JEE Main 2024 · 5 April · morning · Single correct · +4 / −1

If the system of equations
11 x+y+λ z=-5; 2 x+3 y+5 z=3; 8 x-19 y-39 z=μ
has infinitely many solutions, then λ⁴-μ is equal to :

  1. (a) 51
  2. (b) 45
  3. (c) 47
  4. (d) 49
Answer and solution

Answer (c) 47

Solution

Maths sir · 4 steps

Make the determinant zero for λ, then write the third equation as a mix of the first two to get μ.

  1. Determinant must vanish. Infinitely many solutions needs the coefficient determinant D = 0. Expand along the first row.

    D = 11(-117+95) - 1(-78-40) + λ(-38-24) = -124 - 62λ = 0 ⇒ λ = -2

  2. Combine the equations. Look for a and b with a(eq 1) + b(eq 2) matching the left side of eq 3. From x: 11a + 2b = 8. From y: a + 3b = -19. These give a = 2, b = -7. Check z: 2(-2) + (-7)(5) = -39, which matches.
  3. Match the constants. The third equation adds nothing new only if its right side follows the same mix. So μ = 2(-5) + (-7)(3) = -31.
  4. So. λ⁴ - μ = 16 - (-31) = 47, which is option (c).

The idea to keep. For infinitely many solutions, D = 0 alone is not enough. The constants must also obey the same row relation, so one equation is just a combination of the other two.

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 2024.