roots without the modulus gives 1 - 1/3 - 2/3 = 0, so the answer comes out as 0 instead of 24.

The idea to keep. D = 0 only tells you the solution is not unique. Check Dₓ, D_y, D_z for every root: any nonzero one means no solution, and all zero means you have to test further.

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 2023.